STATISTICAL MECHANICS - University Of MarylandSTATISTICAL MECHANICS NOTES: ... mechanics determines...
Transcript of STATISTICAL MECHANICS - University Of MarylandSTATISTICAL MECHANICS NOTES: ... mechanics determines...
STATISTICAL MECHANICS NOTES
Padobedaqve
HHEIRMODYNATUIIE sect
A lot can be denied from general properties of the entropy In
fact if The entropy as a function of the so called extensive variables
is known everything else about the Thermodynamic behavior of the
system can be found
Bottyouggnns D
entropy
-DS = KB ln P
s
instead of P ( phaserArrdoue occupied by the mioocauonirl ensemble )iBoltzmann constant
IT s here for historicalThere will be
reasons
If we measure temperature some refinementsin units of energy ( as we should ) Then 193=1 on This definition
Otherwise later
KB x 138times15 Jq
lt 86 X is eV(degK
A good Thing To reweubeu is
10000 degK plusmn ev
We now list a few properties satisfied by The entropy
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-
info you need
y
extensive quantities arte ones That double
if we double The system for instance
gas wl one molecule type E U N
gas wl several kinds of molecules E
U N Nz
magnet E IN ( magnetic moment )
dielectric E P ( electric dipole moment )elastic medium E
Eij ( strain Tensor )rubber band E
L ( length )black body E
V
Temperature pressure external magnetic fields
are
intensity quantities ( two identical
blocks of ice next To each other have
Twice the volume energy of one block 150T
the same Temperature )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive
S ( d E d V
d N ) = d SCEN N )
T follows fromp =p Tz neglecting
surface effects
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is
concave
SCEHSE VHJV
N tan ) + SCE - DE VIV
N DN )
f
2 SCE V N )
Ss
- SCE )scEtorArr
- - - - -
bullset - iffsees t
- - - - amp
if SCETSE ) + SCE D t )fixE- DE
E
ETDE
E
( we ll see later that this
implies stability )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE
( This property depends on
how we normalize the number of microstates T
quantummechanics determines The night
normalization That implies SCTTDTO
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( This is a direct consequence thatall microstates are equally probable )
Inverting SCEN N ) we get ECS U micro )
If SN Nare changed slowly ( adiabatic ally
) so the system stays in
equilibrium at all intermediate states
DE = 0yen ds +0yen dv + golf DN
-
tin
-lav
-
ten
T - P micro ( chemical potential
-
-
= energy required
heat put into work done
to add one partThe system yen The energy ofthe
system particle added
similar
=
stet
de
tstvkY+yenoyen
PIT
- MT
we used
I
f y1
8Th micro= of we used
Tolan
Eopoundlsifsplusmnh )
yenhyeneyenyenhu
larrayenyendivideseHrArrtlArr s
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
HHEIRMODYNATUIIE sect
A lot can be denied from general properties of the entropy In
fact if The entropy as a function of the so called extensive variables
is known everything else about the Thermodynamic behavior of the
system can be found
Bottyouggnns D
entropy
-DS = KB ln P
s
instead of P ( phaserArrdoue occupied by the mioocauonirl ensemble )iBoltzmann constant
IT s here for historicalThere will be
reasons
If we measure temperature some refinementsin units of energy ( as we should ) Then 193=1 on This definition
Otherwise later
KB x 138times15 Jq
lt 86 X is eV(degK
A good Thing To reweubeu is
10000 degK plusmn ev
We now list a few properties satisfied by The entropy
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-
info you need
y
extensive quantities arte ones That double
if we double The system for instance
gas wl one molecule type E U N
gas wl several kinds of molecules E
U N Nz
magnet E IN ( magnetic moment )
dielectric E P ( electric dipole moment )elastic medium E
Eij ( strain Tensor )rubber band E
L ( length )black body E
V
Temperature pressure external magnetic fields
are
intensity quantities ( two identical
blocks of ice next To each other have
Twice the volume energy of one block 150T
the same Temperature )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive
S ( d E d V
d N ) = d SCEN N )
T follows fromp =p Tz neglecting
surface effects
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is
concave
SCEHSE VHJV
N tan ) + SCE - DE VIV
N DN )
f
2 SCE V N )
Ss
- SCE )scEtorArr
- - - - -
bullset - iffsees t
- - - - amp
if SCETSE ) + SCE D t )fixE- DE
E
ETDE
E
( we ll see later that this
implies stability )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE
( This property depends on
how we normalize the number of microstates T
quantummechanics determines The night
normalization That implies SCTTDTO
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium
( This is a direct consequence thatall microstates are equally probable )
Inverting SCEN N ) we get ECS U micro )
If SN Nare changed slowly ( adiabatic ally
) so the system stays in
equilibrium at all intermediate states
DE = 0yen ds +0yen dv + golf DN
-
tin
-lav
-
ten
T - P micro ( chemical potential
-
-
= energy required
heat put into work done
to add one partThe system yen The energy ofthe
system particle added
similar
=
stet
de
tstvkY+yenoyen
PIT
- MT
we used
I
f y1
8Th micro= of we used
Tolan
Eopoundlsifsplusmnh )
yenhyeneyenyenhu
larrayenyendivideseHrArrtlArr s
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
We now list a few properties satisfied by The entropy
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-
info you need
y
extensive quantities arte ones That double
if we double The system for instance
gas wl one molecule type E U N
gas wl several kinds of molecules E
U N Nz
magnet E IN ( magnetic moment )
dielectric E P ( electric dipole moment )elastic medium E
Eij ( strain Tensor )rubber band E
L ( length )black body E
V
Temperature pressure external magnetic fields
are
intensity quantities ( two identical
blocks of ice next To each other have
Twice the volume energy of one block 150T
the same Temperature )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive
S ( d E d V
d N ) = d SCEN N )
T follows fromp =p Tz neglecting
surface effects
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is
concave
SCEHSE VHJV
N tan ) + SCE - DE VIV
N DN )
f
2 SCE V N )
Ss
- SCE )scEtorArr
- - - - -
bullset - iffsees t
- - - - amp
if SCETSE ) + SCE D t )fixE- DE
E
ETDE
E
( we ll see later that this
implies stability )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE
( This property depends on
how we normalize the number of microstates T
quantummechanics determines The night
normalization That implies SCTTDTO
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium
( This is a direct consequence thatall microstates are equally probable )
Inverting SCEN N ) we get ECS U micro )
If SN Nare changed slowly ( adiabatic ally
) so the system stays in
equilibrium at all intermediate states
DE = 0yen ds +0yen dv + golf DN
-
tin
-lav
-
ten
T - P micro ( chemical potential
-
-
= energy required
heat put into work done
to add one partThe system yen The energy ofthe
system particle added
similar
=
stet
de
tstvkY+yenoyen
PIT
- MT
we used
I
f y1
8Th micro= of we used
Tolan
Eopoundlsifsplusmnh )
yenhyeneyenyenhu
larrayenyendivideseHrArrtlArr s
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is
concave
SCEHSE VHJV
N tan ) + SCE - DE VIV
N DN )
f
2 SCE V N )
Ss
- SCE )scEtorArr
- - - - -
bullset - iffsees t
- - - - amp
if SCETSE ) + SCE D t )fixE- DE
E
ETDE
E
( we ll see later that this
implies stability )
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE
( This property depends on
how we normalize the number of microstates T
quantummechanics determines The night
normalization That implies SCTTDTO
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium
( This is a direct consequence thatall microstates are equally probable )
Inverting SCEN N ) we get ECS U micro )
If SN Nare changed slowly ( adiabatic ally
) so the system stays in
equilibrium at all intermediate states
DE = 0yen ds +0yen dv + golf DN
-
tin
-lav
-
ten
T - P micro ( chemical potential
-
-
= energy required
heat put into work done
to add one partThe system yen The energy ofthe
system particle added
similar
=
stet
de
tstvkY+yenoyen
PIT
- MT
we used
I
f y1
8Th micro= of we used
Tolan
Eopoundlsifsplusmnh )
yenhyeneyenyenhu
larrayenyendivideseHrArrtlArr s
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Inverting SCEN N ) we get ECS U micro )
If SN Nare changed slowly ( adiabatic ally
) so the system stays in
equilibrium at all intermediate states
DE = 0yen ds +0yen dv + golf DN
-
tin
-lav
-
ten
T - P micro ( chemical potential
-
-
= energy required
heat put into work done
to add one partThe system yen The energy ofthe
system particle added
similar
=
stet
de
tstvkY+yenoyen
PIT
- MT
we used
I
f y1
8Th micro= of we used
Tolan
Eopoundlsifsplusmnh )
yenhyeneyenyenhu
larrayenyendivideseHrArrtlArr s
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Its important to notice that S is defined for equilibriummacroscopic states only
So des or DE refer To changes slow
enough such that the system remains in equilibrium at allintermediate stages
For instance if The value is changed
suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be
larger ) Move insight on this will come later when we
discuss the quantum use and the canonical ensemble
equztiousof The relations
state divide
offensive
F=yenetmicroinedividemicro tP=
3eurofmicro=
PCSIYN)
larr
yen=ofyenmicro= - PENyen
TENN )
micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs
of state
They are frequently accessible to
experiments Notice Thet the same quantities as function of other
variables are nyen eqs of state and may not contain The same
information
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
As mentioned before all Thermo dynamical problems can be reduced To
before
~ after
extentFEyenttsot
Eicui Ni Eicuiirz
ELVORN
A little thought implies That
wall
allows
energy
exchange IT
= Tzl
wall allows volume exchange V
= vi
wall allow particle exchange micro
= microi
Non - relativistic non
interact
monatomic gas
Finally let us compute S for one example
Acri pi ) = i
yin
( oc
rick)
PCEMN ) = SHEdri dpi SCE - Eakin )
= microtimes ]
sectdegd Pixdpiydpz dpmicrotimesdpmicroydP
zm Scant yenYpi2 )
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
= VN on 53CR=zmE=
area of a 3N -dimensional
spherical hyper surface wl radius
R=zmEM
N 3Nz= Zm lArr
a yen
S = ka hnp = kBlnV + 3yen bn E
- but ( 3142 ) +312 bytangymicro Stirling approx
Inn
micro Nlnw +
microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]
312= Nka In Vmicroldivide
The expression above violates postulates ( extensively ) and
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising
the behavior of a gas at To is dictated by quantum mechanics
not classical mechanics ( The atom s wavelength is much larger ku
the separation between them )
what is surprising is that
quantum mechanics carrots the calculation above even at target
What is missing from The classical calculation above is that
the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )
So The correct way of
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
counting states is
S = Kalp = ka fETyd4divideg SCE - EE
piyzm )
N
D
avoids double
euro This factor wont mattercounting of microstates now but we ll see its The
with the same particle correct one when we consider
Positions
the classical limit of quantum gases
32=D KB N h
micro - DOO NFZ
From this entropy we can find the eqs of state
t =
feurotfmicro=3yenDyen or E
= 3=143 NT
- Tf
- sectmicro
= -
kBdivide or PV = KBN T
et = If
micro
= shumicroyenh - Eke
Notice That we found The Boyle - Marriott law wo even
discussing Fans colliding against walls or other Kinetic toyconsiderations
Theres a lot behind The assumption of evgodicity
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Euler relation SC dE dv t N ) = d SCE
v N )
yen he
E
Ee + v 0pound +nosectmicro = s
in
in
-
YT - PIT divide
E
= TS - PV + p N
or
S
- Ee - Ny tiff
Recovering The entropy
-energy ) from edivideIf 3 e as are available
TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf
EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N
micro ( SNID
kTulsmicro
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
If only 2 eds
are available ECSN N ) = N
E ( Shu YN i )
k extensity
T TTL microi TCSN M= TCYN YN i )
f-no N
T
and P are intensive
r-
Pizeurofgmicro - PCSW N)=tPC9microMmicro I )
sw
E(9microYmicro )
= fd(4micro ) TCSINMMI ) + EC 04N i )
deg
T Ebslvav
IN
= sectd(4micro ) Tcsln YN ) -
sectDCYN ) PC9Ymicro i ) teaECSINYIN l )
can be found up to one
constant independent ofSN or N
-
If only 1 e as
cant be done
i available
Example non - rd
monoztomic non
interacting gasTKBN
s= ken hVEplusmn E
= ( duke 4=13 eyenaNNSTZ 613
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
+ = yenhmicroidividendivideeyenN- P -
fuelsmicro
= yen5 e 243
Ec sauna n=
Esau) divideydivideP
eyenplusmn- Plan
tea at
= plusmn
euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g
e
yenn -
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Mining EnergyPrinciple
divide s start with an analogy
A circle is the figure with the largestarea for a given perimeter
Of course its also The one with
the smallest perimeter for a given area Even if a given
circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The
final circle minimizes The perimeter for a given area
The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The
one with The largest entropy for a given +0yen energy C the
constraint That the Total energy is unchanged is always there )
But The final equilibrium state can also be characterized as
the state with the smallest energy for a fixed entropy
S
maximum S
Sdeg
ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E
E
E
Convince yourself that 5 as a function of total E and
subsystem energy E
has the general shape depicted above
The minimum energy principle follows from staring at the pictureabove
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Let us see how
the minimum energy principle works in Our
archetypical example
diathermy wall
( energy transfer allowed )actuality
yen pretend
-
-
=
- -
lArrrArr M lArr
E=EtEz fixed
rArr- Sitszfixed
S changes
- E changes
-
+- t
E = Ee ( Si v micro ) +Ez(
IUyNr )
ss
minimum E
forfixeds fldivide=o if oldivide
- Itf =o rate as expected )
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Efficiency of engines
An engine is a machine That Takes energy from some hot
Source ( a furnace say ) uses some of This energy to do work
and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere
We would like To do This over and over again so
The machine should go to a closed cycle That is end up in
the same macroscopic state as it started
For now we will assume
that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic
state is specified by EN N or SN N
IT turns out That its
better To use instead The variables ITS N
Keep N fixed by
having impermeable walls separating The machine from The rest ofthe Universe
A cycle can Then be represented by a closed loopin the its
stages aD
s
Let us stat by considering one specific cycle The G=T cycle
T
t
deg 0yen
- -
ideal gas
toLItaE inside
Ds
Md LEE
hot
coldsource
sdividethe piston is exposed to
five or ice and its motionmoves a wheel
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas in contact w hot source
absorbs heat
To and expands keeping T=Th= cast
In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands
+ 40euro40 wlo absorbing heat cools down and does work
c
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks
rArr z
5 and Transfers heat tote cold sink
9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167
gas isolated from sink ) source contracts
and hzs work dove on it
Forfeited N DE - T ds - PDU so
Sz
DQ
= STDS =
Th ( Sz - S )
T
DQ
si
divide n n
DQ ftds
= - I
Csz - S )~gt
T
sectDDW1
Szt-
J
D
S
DQZ Sz
S
Since after a full cycle The energy of the system goes back
to what it was in The beginning
DQI + DQZ = DW
= area inside the cycle
T heat work done C The Txs plane)
T
absorbed yen The system
A - _
Dwi t Dwz
t tc
-
i
S
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
For practical reasons wed like To
get as much work as possible
from a given amount of heat absorbed ( $15 ) The efficiency
of The engine is defined as
e = I
= Ethos - l -
yen ( l
DQ In (52-9)
The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a
larger
efficiency
For That we would want To maximize The area
inside of The cycle ( DW in red )
and minimize TheTp area below The top come ( DQ
blue
dashedTh
- -
yendivideg - area on the figure ) Its clear
for instance
Tc - -
yenyenpound
-
that the cycle on the left can have its
13
efficiency improved by
changing it
as
shown below
We end op then
sw and DQ wl the Cannot cycle which
increases by the same is the best it can be done
p ( amount E goes up
Thus we have an universal result
In
plusmnlArrdivideg - -
+ yenyenyenFyen euro divideyenasyentFyen_
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Notice that we only used ( implicitly actually ) The fact That
an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands
)
The
main result E
( l - Tyth does not depend on that and is valid
universally
Finally let us revisit
The assumption of adiabatic it namely
that all phases of The cycle were infinitely slow
In each phase
of the cycle the machine goes from one initial state A to a
final state B exchanging a certain amount of heat DQ
( with a source at Temperature T ) and
performing
some work
DW ( positive or negative )
Ill assume The work is done on
a reversible work source namely a system which can exchange work
but not heat ( a paperweight on top of a piston is an example )
Similarlyheat is exchanged with a reversible heat source
- - - -
-
euro3
universe
I
i
Dw gto
5yen
DQ gt0
1- - - -
- - --
iFor given initial and final states A and 13 ( and consequently
given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if
DQ is a minimum
as Ed - Ea = - COWTDQI )
The process with the smallest DQ will correspond To The smallest
possible increase of The entropy of the universe ( system + heat
source The work source was no entropy )
How much is the minimum
amount of heat To be Transferred To the heat Source As
The Total entropy of The universe must increase ( otherwise
The system would stay at A ) we have
SB - SAIIBQ 30 temperatureoftteThat
source
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
That means That The minimum amount of heat wasted on the
heat source happens when the Total entropy of the universe is
unchanged as is
given Q
=
SAI-
r
If more heat tan Syen is delivered to the heat source the universe s
entropy will increase and less work will be delivered to the work source
A similar analysis applies when The system is absorbing heat or
work The main point is That reify Those where
the entropy of The universe does not change lead To the maximum
efficiency Thus our bound on the efficiency of an engine
obtained assuming adiabatic ity is not violated by non adiabatic
processes Notice That The adiabaticity condition means that the
engine works really slowly Practical machines which need
to do Their job fast are not adiabatic and much less
efficient Than The theoretical limit
E f 1- Tatty
would suggest ( roughly a factor 3 less efficient )
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
Thermodynamic stability-
We determined
The equilibrium condition between two
subsystemsTiitz
R=Pz micro
=
microz7
to
7if
datherurl if
movable if porouswall
wall wall
using the maximum entropy principle But we only imposed That
Stsitsz was a critical point it could have been a maximum
a minimum or a saddle point
we ll now see what The
consequences are of demanding that The final equilibrium state
is indeed a maximum
Consider first The case of Two subsystems separated by a diathermyo
well
The condition That 5=4+52 is a maximum implies
allowsenergy
exchange
IT tyenoIdivide=Idivide
divide vv Ez
vuNz fyen lt 0 rArru
DI +325yen lt 0
in
E-Et The quantity 5 sloe has a physical interpretation
final stateof
equilibrium
Hel micro=poundChjyenz yenhmicro
= - Foetal microplusmnn Tspecific heat
D
at constant volk
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
The stability condition is Then
- taut
- yen at
lt o or 1- it gto
ya Mci nai
T =Tz
in equilibrium
By Taking one subsystem to be much longer we conclude that
each one of Ci and Gi should be positive This result is physically
obvious
Suppose Cv lt 0
Then itd get cdyen by absorbing
heat therefore absorbing even move heat
This is an unstable
situation
Let us consider now The ease where both E and U are
exchanged between The Subsystems we have
deg= yen
- 3yen
- 01
OEZM
o= divide = st
- IIdivide iv
N Ei
Vi Nz
in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01
plusmn
of oeiavi
ovizequilibrium
+ 1 larr
we need This To be
negative for any dehdu
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
For The second variation Is to be negative for any E
U
its
necessary That The eigeuvcwes of01
EeiyenpFoe
be negative
Let us wore out a general condition for this to
be
tv j
o
d
- catch tae 5 o
teddies
t= at
plusmn(adivideg2-ceaT
co
the eigewawesd ate lt o and ac -5 )
0
irArr a lt 0 and
c - I ( 0
a
torus at Yah micro
= EecrArrhmicro
= - ttz seekmicro=dividezmicroodividedivide
me condition
2
Cr ) 0 as
o ) c - plusmn
= Is - (zEsDyeny
beforeOV
ENotthis can be written in
a pretty way wl a TransparentIt CyenlEyenfeh)
physical meaning
-- - -
IeuroQplusmnejhiyenyen
aspyent=crArr
DV
=xtyenhyeneyenwtimestyenh
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one
z
2
= Buy + t Ile 1-
I
2
oeov
+zzeu one
so osmyent=divide stdivide[dividesdividetI=dividedivide
isothermal
compressibility
The stability condition implies That the isothermal compressibilityis positive
This is not so obvious A negative compressibility
means That the system expands
as The pressure increases
That would
increase The pressure of the other system leading to a further expansionand so on
What I dont find obvious is that The isotevwd
compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs
)is the relevant one