Lecture 05

7/17/2019 Lecture 05

http://slidepdf.com/reader/full/lecture-05-568bef1800b38 1/41

Chapter 5Simple Applications of Macroscopic

Thermodynamics

7/17/2019 Lecture 05


Preliminary DiscussionClassical, Macroscopic,

Thermodynamics• Now, we drop the statistical mechanics

notation for average quantities. So that now, All Variables are Averages Only! • We’ll discuss relationships between

macroscopic variables using

The Laws of Thermodynamics

7/17/2019 Lecture 05


• Some Thermodynamic Variables of Interest:

Internal ner!y " , ntropy " S

Temperature " T• Mostly for Gases:

(but also true for any substance!

#ternal Parameter " V$enerali%ed &orce " p'V " (olume, p " pressure)

• For a General System:#ternal Parameter " #

$enerali%ed &orce " *

7/17/2019 Lecture 05


• "ssume that the #ternal Parameter " Volume

V in order to have a specific case to discuss. #or

systems with another e$ternal parameter #, theinfinitesimal wor% done + " *d#. &n this case,

in what follows, replace p by * ' dV by d#.• #or infinitesimal, quasistatic processes!

-st . /nd 0a1s of Thermodynamics-st 0a1: +2 " d 3 pdV

/nd 0a1: +2 " TdSombined st " #nd Laws

TdS " d 3 pdV

7/17/2019 Lecture 05


ombined st " #nd Laws

TdS " d 3 pdV• Note that, in this relation, there are

5 Variables: T, S, , p, V• &t can be shown that!

Any $ of these can always be e%&ressed

as f'nctions of any # others(• )hat is, there are always # inde&endent

variables " $ de&endent variables( )hich

# are chosen as inde&endent is arbitrary(

7/17/2019 Lecture 05


*rief , Pure Math Discussion

• *onsider + variables! #, y, %4 Suppose we

%now that # . y are +nde&endent Variables (

)hen, +t M'st *e ,ossible to e$press % as a

function of # . y. )hat is,

There M'st be a F'nction % " %'#,y).

• #rom calculus, the total differential of %'#,y)

has the form!

d% ≡ '%6#)yd# 3 '%6y)#dy 'a)

7/17/2019 Lecture 05


• Suppose that, in this e$ample of + variables! #, y, %, we

want to ta%e y ' % as independent variables instead of #

' y. )hen,

There M'st be a F'nction # " #'y,%).• #rom calculus, the total differential of #'y,%) is!

d# ≡ '#6y)%dy 3 '#6%)yd% 'b)• sing 'a) from the previous slide

-d%≡

'%6#)yd# 3 '%6y)#dy 'a)

' 'b) together, the partial derivatives in 'a) ' those in 'b)

can be related to each other.• We always assume that all functions are analytic.

So- the #nd cross derivatives are e.'al

Such as! '/

%6#y)≡

'/

%6y#), etc.

7/17/2019 Lecture 05


Mathematics Summary• *onsider a function of / independent variables!

f " f'#-,#/)4• &t’s e$act differential is

df ≡ y-d#- 3 y/d#/

' by definition!

• 0ecause f'#-,#/) is an analytic function, it is always true

that!

/ 1

/ 1

1 / x x

y y

x x

∂ ∂= ÷ ÷∂ ∂

• 2ost Ch4 5 applications use this with the

ombined st " #nd Laws of Thermodynamics

TdS " d 3 pdV

7/17/2019 Lecture 05


Some Methods Methods . /sef'l Math Tools/se f'l Math Tools for

Transformin! Deri(ati(esTransformin! Deri(ati(es

Deri(ati(e In(ersion

Triple Product '#y%7- rule)

Chain 8ule #pansion to Add Another Variable

Ma#1ell 8eciprocity 8elationship

x x F y y

F

∂∂

=

∂∂ 1

T T S P P

S

∂∂

=

∂∂ 1

1−=

∂∂

∂∂

∂∂

x F y F y

y x

x F

1−=

∂∂

∂∂

∂∂

T H P H

P

P

T

T

H

x x x y

F

y

F

∂Φ∂ Φ∂∂= ∂∂ T C T

C

H

T

T

S

H

S

P

P

P P P

11

== ∂∂ ∂∂= ∂∂

( ) ( )

y

x

x

y

x

y F

y

x F

∂

∂∂∂=

∂

∂∂∂

yx xy

F F =

7/17/2019 Lecture 05


Pure Math: 9acobian Transformations

•" 3acobian )ransformation is often used to

transform from one set of independentvariables to another.•#or functions of / variables f ' % , y) ' g ' % , y) it is!

( )

( ) y x x y

x y

x y

x

g

y

f

y

g

x

f

y

g

x

g

y f

x f

y x

g f

∂∂

∂∂

−

∂∂

∂∂

=

∂

∂

∂

∂

∂∂

∂∂

≡∂∂

,

,

Determinant

7/17/2019 Lecture 05


Transposition

In(ersion

Chain 8ule#pansion

( )( )

( )( ) y x

f g

y x

g f

,

,

,

,

∂∂−=

∂∂

( )

( ) ( )( ) g f

y x y x

g f

,

,

1

,

,

∂∂=∂

∂

( )( ) ( )( ) ( )( ) y xw z

w z g f

y x g f

,,

,,

,, ∂∂∂∂=∂∂

9acobian Transformations9acobian Transformations

;a(e Se(eral /sef'l ,ro&erties/se f'l ,ro&erties

7/17/2019 Lecture 05


• Suppose that we are only interested in the first

partial derivative of a function f'%,!) with respect

to % at constant !:

( )

( ) g z

g f

z

f

g ,

,

∂∂=

∂∂

( )

( )( )

( ) y x

g z

y x

g f

z

f

g

,

,

,

,

∂

∂∂∂

=

∂∂

• )his e$pression can be simplified using the chain

rule e$pansion ' the inversion property

7/17/2019 Lecture 05


d " TdS 7 pdV '-)

#irst, choose S ' V as inde&endent variables:

≡ 'S,V)

Properties of the Internal ner!y

dV V

U dS

S

U dU

S V

∂∂

+

∂∂

=

T S

U

V

=

∂∂

pV

U

S

−=

∂∂

*omparison of '-) ' '/) clearly shows that

d

'/)

"pplying the general result with /nd cross derivatives gives!

V S S

p

V

T

∂

∂−=

∂

∂ Ma%well 0elation Ma%well 0elation II

and

7/17/2019 Lecture 05


&f S ' p are chosen as inde&endent variables, it is

convenient to define the following energy!

; ≡ ;'S,p) ≡ 3 pV ≡ 1nthal&y 1nthal&yse the combined -st . /nd 0a1s. 4ewrite them in terms of d;: d

" TdS 7 pdV " TdS 7 <d'pV) 7 Vdp= or

d; " TdS 3 Vdp

*omparison of '-) ' '/) clearly shows that

'-)

'/)

"pplying the general result for the /nd cross derivatives gives!

pS S

V

p

T

∂

∂=

∂

∂

0ut, also!

and

Ma%well 0elation Ma%well 0elation IIII

7/17/2019 Lecture 05


&f T ' V are chosen as inde&endent variables, it is


& ≡ &'T,V) ≡ > TS≡ 2elmholt3 Free 2elmholt3 Free

1nergy 1nergy• se the combined -st . /nd 0a1s. 4ewrite them in terms of d&:

d " TdS 7 pdV " <d'TS) 7 SdT= 7 pdV or

d& " >SdT 7 pdV '-)

• 0ut, also! d& ? '

&6

T)VdT 3 '

&6

V)TdV '/)• *omparison of '-) ' '/) clearly shows that

' &6 T)V ? >S and ' &6 V)T ? >p

• "pplying the general result for the /nd cross derivatives gives!

Ma%well 0elation Ma%well 0elation IIIIII

7/17/2019 Lecture 05


&f T ' p are chosen as inde&endent variables, it is


$ ≡ $'T,p) ≡ 7TS 3 pV ≡ Gibbs Free 1nergyGibbs Free 1ner gy• se the combined -st . /nd 0a1s. 4ewrite them in terms of d;:

d " TdS 7 pdV " d'TS) > SdT 7 <d'pV) 7 Vdp= or

d$ " >SdT 3 Vdp '-)

• 0ut, also! d$ ? '

$6

T)pdT 3 '

$6

p)Tdp '/)

• *omparison of '-) ' '/) clearly shows that

' $6 T)p ? >S and ' $6 p)T ? V

• "pplying the general result for the /nd cross derivatives gives!

Ma%well 0elation Ma%well 0elation IVIV

7/17/2019 Lecture 05


-4 Internal ner!y: ≡ 'S,V)

/4 nthalpy: ; " ;'S,p) ≡ 3 pV

@4 ;elmholt% &ree ner!y: & " & 'T,V) ≡ 7 TS

4 $ibbs &ree ner!y: $ " $'T,p)≡

7 TS 3 pV

Summary: 1nergy F'nctions 1ner gy F'nctions

ombined ombined st st

"" ##nd nd Laws Laws

-4 d " TdS 7 pdV

/4 d; " TdS 3 Vdp @4 d& " > SdT 7 pdV

4 d$ " > SdT 3 Vdp

7/17/2019 Lecture 05


dy y

zdx

x

zdz Ndy Mdx

x y

∂∂+

∂∂==+

y x x

N

y

M

∂∂

=

∂∂

pS S

V pT ∂∂= ∂∂V S S

p

V

T

∂∂−= ∂∂

V T T

p

V

S

∂

∂=

∂

∂

pT T

V

p

S

∂

∂−=

∂

∂

-4 /4

@4 4

Another Summary: Ma#1ellBs 8elations

'a) " 2 3

'b) S " '2res6T)

'c) ; " 3 pV

'd) & " 7 TS

'e) $ " ; > TS

-4 d " TdS 7 pdV

/4 d; " TdS 3 Vdp@4 d& " >SdT > pdV

4 d$ " >SdT 3 Vdp

7/17/2019 Lecture 05


Ma#1ell 8elations: The Ma!ic SEuareFG

V & T

$

P;

S

5ach side is labeled with an

ner!y ', ;, &, $).)he corners are labeled with

Thermodynamic Variables

'p, V, T, S)4 6et theMa#1ell 8elations

by 7wal%ing8 around the

square. 9artial derivativesare obtained from the sides.

)he Ma#1ell 8elations

are obtained from the corners.

7/17/2019 Lecture 05


Summary

The Most ommon Most ommonMa#1ell 8elations:Ma#1ell 8elations:

P T P S

V T V S

T

V

P

S

S

V

P

T

T

P

V

S

S

P

V

T

∂∂=

∂∂−

∂∂=

∂∂

∂∂= ∂∂ ∂∂−= ∂∂

7/17/2019 Lecture 05


Ma#1ell 8elations: Table ' H )

i

7/17/2019 Lecture 05


Internalner!y

;elmholt%

&ree ner!y

nthalpy

$ibbs &reener!y

Ma#1ell 8elationsMa#1ell 8elations from d, d&, d;, . d$

S C M bl P ti

7/17/2019 Lecture 05


Some Common Measureable Properties

;eat Capacity at Constant Volume:

;eat Capacity at Constant Pressure:

M C M bl P ti

7/17/2019 Lecture 05


More Common Measureable Properties

Volume #pansion Coefficient:

Isothermal Compressibility:

4ote!! 8eifBsnotation for

this is J

The KulL Modulus is the

in(erse of the Isothermal

Compressibility

K ≡ ')>-

7/17/2019 Lecture 05


Some Sometimes seful 8elationshipsSummary of 8esults

:erivations are in the te$t and;or are left to the student<

ntropy:

dT RT

H dP

RT

V

RT

Gd

2−=

nthalpy:

$ibbs &ree

ner!y:

T i l l

7/17/2019 Lecture 05


Typical #ample• 6iven the entropy S as a function of temperature

T ' volume V, S " S'T,V), find a convenient

e%&ression for '

S6

T)P, in terms of some

meas'reable &ro&erties(

• Start with the e$act differential!

• se the triple product rule ' definitions!

7/17/2019 Lecture 05


• se a Ma#1ell 8elation:

• *ombining these e$pressions gives!

• *onverting this result to a partial derivative gives!

)hi b i

7/17/2019 Lecture 05


• )his can be rewritten as!

• )he triple product rule is!

• Substituting gives!

7/17/2019 Lecture 05


4ote again the definitions:

• Volume #pansion Coefficient

N≡

V>-'

V6

T)p

• Isothermal Compressibility

≡

>V>-'

V6

p)T

• 4ote again!! 4eif’s notation for the

Volume #pansion Coefficient is J

7/17/2019 Lecture 05


• sing these in the previous e$pression

finally gives the desired res'lt:

•sing this result as a starting point,

A G1410AL 01LAT+O4S2+, between the

;eat Capacity at Constant Volume CV

' the

;eat Capacity at Constant Pressure Cp

can be found as follows!

7/17/2019 Lecture 05


• sing the definitions of the isothermal

compressibility and the volume e$pansion

coefficient , this becomes

$eneral 8elationship

bet1een C( . Cp

7/17/2019 Lecture 05


Simplest Possible #ample: The Ideal $as

P

RTP

RT

vP

RT

P

RT

P v P

v

v

T

RT

R

vP

R

P

RT

T vT

v

v

T T

P P

1

11

1

11

/

=

==

∂∂−=

∂∂−=

=

==

∂∂=

∂∂=

κ

κ

β

β

• #or an Ideal $as, it’s easily shown (4eif that the

1.'ation of State (relation between pressure P, volume V,temperature T is (in per mole units<! PO " 8T. O " 'V6n)

• With this, it is simple to show that the volume e$pansion

coefficient N ' the isothermal compressibility are!

and

7/17/2019 Lecture 05


and

• So, for an Ideal $as, the volume e$pansion coefficient

' the isothermal compressibility have the simple forms!

• We =ust found in general that the heat capacities at

constant volume ' at constant pressure are related as

• So, for an Ideal $as, the specific heats per mole

have the very simple relationship!

th S ti f l i

7/17/2019 Lecture 05


ther, Sometimes seful, #pressions

T CONSTANT dV V

R

T

P S

T CONSTANT dP

P

R

T

V S

T CONSTANT dP T V T V H

P

P V

T V

P

P P

T P

P

P P

T P

∫ ∫

∫

=

=

=

−

∂∂

−=

−

∂

∂−=

∂∂−=

0

.

0

.

0

.

M A li ti i th C bi d

7/17/2019 Lecture 05


More Applications: sin! the Combined

-st . /nd 0a1s 'The TdS 1.'ationsF)

Calorimetry A!ain• *onsider T1o Identical bQects, each of mass m, '

specific heat per %ilogram cP. See figure ne$t page.

bQect - is at initial temperature T-.bQect / is at initial temperature T/.

"ssume T/ R T-.

• When placed in contact, by the #nd Law, heat 2 flows from the hotter (bQect / to the cooler

(bQect -, until they come to a common

temperature, Tf .

7/17/2019 Lecture 05


• T1o Identical bQects, of mass m, ' specific heat per

%ilogram cP. bQect - is at initial temperature T-. bQect / is

at initial temperature T/.

• T/ R T-. When placed in contact, by the #nd Law, heat 2 flows from the hotter (bQect / to the cooler (bQect -,

until they come to a common temperature, Tf .

bQect -Initially

at T-

bQect /Initially

at T/

2⇒

2eat Flows

/

/1 T T T f

+=

•"fter a long enough time, the two ob=ects are at the sametemperature Tf . Since the / ob=ects are identical, for this case,

#or some timeafter initial

contact!

7/17/2019 Lecture 05


• )he 1ntro&y hange S for this process can also

be easily calculated!

+=∆

=

=

=

+

=

+=∆

∫ ∫

/1

/1

/1

/

/1/1

/

/1

/ln/

ln/lnln

lnln1 /

T T T T mcS

T T

T mc

T T

T mc

T T

T mc

T

T

T

T mc

T

dT

T

dT mcS

P

f

P

f

P

f

P

f f

P

T

T

T

T P

f f

• >f course, by the #nd Law,

the entropy change S m'st

be &ositive!! )his requires

that the temperatures satisfy! ?(

?/

@/

/

//1

/1//

/1

/1/1//

/1

/1/1

>−

>−+>++

>+

T T

T T T T

T T T T T T

T T T T

Some seful TdS EuationsF

7/17/2019 Lecture 05


Some seful TdS Euations• 4OT1: &n the following, various quantities are

written in per mole units< Wor% with the

ombined st " #nd Laws:

Definitions:• ≡ Number of moles of a substance.

• O≡

'V6)≡

Aolume per mole.• u ≡ '6) ≡ &nternal energy per mole.

• h ≡ ';6) ≡ 5nthalpy per mole.

• s≡

'S6)≡

5ntropy per mole.• c( ≡ 'C(6) ≡ const. volume specific heat per

mole.

• cP ≡

'CP6)≡

const. pressure specific heat per mole.

6i th d fi iti it b h th t

7/17/2019 Lecture 05


dP c

dvv

cdP P

T cdvv

T cTds

dP TvdT cdP T

vT dT cTds

dvT

dT cdvT

P T dT cTds

v P

v

v

P

P

P

P

P

v

v

v

β

κ

β

β

κ

β

+= ∂∂

+ ∂∂

=

−=

∂∂−=

+=

∂∂

+=

• 6iven these definitions, it can be shown that

the ombined st " #nd Laws 'TdS) can be

written in at least the following ways!

• Student e$ercise to show that starting with the previous

7/17/2019 Lecture 05


+nternal 1nergy

u'T,O):

dv P

v

udT cTds

dvvudT

T udu

T

v

T v

+

∂

∂+=

∂∂+

∂∂=

1nthal&y

h'T,P):

Student e$ercise to show that, starting with the previous

e$pressions ' using the definitions (per mole of internal

energy u ' enthalpy h gives!

• Student e$ercise also to show that similar manipulations

7/17/2019 Lecture 05


v

v

v

vvvvv

P v

P

T

T

c

P

s

P

T

T

sT

T P

T

T

s

P

s

dvv

sdP

P

sds

v P s s

∂∂

=

∂∂

∂∂

∂∂

=

∂∂

∂∂

=

∂∂

∂∂

+

∂∂

=

=

1

,( *onsider

P

P

P

p P P P P

P v

v

T

T

c

v

s

v

T

T

sT

T v

T

T

s

v

s

dv

v

sdP

P

sds

∂∂

=

∂∂

∂∂

∂∂

=

∂∂

∂∂

=

∂∂

∂

∂+

∂

∂=

1

dvv

T cdP

P

T cTds

dvv

T

T

cdP

P

T

T

cds

dvv

sdP

P

sds

P

P

v

v

P

P

v

v

P v

∂∂+

∂∂=

∂∂

+

∂∂

=

∂∂+

∂∂=

• Student e$ercise also to show that similar manipulations

give at least the following different e$pressions for the

molar entropy s! 1ntro&y s'T,O):

Lecture 05

Documents

Transcript of Lecture 05