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04/19/23 PHY 113 -- Lecture 20 1
Announcements
1. Please bring your laptops to lab this week.
2. Physics seminar this week – Thursday, Nov. 20 at 4 PM – Professor Brian Matthews will discuss relationships between protein structure and function
3. Schedule –
Today, Nov. 18th: Examine Eint especially for ideal gases
Discuss ideal gas law
Continue discussion of first law of thermodynamics
Thursday, Nov. 20th: Review Chapters 15-20
Tuesday, Nov. 25th: Third exam
04/19/23 PHY 113 -- Lecture 20 2
From The New Yorker Magazine, November 2003
04/19/23 PHY 113 -- Lecture 20 3
First law of thermodynamics
Eint = Q W
For an “ideal gas” we can write an explicit relation for Eint.
What we will show:
is a parameter which depends on the type of gas (monoatomic, diatomic, etc.) which can be measured as the ratio of two heat capacities: =CP/CV.
f
i
V
VPdVW
Tk-N
RT-n
E B1γ1γ)gas ideal(
int
04/19/23 PHY 113 -- Lecture 20 4
Thermodynamic statement of conservation of energy –
First Law of Thermodynamics
Eint = Q - W
“Internal” energy of system
Heat added to system
Work done by system
Eint = Eint(f)int(i) independent on path
Qif depends on path
Wif depends on path
04/19/23 PHY 113 -- Lecture 20 5
How is temperature related to Eint?
Consider an ideal gas
Analytic expressions for physical variables
Approximates several real situations
Ideal Gas Law: P V = n R T
pressure (Pa)volume (m3)
number of moles
temperature (K)gas constant (8.31 J/(moleK))
04/19/23 PHY 113 -- Lecture 20 6
P V = n R T
Ideal gas – P-V diagram at constant T
Vi Vf
Pf
Pi
Pi Vi = Pf Vf
04/19/23 PHY 113 -- Lecture 20 7
04/19/23 PHY 113 -- Lecture 20 8
Microscopic model of ideal gas:
Each atom is represented as a tiny hard sphere of mass m with velocity v. Collisions and forces between atoms are neglected. Collisions with the walls of the container are assumed to be elastic.
04/19/23 PHY 113 -- Lecture 20 9
Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it:
average over atoms
What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by:
avgavgK
VN
vmVN
P32
32 2
21
tvm
tp
F ixiixix
2
d
x
ixvdt /2
22
2
/22
xii
ixi
i
ix
ixi
ix
ixiix
vmVN
dAvm
AF
P
dvm
vdvm
F
vix
-vix
number of atoms
volume
04/19/23 PHY 113 -- Lecture 20 10
222
21
32
31
vmVN
vmVN
vmVN
P x
Ideal gas law continued:
Recall that
Therefore:
Also:
TNkvmNPV
TNkTNR
NnRTPV
B
B
A
2
21
32
:model cMicroscopi
:relation cMacroscopi
mTk
vv
Tkvm
Brms
B
3
23
21
2
2
TkNvmNE
Tkvm
B
B
23
21
23
21
2int
2
04/19/23 PHY 113 -- Lecture 20 11
RT-n
Tk-N
TkNvmNE BB 1γ1γ23
21
2int
Internal energy of an ideal gas:
derived for monoatomic ideal gas more general relation for
polyatomic ideal gas
Gas (theory) exp)
He 5/3 1.67
N2 7/5 1.41
H2O 4/3 1.30
Big leap!
04/19/23 PHY 113 -- Lecture 20 12
Determination of Q for various processes in an ideal gas:
Example: Isovolumetric process – (V=constant W=0)
In terms of “heat capacity”:
WQTR-n
E
RT-n
E
1γ
1γ
int
int
fififi QTR-n
E 1γ
int
1γ
1γ
-R
C
TnCTR-n
Q
V
fiVfifi
04/19/23 PHY 113 -- Lecture 20 13
Example: Isobaric process (P=constant):
In terms of “heat capacity”:
Note: = CP/CV
fifififi WQTR-n
E 1γ
int
1γγ
1γ
1γ1γ
-R
R-R
C
TnCTnRTR-n
VVPTR-n
Q
P
fiPfifiifififi
04/19/23 PHY 113 -- Lecture 20 14
More examples:
Isothermal process (T=0)
T=0 Eint = 0 Q=W
WQTR-n
E
RT-n
E
1γ
1γ
int
int
i
fV
V
V
V V
VnRT
VdV
nRTPdVWf
i
f
i
ln
04/19/23 PHY 113 -- Lecture 20 15
Even more examples:
Adiabatic process (Q=0)
TnRVPPV
nRTPV
VPTR-n
WE
1γ
int
γγγ
γ
lnln
γ
1γ
ffiii
f
i
f VPVPP
P
V
V
PP
VV
VPPVVP-TnR
04/19/23 PHY 113 -- Lecture 20 16
VVP
P
VVP
P
ii
ii
:Isotherm
:Adiabat
04/19/23 PHY 113 -- Lecture 20 17
Peer instruction question
Suppose that an ideal gas expands adiabatically. Does the temperature
(A) Increase (B) Decrease (C) Remain the same
1-γ
1-γ1-γ
γγ
f
iif
ffii
i
iiiii
ffii
V
VTT
VTVT
V
TnRPnRTVP
VPVP
04/19/23 PHY 113 -- Lecture 20 18
Review of results from ideal gas analysis in terms of the specific heat ratio CP/CV:
For an isothermal process, Eint = 0 Q=W
For an adiabatic process, Q = 0
1γ ;
1γ int -
RCTnCTR
-n
E VV
1γγ-R
CP
i
fii
i
fV
V V
VVP
V
VnRTPdVW
f
i
lnln
1-γ1-γ
γγ
ffii
ffii
VTVT
VPVP
04/19/23 PHY 113 -- Lecture 20 19
Extra credit:
Show that the work done by an ideal gas which has an initial pressure Pi and initial volume Vi when it expands adiabatically to a volume Vf is given by:
1γ
11γ f
iV
V
ii
V
VVPPdVW
f
i
04/19/23 PHY 113 -- Lecture 20 20
Peer instruction questions
Match the following types of processes of an ideal gas with their corresponding P-V relationships, assuming the initial pressures and volumes are Pi and Vi, respectively.
1. Isothermal
2. Isovolumetric3. Isobaric
4. Adiabatic
(A) P=Pi (B) V=Vi (C) PV=PiVi (D)PV=PiVi
04/19/23 PHY 113 -- Lecture 20 21
P (
1.01
3 x
105 )
Pa
Vi Vf
Pi
Pf
A
B C
D
Examples process by an ideal gas:
AB BC CD DA
Q
W 0 Pf(Vf-Vi) 0 -Pi(Vf-Vi)
Eint
1-γ
)( ifi PPV 1-γ
)(γ iff VVP 1-γ
)( iff PPV 1-γ
)(γ ifi VVP-
1-γ
)( ifi PPV 1-γ
)( iff PPV 1-γ
)( iff VVP 1-γ
)( ifi VV-P
Efficiency as an engine:
e = Wnet/ Qinput
04/19/23 PHY 113 -- Lecture 20 22