Thermodynamics and Statistical
Mechanics
Does the Internal Energy of a Real Gas
Depend Only on Temperature?
Thermo & Stat Mech Spring 2006 Class 4
1
Change of Phase
First Law is: du = đq – Pdv
For a finite change at constant temperature
Du = q – PDv
At transition, u2 – u1 = l12 – P(v2 – v1)
l12 = (u2 + Pv2) – (u1 + P v1)
l12 = h2 – h1
Thermo & Stat Mech - Spring 2006
Class 4
2
Enthalpy and Latent Heat
Enthalpy is defined as: h = u + Pv
 12  h  h
 23  h  h
solid to liquid (fusion)
 13  h  h
solid to vapor(sublimation)
liquid to vapor(vaporization)
Thermo & Stat Mech - Spring 2006
Class 4
3
Enthalpy is a State Function
Thermo & Stat Mech - Spring 2006
Class 4
4
Does u depend on Volume?
If u = u(T, v), then,
u 
u 


du    dT    dv
 T v
 v T
Thermo & Stat Mech - Spring 2006
Class 4
5
From Cyclical Relation
 u 
 u 
 
 
 T     v T    v T
 
cv
 u 
 v u
 
 T v
 u   c  T 
 

v
 v T
 v u
Thermo & Stat Mech - Spring 2006
Class 4
6
Gay-Lussac–Joule Experiment
Thermo & Stat Mech - Spring 2006
Class 4
7
Gay-Lussac–Joule Results
 T   0
 
 v u
T 

-3
     0.001 K kilomole m
 v u
Thermo & Stat Mech - Spring 2006
Class 4
8
Joule-Thompson Throttling
Process
A gas passes through a constriction
from a region where it is at high
pressure to a region where it is at lower
pressure. The gas expands, and the
temperature of the gas can be lowered.
This is an important tool in low
temperature physics.
Thermo & Stat Mech - Spring 2006
Class 4
9
Joule-Thompson Throttling
Process
Thermo & Stat Mech - Spring 2006
Class 4
10
Joule-Thompson Throttling
Process
On high pressure side: Wi = – PiVi
On low pressure side: Wf = + PfVf
Total work: W = PfVf – PiVi
Q = 0 = DU + W
0 = Uf – Ui + PfVf – PiVi
0 = (Uf + PfVf ) – (Ui + PiVi )
Thermo & Stat Mech - Spring 2006
Class 4
11
Enthalpy
H  U  PV
dH  dU  PdV  VdP
(First law) dU  dQ  PdV
so dU  PdV  dQ
dH  dQ  VdP
dq  dh  vdP
Thermo & Stat Mech - Spring 2006
Class 4
12
Enthalpy
dq  dh  vdP
 dq    dh   c
   
P
 dT  P  dT  P
Thermo & Stat Mech - Spring 2006
Class 4
13
Enthalpy
For h  h(T , P)
h 
h 
h 



dh    dT    dP  cP dT    dP
 T  P
 P T
 P T
For dh  0,
dT   h 
h 
dT 



0  cP      or    cP  
 dP  h  P T
 P T
 dP  h
Thermo & Stat Mech - Spring 2006
Class 4
14
Enthalpy
 dT    Joule - Thomson coefficien t
 
 dP  h
 h   c 
 
P
 P T
If   0, h  h( P)
True for ideal gas.
Thermo & Stat Mech - Spring 2006
Class 4
15
Ideal Gas
dh 
dh

cP    
 dT  P dT
du
cV 
dT
Thermo & Stat Mech - Spring 2006
Class 4
16
Joule-Thompson Throttling
Process
Thermo & Stat Mech - Spring 2006
Class 4
17