Does the Internal Energy of a Real Gas Depend
Only on Temperature?
Does
u
depend on Volume?
dv
v
u
dT
T
u
du
T
v
If
u
=
u
(
T
,
v
), then,
0
u
u
v
T
u
u
dT
dv
T
v
v
T
v
u
u
u
u
T
T
c
v
T
v
v
Gay-Lussac–Joule Results
-3
1
0,
0.001 K kilomole m
u
v
T
u
u
T
u
v
c
v
T
T
v
v
is
Gay-Lussac–Joule Coefficient
Usually is called Joule Coefficient
For perfect Ideal gas
=0
0,
0,
0
constant
Q
U
W
U
U
Q
W
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
Joule-Thompson Throttling Process
x
1
x
2
P
1
=F
1
A
P
2
=F
2
A
Joule-Thompson Throttling Process
0 = U
2
– U
1
+ P
2
V
2
– P
1
V
1
0 = (U
2
+ P
2
V
2
) – (U
1
+ P
1
V
1
)
0=H
–H
, H is constant
On high pressure side:
W
1
= – P
1
V
1
On low pressure side:
W
2
= + P
2
V
2
Total work:
W
=
P
2
V
2
– P
1
V
1
Enthalpy
For
0, 0
Por
Ph T T h
dT
h
h
dT
dh
c
c
dP
P
P
dP
For
( , )
P
P T T
h
h T P
h
h
h
dh
dT
dP
c dT
dP
T
P
P
Joule-Thomson coefficient
h
dT
dP
,
If
0,
( )
0
P
T
h
c
h
h P
For perf
P
ect ideal gas
Enthalpy
....
dQ dH VdP
First law
dQ dU PdV
H
U
PV
dH
dU
PdV VdP
(First law
)
Joule-Thompson Throttling Process
T and V independent
v
T
u
u
dq
dT
P dv
T
v
đq = du + Pdv
First Law
v
T
u
u
du
dT
dv
T
v
Constant Pressure
p v
T P
u
v
c
c
P
v
T
p
v
T
P
dq
u
u
v
P
dT
T
v
T
p
v
T
P
u
v
c
c
P
v
T
P
P
P
v
T
u
u
dq
dT
P dv
Ideal Gas
u
is not a function of
v
.
p
v
T
P
u
v
c
c
P
v
T
0
p
v
R
c
c
P
P
Adiabatic Process
0
s
s
v
T
u
u
dq
dT
P dv
T
v
0
v
s
s
T
u
c dT
P dv
v
v
s
T
T
u
c
P
V
v
T and P independent
p
h
h
dq
dT
v dP
T
p
dq = dh –vdP First Law
p
T
h
h
dh
dT
dp
T
p
T and P independent
p
T
h
h
dq
dT
v dP
T
p
P
T
h
dq
c dT
v dP
p
v
v
P
v
v
T
h
c dT
c dT
v dP
p
p
v
h
P
c
c
v
p
T
Adiabatic Process
0
s
s
p
T
h
h
dq
dT
v dP
T
p
0
p
s
s
T
h
c dT
v dP
p
p
T
h
c
v
P
p
Adiabatic Process
p
s
T
T
h
c
v
P
p
v
s
T
T
u
c
P
V
v
p
T
s
v
T
h
v
p
c
v
c
P
u
P
v
h
u
v
p
p
p
T
T
T
h
u
pv
v
h
u
v
u
v
v
p
p
p
v
T
p
p
T
T
T
T
T
u
v
v
P
v
P
P
Adiabatic Process
,
p
p
s
T
s
T
v
v
c
v
v
P
P
c
c
P
P
v
v
c
constant
s
T
P
P
P
for ideal gas
v
v
v
Pv
without knowing anything aboutU
Adiabatic Process
We derived previously
(
p v)
v T
h
T
v
c
c
p
P
(
p v)
T P
u
T
P
c
c
v
v
p T s T v Th
v
p
c
v
v
c
P
u
P
P and V independent
third way of writing first law:
,
,
... .... ..???...
v
p
dq
du
pdv
dq
dh vdp dq
c
c
p v
u
u
du
dv
dp
v
p
v pu
u
dq
dp
p dv
p
v
v vv v v
u
u
T
T
c
p
T
p
p
p
p p p p p
u
h
h
T
T
u
pv
h
p
c
v
v
T
v
v
v pT
T
dq
c
dp
c
dv
