388 ENGINEERING THERMODYNAMICS
dharm
\M-therm\Th8-1.pm5
We have
p a
v
HFG +^2 KJI (v – b) = RT [From eqn. (8.15)]
Keeping p constant and differentiating with respect to T, we get
p a
v
ab
v
RS −+
T
U
V
(^23) W
2 dv
dT p
F
HG
I
KJ = R
or dv
dT p
F
HG
I
KJ
R
p a
v
ab
v
RS −+
T
U
V
(^23) W
2
Substituting this value of
dv
dT p
F
HG
I
KJ in the equation cpμ = T^
dv
dT p
F
HG
I
KJ – v (where μ is a measure
of cooling effect), we get
cpμ = RT
p a
v
ab
v
RS −+
T
U
V
(^23) W
2
- v
and substituting for RT from equation (8.19) this reduces to
cpμ =
−+ −
−+
bp a
v
ab
v
p a
v
ab
v
23
2
2
23
The denominator of this expression is always positive, since it is R
dT
dv p
F
HG
I
KJ. Hence the
cooling effect, μ, is positive if
bp <^2 a
v
- 3
2
ab
v
...(8.31)
and negative if
bp >^2 a
v
-^3 ab 2
v
...(8.32)
and inversion occurs when
bp =
2 a
v
- 3
2
ab
v
or p =
a
b^
23
v^2
b
v
F −
HG
I
KJ
...(8.33)
In order to get the temperature of inversion this equation must be combined with the origi-
nal equation. Thus
2 a
b^
1
2
F −
HG
I
KJ
b
v = RT ...(8.34)
Since v is necessarily always greater than b, it will be seen that as v increases so also does
the temperature of inversion.