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.