
The Beginnings 31
Which is incidentally the same equation for adiabatic changes in modern thermody-
namics too. Therefore, this particular equation does not care what the nature of heat
is.
Let us evaluate
∂
Q
∂
V
T
for an ideal gas directly from eqn.(1.34):
∂
Q
∂
V
T
=
∂
Q
∂
V
P
+
∂
Q
∂
P
V
∂
P
∂
V
T
=
P
R
C
P
−
P
R
C
V
(1.38)
Combining this with the expression for
μ
(T), one gets the remarkable equality
C
P
−C
V
=
R
μ
(T)T
(1.39)
This is the mathematical derivation of Carnot’s result for the specific heats; and the
difference can only depend on temperature, with C
P
always greater than C
V
. Carnot
had concluded that if C
P
−C
V
was a constant, the specific heats must have a logarth-
mic dependence ...