
76 The Principles of Thermodynamics
Let us first illustrate getting a relation between partial derivatives of a different type
than what we have obtained so far. From eqn.(3.36) it is easy to get
∂
U
∂
V
T
= T
∂
S
∂
V
T
−P (3.39)
It is to be noted that we got this even though neither of the independent variables in
eqn.(3.36) has been held fixed. Once again, the integrability condition for the first of
eqn.(3.38) is
∂
∂
V
C
V
T
T
=
⎛
⎝
∂
∂
T
P+
∂
U
∂
V
T
T
⎞
⎠
V
(3.40)
On using the integrability condition for dU,
∂
∂
V
∂
U
∂
T
V
T
=
∂
∂
T
∂
U
∂
V
T
V
and
C
V
=
∂
U
∂
T
V
, the above equation reduces to
∂
U
∂
V
T
= T
∂
P
∂
T
V
−P (3.41)
Eqn.(3.41) is a very important equation in thermodynamics from which