Difference in Cp and Cv
From the definitions of Cp and Cv ,
Cp − Cv =
dq
dT
−
P
dq
dT
(1)
V
Substitution of the definition of entropy gives
⎡ ∂S
Cp − Cv = T ⎢
⎣ ∂T
∂S ⎤
⎥
∂T V ⎦
−
P
(2)
These partials are converted from the total differential obtained from S = f (T , V )
dS =
∂S
∂T
∂S
∂T
−
P
dT +
V
∂S
∂T
=
V
∂S
∂V
∂S
∂V
dV , which when divided by dT at dP = 0 becomes
T
T
∂V
∂T
(3)
P
Substituting Eq. (3) into Eq. (2) gives
Cp − Cv = T
∂S
∂V
T
∂V
∂T
∂S
∂V
The Maxwell Relationship
Cp − Cv = T
(4)
P
∂P ∂V
∂T V ∂T
=
T
∂P
further transforms Eq. (4) to
∂T V
(5)
P
The total differential from P = (T , V ) at dP = 0 gives 0 =
∂P
∂P
=−
∂T V
∂V
T
∂V
∂T
=−
P
∂V
∂T
P
∂V
∂P
∂P ⎞ ∂V ⎞ ∂P ⎞
⎟
⎟ +
⎟ and
∂V ⎠T ∂T ⎠ P ∂T ⎠V
= α β where α ≡
T
1 ∂V
V ∂T
β ≡−
P
1 ∂V
V ∂P
T
Therefore, Eq. (5) may be written
α 2TV
Cp − Cv =
β
(6)
Note that substituting the definition of entropy into Eq. (1) does not introduce an additional constraint of
reversibility since heat capacities are state functions.