CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 91
3.5.1.2 Cpdependence on pressure
Cp(T, p 2 ) =Cp(T, p 1 )−∫p 2p 1T
(
∂^2 V
∂T^2)pdp , [T]. (3.63)3.5.1.3 CV dependence on volume
CV(T, V 2 ) =CV(T, V 1 ) +
∫V 2V 1T
(
∂^2 p
∂T^2)VdV , [T]. (3.64)3.5.1.4 Relations between heat capacities.
Cp = CV+[
p+(
∂U
∂V)T](
∂V
∂T)p=CV+T
(
∂p
∂T)V(
∂V
∂T)p= CV−T
(
∂V
∂T) 2( p
∂V
∂p)T=CV−T
(
∂p
∂T) 2( V
∂p
∂Vm)T. (3.65)
The following relation applies between heat capacities and the coefficients of isothermal
compressibilityβ[see2.1.7] and thermal expansionα[see2.1.6]
Cp=CV+T Vα^2
β. (3.66)
3.5.1.5 Ideal gas.
The heat capacities of an ideal gas are functions of temperature only. Relations (3.65) rearrange
to theMayer relation
C◦pm(T) =CV m◦ (T) +R. (3.67)