1086 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics
The Heat Capacity of a Dilute Gas
The heat capacity at constant volume is given by Eq. (2.4-4) for a simple systemCV
(
∂U
∂T
)
N,V(
∂
∂T
[
NkBT^2(
∂ln(z)
∂T)
V])
N,VCV 2 NkBT(
∂ln(z)
∂T)
V+NkBT^2(
∂^2 ln(z)
∂T^2)
V(26.1-13)
The heat capacity at constant pressure for our dilute gas can be obtained fromCV. From
Eq. (2.5-9) and Eq. (2.5-1)CP
(
∂H
∂T
)
P,n(
∂U
∂T
)
P,n+P
(
∂V
∂T
)
P,n(26.1-14)
Equation (B-7) of Appendix B is an example of thevariable-change identity:
(
∂U
∂T)
P,n(
∂U
∂T
)
V,n+
(
∂U
∂V
)
T,n(
∂V
∂T
)
P,n(26.1-15)
We substitute this equation into Eq. (26.1-14) and use the fact thatCV(∂U/∂T)V,n
to writeCPCV+
[(
∂U
∂V
)
T,n+P
](
∂V
∂T
)
P,n(26.1-16)
From Eq. (26.1-12) a dilute gas obeys the ideal gas equation of state so that
(
∂V
∂T)
P,nnR
P(dilute gas) (26.1-17)We assume that our dilute gas shares another property of an ideal gas:(∂U/∂V)T,n0 (ideal gas) (26.1-18)so thatCPCV+nRCV+NkB (26.1-19)The Enthalpy of a Dilute Gas
The enthalpy is defined by Eq. (2.5-1):HU+PV (26.1-20)Since our dilute gas obeys the ideal gas equation of state,HU+PVNkBT^2(
∂ln(z)
∂T)
V+NkBT (26.1-21)