Physical Chemistry Third Edition

26.2 Working Equations for the Thermodynamic Functions of a Dilute Gas 1093

The Heat Capacity of a Dilute Gas

The heat capacity at constant volume is given by Eq. (26.1-13), but it is easier to

differentiate the formulas for the internal energy.

CV,tr

(

∂Utr

∂T

)

V,N



3

2

NkB

3

2

nR (26.2-10a)

CV,el

(

∂Uel

∂T

)

N

≈0 (most substances) (26.2-10b)

CV,rot

(

∂Urot

∂T

)

N

NkBnR

⎛

⎝

diatomic or linear

polyatomic

substances

⎞

⎠ (26.2-10c)

CV,rot

(

∂Urot

∂T

)

N



3

2

NkB

3

2

nR

⎛

⎝

nonlinear

polyatomic

substances

⎞

⎠ (26.2-10d)

CV,vib

d

dT

(

Nhν

ehν/kBT− 1

)



Nhν

(ehν/kBT−1)^2

ehν/kBT

(

hν

kBT^2

)

NkB

(

hν

kBT

) 2

ehν/kBT

(ehν/kBT−1)^2

NkB

(

hν

kBT

) 2

ehν/kBT

(ehν/kBT−1)^2

CV,vibNkB

(

hcν ̃

kBT

) 2

ehcν/k ̃ BT

(ehc ̃ν/kBT−1)^2

(

diatomic

substances

)

(26.2-10e)

CV,vibNkB

3 n∑−5(6)

i 1

(

hc ̃νi

kBT

) 2

ehcν ̃i/kBT

(ehcν ̃i/kBT−1)^2

(

polyatomic

substances

)

(26.2-10f )

The heat capacity at constant pressure is given by Eq. (26.1-19). We add the termNkB

to the translational contribution. The other contributions toCPare the same as the

contributions toCV.

CP,trCV,tr+NkB

5

2

NkB

5

2

nR (26.2-10g)