Physical Chemistry Third Edition

1098 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

Sm,vib◦ 

Um,vib

T

+Rln(zvib)



481 .92 J mol−^1

298 .15 K

+(8.3145 J K−^1 mol−^1 )ln(1.072) 2 .194 J K−^1 mol−^1

Sm,el◦ 

Um◦,el

T

+Rln(zel)≈ 0 + 0  0

S◦m 161 .831 J K−^1 mol−^1 + 58 .627 J K−^1 mol−^1 + 2 .194 J K−^1 mol−^1

 222 .652 J K−^1 mol−^1

The experimental value is 223.079 J K−^1 mol−^1.

C◦V,m,tr

3

2

R 12 .4718 J K−^1 mol−^1

C◦V,m,rotR 8 .3145 J K−^1 mol−^1

C◦V,m,vibR

(

hν

kBT

) 2

ehν/kBT

(ehν/kBT−1)^2

Rx^2

ex

(ex−1)^2

 4 .680 J K−^1 mol−^1

C◦V,mel≈ 0

CV,m◦  25 .466 J K−^1 mol−^1

C◦P,m 25 .466 J K−^1 mol−^1 + 8 .3145 J K−^1 mol−^1  33 .780 J K−^1 mol−^1

The experimental value is 33.949 J K−^1 mol−^1 at this temperature. The discrepancies between

our values and the experimental values are presumably due to the inadequacy of the harmonic

oscillator–rigid rotor energy level expression.

The Helmholtz Energy of a Dilute Gas

The contributions to the Helmholtz energy are given by the formulas

Atr−NkBTln

(

Ztr

N

)

−NkBT (26.2-16a)

Arot−NkBTln(Zrot) (26.2-16b)

Avib−NkBln(Zvib) (26.2-16c)

Aelec−NkBln(Zelec) (26.2-16d)

where we place the divisorNand the termNkBTwith the translational contribution.

EXAMPLE26.9

Find the vibrational contributions to the thermodynamic energy, Helmholtz energy,

and entropy of 1.000 mole of water vapor at 1.00 atm and 100◦C. There are three

vibrational normal modes, with frequencies 1.0947× 1014 s−^1 ,4. 7817 × 1013 s−^1 , and

1. 1260 × 1014 s−^1.