Physical Chemistry Third Edition

26.2 Working Equations for the Thermodynamic Functions of a Dilute Gas 1097

Solution

ztr

(

2 πmkBT

h^2

) 3 / 2

V



(

2 π(1. 16134 × 10 −^25 kg)(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

(0.02479 m^3 )

 1. 4028 × 1031

ztr

N



1. 4028 × 1031

6. 022 × 1023

 2. 329 × 107

The rotational partition function was already calculated in Example 25.7:

zrot 424. 7

The vibrational partition function was already calculated in Example 25.8:

zvib 1. 072

zel≈ 1. 000

zztrzrotzvibzel(1. 4028 × 1031 )(424.7)(1.072)(1.000) 6. 3867 × 1033

z

N



6. 3867 × 1033

6. 02214 × 1023

 1. 0605 × 1010

U◦m,tr

3

2

RT(1.500 mol)(8.3145 J K−^1 mol−^1 )(298.15 K) 3718 .45 J mol−^1

Um,rot◦ RT(1.000 mol)(8.3145 J K−^1 mol−^1 )(298.15 K) 2478 .97 J mol−^1

Um,vib◦ 

NAvhν

ehν/kBT− 1



NAvhcν ̃

ehcν/k ̃ BT− 1

 481 .92 J mol−^1

Um,el◦ ≈ 0

Um◦ 3718 .45 J mol−^1 + 2478 .97 J mol−^1 + 481 .92 J mol−^1

 6679. 34 .42 J mol−^1

 6 .67934 kJ mol−^1

Hm◦Um◦+PVmUm◦+RT 6679 .34 J mol−^1 + 2478 .97 J mol−^1

 9158 .31 J mol−^1  9 .15831 kJ mol−^1

These values are relative to the ground-state energy. The experimental value of H◦mis

9.181 kJ mol−^1 relative to this zero of energy.

Sm,tr◦ (8.3145 J K−^1 mol−^1 )

(

9. 2100 +

5

2

ln( 298. 15 )+

3

2

ln( 0. 0699375 )+ln(1)

)

 161 .831 J K−^1 mol−^1

Sm,rot◦ 

Um, rot

T

+Rln(zrot)R+Rln(zrot)

 8 .3145 J K−^1 mol−^1 +(8.3145 J K−^1 mol−^1 )ln(424.7) 58 .627 J K−^1 mol−^1