Physical Chemistry Third Edition

1112 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics



(

2 π(1. 674 × 10 −^27 kg)(1. 3807 × 10 −^23 JK−^1 )(500.0K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

(2)

 2. 127 × 1030 m−^3 (2) 4. 254 × 1030 m−^3

The partition function of HBr has a translational and a rotational factor:

z′′tr,Br

(

2 πmHBrkBT

h^2

) 3 / 2



(

2 π(1. 327 × 10 −^25 kg)(1. 3807 × 10 −^23 JK−^1 )(500.0K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

 1. 501 × 1033 m−^3

zrot,HBr

kBT

hcB ̃e



(1. 3807 × 10 −^23 JK−^1 )(500.0K)

(6. 6261 × 10 −^34 J s)(2. 9979 × 1010 cm s−^1 )(8.4649 cm−^1 )

 41. 05

z′′HBr(1. 501 × 1033 m−^3 )(41.05) 6. 161 × 1034 m−^3

The partition function of the activated complex has a translational, a rotational, and a vibra-

tional factor:

z‡HHBr′′ z‡HHBr,tr′ z‡HHBr,rot′ z‡HHBr,vib′

z‡tr,HHBr′′ 

(

2 πmHHBrkBT

h^2

) 3 / 2



(

2 π(1. 360 × 10 −^25 kg)(1. 3807 × 10 −^23 JK−^1 )(500.0K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

 1. 558 × 1033 m−^3

To obtain the rotational partition function, we require the moment of inertia of the activated

complex. The center of mass of three objects is found as in Eq. (D-25) of Appendix D.

xc

m 1 x 1 +m 2 x 2 +m 3 x 3

m 1 +m 2 +m 3

If the origin of coordinates is taken at the hydrogen nucleus at the end of the activated complex,

we have

xc

0 +mH(150 pm)+mBr(292 pm)

mH+mH+mBr



(1.008 amu)(150 pm)+(79.904 amu)(292 pm)

2 .016 amu+ 79 .904 amu

 286 .7pm

The moment of inertia is

IemH(286.7pm)^2 +mH(136.7pm)^2 +mBr(5.3pm)^2

(1.008 amu)(286.7pm)^2 +(1.008 amu)(136.7pm)^2 +(79.904 amu)(5.3pm)^2

(1. 039 × 104 amu pm^2 )

(

0 .001kg

6. 02214 × 1023 amu

)(

1m

1012 pm

) 2

 1. 73 × 10 −^46 kg m^2