Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

PROBLEMS AND EXAMPLES 121

For Na 2 (g) molecules, a similar calculation gives a result that is not very different

from the result for atoms. At 1300 K, (Na 2 (g))= 7 .14 pm. The dimensions for both

thermal wavelengths are picometers

(

10 −^12 m

)

,sothevolume 

3 is of the order of

10 −^33 m^3. This dimension confirms our prior supposition that translational energy

levels are very close together. They can be regarded as a continuum if mathematically

convenient, and they are fully populated by a mole of particles.

Example 8.2 The Translational Equilibrium Constant

Given that the energy separation of ground states (Section 8.3) is ε 0 =

70 .5kJmol−^1 (from spectroscopic measurements), calculate what the equilibrium

constant would be for translational motion only (ignoring all internal modes of

motion).

Solution 8.2 The equilibrium expression is

Keq=

QNa/NA^2

QNa 2 /NA

e−ε^0 /RT

The exponents 2 and 1 arise from the stoichiometric coefficientsξin the equilibrium

expression for the reaction written as a dissociation

Keq=

(pNa)^2

pNa 2

Expressing the partition functions in terms of the thermal wavelengths, we obtain

Keq=

(

(^3) Na 2
(^6) Na

)

Vm

NA

e−ε^0 /RT=

3. 64 × 10 −^34

1. 06 × 10 −^66

0. 082

6. 022 × 1023

e−^6.^51

= 4. 67 × 107 e−^6.^51 = 6. 95 × 104

The exponential value−ε 0 /RT=− 70. 4 /R( 1300 )=− 6 .51 comes from spec-

troscopic data. The molar volume is 0.0820 m^3 mol−^1 ande−^70.^4 /RT=e−^6.^51 =

1. 488 × 10 −^3.

But things are not quite that simple. The Na 2 molecule has internal modes of

motion, one for vibration along the molecular axis and one for rotation about its

center of mass. Both partition functions can be determined from spectroscopic data.

We have already seen how the value ofQvibarises from the resonance frequency for

vibrational motion. The quantized levels for rotation of Na 2 (g) molecules are found

in essentially the same way. The two calculations give

Qvib= 6. 254 , Qrot= 3. 132 × 103

The partition functions are unitless. One more factor comes into the calculation:

integer 4, which is the multiplicity of the sodium atom electronic structure.