Concise Physical Chemistry

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

COMPUTATIONAL STATISTICAL THERMODYNAMICS 119

The thermal wavelength has the units of length, m [or picometers (pm) in

atomic–molecular problems]. If we let the volume chosen be the molar volume

a^3 =Vm, then the volume per particle isVm/NA, whereNAis Avogadro’s number.

Now is only a collection of constants times the inverse square root of the mass of

the particle and the temperature:

=

(

h^2 β

2 πm

)^12

=h

(

1

2 πmkBT

) (^12)
=h

(

1

2 πm

) 12 (

1

kBT

) (^12)

= 7. 113 × 10 −^23

1

√

mT

The partition function is a measure of the accessibility of quantum states to

particles distributed within the system. Because the cube of the thermal wavelength

is inversely related to the partition function through the volume of the container in

which the particles are trapped, gives us an idea of how big a container has to be for

its quantum states to be fully populated. For atoms and small molecules the volume



3 is of the order of 10−^30. Any imaginable laboratory container is very much larger

than this, so we conclude that alltranslational levels are accessible. Bear in mind

that this volume restriction relates to translational motion only and not to any other

mode of motion.

8.9 COMPUTATIONAL STATISTICAL THERMODYNAMICS

Further insight into this calculation can be gained from a computational solution

forKeq.

Several computer programs give calculated values for the thermodynamic func-

tions and the related partition functions. The functions are usually broken up into

their individual contributions as described above. An edited quantum mechanical

output of this kind for the sodium atom is given in Table 8.2. These values were taken

from a much larger output file from the program GAUSSIAN 03©C. Computed values

should be used with some caution because they often rely on approximations like the

harmonic oscillator approximation.

One can take the total partition functions, plug them into the equilibrium constant

quotient and multiply by the difference in ground state energies of the atoms relative

to the molecules to obtainKeq.TheQvalues chosen are fromV=0, the vibrational

ground state.

Keq=

Q^2 Na

QNa 2

e−ε^0 /RT=

(

3. 44 × 108

) 2

9. 531 × 1012

e−^6.^51 = 18. 5