Physical Chemistry , 1st ed.

qexp

exp

h^2 (n

8

x

2

m



V

n

2/

y

2

(^3) k



T

nz^2 )



q

nz

exp

8 m



V

h

2

2

/

n

3

x^2

kT



ny

exp

8



m

h

V

(^2) n
2
y^2
/3kT
nz
exp
8 m



V

h

2

2

/

n

3

z^2

kT

 (17.51)

where the individual summations are now over the x,y, and ztranslational

quantum numbers. All levels of one-dimensional particles-in-boxes are singly

degenerate.

Since there is usually no preferential dimension for a three-dimensional sys-

tem (a condition described as “isotropic”), then nxis calculationally equivalent

to ny, which is calculationally equivalent to nz. Also, because the system is as-

sumed to be cubic, the quantized energies will be the same in all three dimen-

sions. This means that the three terms in equation 17.51 can be combined into

the third power of a single term:

q

n

exp

8 m



V

h

2

2

/

n

3

2

kT



3

(17.54)

We have removed the subscript from the general translational quantum

number n.

We now take the mathematical leap inferred by equation 17.47: if the indi-

vidual terms in 17.52 are close enough together, we can approximate the (in-

finite) sum as an integral:

q



n 0

exp

8 m



V

h

2

2

/

n

3

2

kT

dn

3

(17.53)

The variable in the integral is n, the translational quantum number. This inte-

gral has the form and solution



x 0

eax

2

dx

1

2





a



1/2

(see Appendix 1), where in the case of equation 17.53 the expression for the

constant a is



8 mV

h

2

2

/3kT

We can therefore substitute a definite expression for the integral. We have

q





1

2





1/2

3

Rearranging all of the terms and distributing through the powers, we get

qtrans

2 

h

m

2

kT



3/2

V (17.54)

where the volume variable Vhas been algebraically removed from the paren-

theses. The “trans” subscript has been added to remind ourselves that we are

determining the partition function with respect to the translation of the atoms.

Again, in equation 17.54 the variable mrepresents the mass of the individual

gas particle.





8 mV

h

2

2

/3kT



8 m

h

V

2

2/3(nx

(^2) ny (^2) nz (^2) )

kT
606 CHAPTER 17 Statistical Thermodynamics: Introduction