Physical Chemistry , 1st ed.

Equation 17.54 is a useful conclusion. The (translational) partition func-
tion, originally defined as an infinite sum of negative exponentials of the en-
ergy levels, is equal to an expression in terms of the mass of the gas particles,
the absolute temperature, the system volume, and some fundamental univer-
sal constants. This expression lets us calculate explicit values for q, which can
then be used to determine values for energy, entropy, heat capacity, and so on.
These calculated values—determined from a statistical rather than a phenom-
enological perspective—can then be compared to experimental values. We will
thus get the first chance to see how well a statistical approach to thermo-
dynamics compares with experiment.
Remember that qis unitless, since it is simply the sum of exponential func-
tions, so all of the units in equation 17.54 will ultimately cancel. However, we
will have to convert some units, particularly units of volume. Typically, vol-
umes are expressed in units of liters. In order for the units to cancel properly,
it is easiest if volume is expressed in units of cubic meters, m^3. Recall that the
liter can be defined as a cube having sides 1.00 decimeter (1.00 dm) in length.
Since a decimeter is 0.1 meter, we will take advantage of the conversion factor

1 L 0.001 m^3 (17.55)

Units will also work out if we express (molar) mass quantities in units of kg, not g.

Example 17.4

Calculate qtransfor 1 mole of He at standard thermodynamic conditions

(T298 K,V24.5 L). The molar mass of He is 4.0026 g.

Solution

Keeping equation 17.55 in mind, the volume of 1 mole of He in cubic meter

units is 0.0245 m^3. Also, we should express the mass of one atom of He in kg

units: (0.0040026 kg)/(6.02  1023 ) 6.65  10 ^27 kg. For qtrans,we get

q

3/2

0.0245 m^3

First we will work out the numbers. When we combine all of the numerical

values (not forgetting, of course, the 3/2 power on the brackets), we get

number 1.90  1029

Now we will examine the units. They are



3/2

m^3

Consider the units inside the parentheses first. The units of K cancel in the

numerator, and one of the J units cancels in the numerator and the denom-

inator. We have remaining



J

k

s

g

 2

Remember, however, that the unit joule is a compound unit and equal to

(kgm^2 )/s^2. Substituting:



kg

k



g

m^2



m

1

 2

kg





kg

s



2

m^2

s^2

kg 

K

J

K



(Js)^2

2 3.14159(6.65  10 ^27 kg)(1.381  10 ^23 J/K)298 K



(6.626  10 ^34 Js)^2

17.6 The Partition Function: Monatomic Gases 607