Physical Chemistry , 1st ed.

where in the second equality we have made a substitution using pmv.

Comparing equations 17.64 and 17.65, we find that the classical momentum p

is equivalent to (2mkT)1/2. Actually, this equivalency is only suggestive, not

exact, since the expression for is ultimately derived from the most probable

distribution and is related more to the average momentum,p , of the gas par-

ticles. A similar relationship can be determined from the expressions for the

energies of the gas particles. For Nparticles, equation 17.56 says that

E^32 NkT

Classically, the energy of motion for Nparticles is

EN

2

p

m

2



Equating the two expressions and solving for p, we find that pis equivalent to

(3mkT)1/2, which is almost the same expression. Again, this is only suggestive,

not exact, but such relationships are expected if two widely different perspec-

tives on thermodynamics predict similar values for measurable quantities.

Example 17.7

At 298 K, the most probable velocity of an Ar atom is 352.4 m/s. Calculate ,

the thermal de Broglie wavelength, and the most probable value of,the

(normal) de Broglie wavelength of an Ar atom. Ar has a molar mass of 39.9 g.

Solution

The most probable value for the de Broglie wavelength is inversely propor-

tional to its momentum. We will have to express the mass of a single Ar atom,

in kg units:



m

h

v



2.8369  10 ^11 m

Now let us calculate the thermal de Broglie wavelength:

6.626  10 ^34 Js



39.9 m

g

ol



1

1

00

k

0

g

g



6.02

1



mo

1

l

023

 352.4 

m

s



612 CHAPTER 17 Statistical Thermodynamics: Introduction



2 

h

m

2

kT



1/2





(6.626  10 ^34 Js)^2 1/2



2 3.1415926 39.9 

m

g

ol



1

1

00

k

0

g

g



6.0

1

2 

mo

1

l

023

  1.381  10 ^23 

K

J298 K

Note that we have again had to determine the mass of a single Ar atom, in kg

units. Solving:

1.6005  10 ^11 m

The two answers are not that far off from each other (in fact, they differ by

a factor of1/2).

The thermal de Broglie wavelength actually has some utility in a statistical

approach to the behavior of matter. In order for the equations of the Boltzmann

distribution to apply to a system, it is necessary that the thermal de Broglie