Physical Chemistry , 1st ed.

Solving, we find that

vavg 1282 

m

s



or more than 1 km per second. This is a substantial velocity in terms of per-

sonal experience, but not for atoms in the gas phase. Notice that we have con-

verted all units to standard units so that they would cancel.

There is also a simple relationship between the pressure of a monatomic gas

and its kinetic energy, which can be considered solely as energy of translation.

(We are ignoring electronic and nuclear energies, as we did in our original dis-

cussion of partition functions of monatomic gases.) Since the classical expres-

sion for kinetic energy is

kinetic energy ^12 mv^2

we can suggest the following analogous equation for the averagekinetic energy

of the gas particles:

Eavg^12 mv^2 avg (19.9)

Substituting for mv^2 avgin equation 19.8, we find that

p

2 N

3

E

V

avg

Rewriting this by bringing the volume variable over to the other side, we have

pV^23 NEavg (19.10)

This is beginning to look like part of the ideal gas law. In fact, if we relate equa-

tion 19.10 to the ideal gas law, which says that pVnRT, we find that

NEavg^32 nRT

Finally, if we are considering 1 mole of an ideal gas,Nis Avogadro’s number

and NEavgis a molar energy,E. The variable nin the above equation is 1 mole,

so we have for an ideal gas

E^32 RT (19.11)

where we have defined NAEavgas E, the molar energy of the gas. In Chapter 17,

when we considered the translational energy of a monatomic gas, we found that

E^32 NkT

For 1 mole of gas,NNA. Equating the two equations for energy, we find that

RNAk (19.12)

which shows the relationship between Boltzmann’s constant and the ideal gas

law constant.

19.3 Definitions and Distributions

of Velocities of Gas Particles

In the previous section, we were able to relate the average velocity of a sample

of ideal gas to the pressure of that gas. So far, we have not defined what we

mean by “average velocity.”

Actually, there are several ways to define this quantity. First of all, the kinetic

theory of gases does not make any presumption about the velocity of any par-

656 CHAPTER 19 The Kinetic Theory of Gases