Physical Chemistry , 1st ed.

Using the form ofgxin equation 19.24, this means that





vx

Ae(1/2)Kvx

2

dvx 1

We can use the table of integrals in Appendix 1 to show that

A 



2

K



1/2

(19.25)

A complete understanding ofgx—and by extension, the entire three-dimensional
probability function—depends on determining the constant K.
To determine K, we first use the idea that the velocity in each dimension is
equivalent, that is, the average squared velocity in the xdimension is equal to
the average squared velocity in the ydimension, which is equal to the average
squared velocity in the zdimension:

v^2 avg,xv^2 avg,yv^2 avg,z

(We used this idea in the derivation of equation 19.8.) Since the average ki-
netic energy of a single gas particle is

Eavg^12 mv^2 avg

we can use the definition of the average squared velocity to get

Eavg^12 m(v^2 avg,xv^2 avg,yv^2 avg,z)

or, by using the equivalence of the velocity components:

Eavg^32 mv^2 avg,x

where we are arbitrarily using the xcomponent of the average squared veloc-
ity. Comparing this with equation 19.11:

Eavg

3

2

mv^2 avg,x

N

1



3

2

RT

IfNwere Avogadro’s number of particles, the above equation would give us

^32 mv^2 avg,x^32 kT

(where we have taken advantage of the relationship between Rand k).
Rearranging, we get

v^2 avg,x

k

m

T

 (19.26)

which we can use to determine the constant K.
In finding K, we use an idea that we developed in statistical thermodynamics
about how to calculate an average value for a variable. Recall the definition of
an average value as defined in equation 17.3:

possible

u (19.27)

where ujis the particular value of the variable u,Pjis the probability that this
particular value shows up in a group of values, and urepresents the average
value of the variable u. (See the problem worked out near the end of section



values

j 1

ujPj





j

Pj

19.3 Definitions and Distributions of Velocities of Gas Particles 661