Physical Chemistry , 1st ed.

17.2 for an example of how to use this formula.) For a well-behaved probabil-

ity function, the summation of all probabilities,

j

Pjin the denominator,

equals 1, so equation 19.27 becomes

possible

u

values

j 1

ujPj

Finally, for a function that can have many possible values which are ex-

pressed by a smoothly varying probability function, the above summation can

be replaced with an integral, so we have

u 

max

min

ujPjdu (19.28)

Equation 19.28 was derived using the same conditions describing the distrib-

ution of gas velocities. Therefore, we can use equation 19.28 to set up an inte-

gral for the average squared velocity in the xdimension. The variable is vx^2 , and

the probability function Pjis given by equation 19.24 with the subsequent de-

termination of the pre-exponential constant A. Substituting into equation

19.28, we have for the average squared velocity:

v^2 avg,x 





vx^2  



2

K



1/2

e(1/2)Kvx

2

dvx (19.29)

Since equation 19.29 is an even function of the variable vx, we can divide the

range in half, from 0 to rather than to , and multiply the value

of that integral by 2. Therefore,

v^2 avg,x 2 



2

K



1/2





0

vx^2 e(1/2)Kvx

2

dvx (19.30)

where all constants have been removed to outside the integral. The integral in

equation 19.30 has a known form; from Appendix 1, we use  0 x^2 ebx

(^2) /2
dx
(where in equation 19.30,xvx) equals 1/2/[21/2(K)3/2]. Substituting into
equation 19.30:
v^2 avg,x 2 



2

K



1/2



2 1/2(



1/

K

2

)3/2

 which must equal 

k

m

T



where the last part is taken from equation 19.26. Most of the terms on the K

side cancel. Solving:





1

K



k

m

T



K

k

m

T

 (19.31)

We therefore have for the distribution function gx:

gx 

2

m

kT



1/2

emvx

(^2) /2kT
(19.32)
Similar one-dimensional distribution functions are easily written for gyand gz.
Equation 19.32, and the two parallel functions for the yand zdimensions,
do not directly give the three-dimensional probability function. We have de-
fined the product of the three unidimensional probabilities as :
(v) gx(vx) gy(vy) gz(vz)
662 CHAPTER 19 The Kinetic Theory of Gases