Physical Chemistry , 1st ed.

Canceling all the units but kelvins and solving for T:

T264 K

The appropriate units for each variable should be used so that everything

cancels properly, leaving kelvins as the only remaining unit.

A root-mean-square speed for gas particles is easy to define but should not

obscure a key point: gas particles do not all move at the same speed. Also, as

implied at the beginning of this section, not all possible speeds are equally

probable. Rather, there is a particular distribution of different gas speeds in any

sample. What is the mathematical expression that gives us the distribution of

gas speeds?

We start by pointing out that if each dimension is independent, then we can

consider distribution functions for each dimension separately. In the x,y, and

zdimensions, we can define distribution functions (which are also probability

functions) gx(vx),gy(vy), and gz(vz), in which each function focuses on only one

of three dimensions. In terms of these functions, the probability of any gas par-

ticle having a particular three-dimensional velocity is the product of the one-

dimensional probabilities. That is,

probability gx(vx) dvxgy(vy) dvygz(vz)dvz (19.14)

If these probability functions are in fact functions of velocities, then the pos-

sible range is →, since we must consider direction as well as magni-

tude. We will require that each individual probability function sum up to 100%

over the entire range of the variable. For the xdimension, this is written math-

ematically as





vx

gx(vx) dvx 1 (19.15)

and similar expressions can be written for gyand gz.

A common way to determine the form of the functions represented by

equation 19.15 is to understand that if the three dimensions can be considered

equivalent, then the overall three-dimensional probabilityis a function of the

overall three-dimensional velocity. If we use the symbol (v) to indicate the

three-dimensional probability function, then this statement implies that

gx(vx) gy(vy) gz(vz)     (v) (19.16)

In the left side of equation 19.16, the three one-dimensional probability func-

tions are multiplied together to get the overall three-dimensional probability,

but the right side implies that this product must be some function of the three-

dimensional velocity v. Furthermore, we know a relationship between the over-

all velocity and its components:

v^2 vx^2 vy^2 vz^2 (19.17)

We focus on a single dimension, the xdimension, and determine the func-

tion gx(vx). If we were to take the derivative of equation 19.16 with respect to

vx, we would still have an equality. The problem is that the left side and the

right side of equation 19.16 are written in terms of different variables. However,

calculus has something called the chain rule, and equation 19.17 gives us a re-

lationship between vand vx, the two variables of interest here. Taking the de-

rivative of each side of equation 19.16 with respect to vx:

658 CHAPTER 19 The Kinetic Theory of Gases