Physical Chemistry , 1st ed.

Now that the probability distribution function has been found, different

“average” speeds can be defined. For example, each plot in Figure 19.4 peaks

at some maximum, implying that there is some speed that has the highest pop-

ulation among the gas particles at any particular temperature. We can find an

expression for this most probable speedby taking the derivative of equation

19.33 with respect to v, setting the derivative to zero (because the slope at a

maximum is equal to zero), and solving for the velocity at this maximum

point. We get

vmost prob 

2

m

kT



1/2

(19.34)

where mis the mass of a single gas particle. In molar quantities, this equation is

vmost prob 

2

M

RT



1/2



2

M

RT

 (19.35)

where Mis the molar mass of the gas particles.

Example 19.3

What is the most probable speed of He atoms if the gas temperature is

264 K? (Notice that this is the temperature calculated from Example 19.2.)

Solution

Using the molar mass of He (in kg units) and equation 19.35:

vmost prob

1/2

After decomposing the J unit into its base units, we have

vmost prob 1,097,448 

m

s^2

2



1/2

vmost prob1048 m/s 1.048  103 m/s

The most probable speed is always a little lower than the root-mean-square

speed, as a comparison between equations 19.13 and 19.35 shows. Both defin-

itions of average speeds have only the mass of the particle and the absolute

temperature of the gas as variables. The other terms are constants.

Finally, now that we have a distribution function G(v) for the velocities, we

can determine another average speed. Again, we will use equation 19.28, from

statistics, to determine another average value for the speed. Using the idea that

u 

max

min

ujPjdu

we will use G(v) as our probability function Pjand v, the velocity, for the vari-

able uj. The average(or mean) speed, v, is found by solving the expression

v 



v 0

v 4    

2

m

kT



3/2

v^2 emv

(^2) /2kT
dv
2 8.314 
mo

J

lK

 (264 K)



0.00400 

m

kg

ol



664 CHAPTER 19 The Kinetic Theory of Gases