Physical Chemistry , 1st ed.

Again, this integral can be evaluated using the integral table in Appendix 1. We
find that

v 

8

k

m

T



1/2

 

8

R

M

T



1/2



8

R

M

T

 (19.36)

where again mis the mass of a single gas particle and Mis the molar mass of
the gas sample. This definition of an average speed also varies only with the
mass of the particle and the absolute temperature of the gas.

Example 19.4

Consider a sample of Ar gas. Determine the temperature of the gas if the fol-

lowing velocities were equal to 500.0 m/s:

a.vrms

b.vmost prob

c.v

d.Do the relative temperatures meet expectations?

Solution

We will need to solve for temperature using each of the expressions for the

average velocity.

a.For vrms, we have

500.0 

m

s





T400.4 K

b.For vmost prob, we have

500.0 

m

s





T600.6 K

c.For v, we have

500.0 

m

s





T471.7 K

d.These results show that the most probable speed requires the highest tem-

perature, and the average and root-mean-square speeds require lower tem-

peratures to have the same value for the “average.”

Figure 19.5 shows a plot of the probability distribution function of Ar gas
at the same temperature, 500 K. On the plot, the vrms,vmost prob, and vvalues
are marked. The relative ordering of average speeds is similar for all gases, and
illustrates the slightly different definitions for each of these quantities. What

8    8.314 

mo

J

lK

 T



0.03995 

m

kg

ol



2    8.314 

mo

J

lK

 T



0.03995 

m

kg

ol



3    8.314 

mo

J

lK

 T



0.03995 

m

kg

ol



19.3 Definitions and Distributions of Velocities of Gas Particles 665