bei48482_FM

Example 9.3

Verify that the rms speed of an ideal-gas molecule is about 9 percent greater than its average speed.

Solution

Equation (9.14) gives the number of molecules with speeds between and din a sample

of Nmolecules. To find their average speed , we multiply n()dby , integrate over all values

of from 0 to , and then divide by N. (See the discussion of expectation values in Sec. 5.5.)

This procedure gives

 

n() d 4 

3  2



^3 em^2 ^2 kTd

If we let a m 2 kT, we see that the integral is the standard one



0

x^3 eax

2

dx

and so 
4 

3  2



2



Comparing with rmsfrom Eq. (9.15) shows that

rms   1.09

Because the speed distribution of Eq. (9.14) is not symmetrical, the most probable

speed pis smaller than either or rms. To find p, we set equal to zero the derivative

of n() with respect to and solve the resulting equation for . The result is

p (9.16)

^2 kT

m

Most probable

speed

3 



8

3 kT



m

8 kT



m

2 kT



m

1



2

m



2 kT

1



2 a^2

m



2 kT

1



N

304 Chapter Nine

Figure 9.3Maxwell-Boltzmann speed distribution.

v^2 = root-mean-square speed = 3kT/m

vp = most probable speed = 2kT/m

v = average speed = 8kT/πm

v

n(

v)

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