bei48482_FM

From a table of definite integrals we find that



0

xeaxdx 

Here a 1 kT, and the result is

N (kT)^3 ^2

C (9.10)

and, finally,

n() d ekTd (9.11)

This formula gives the number of molecules with energies between and din a

sample of an ideal gas that contains Nmolecules and whose absolute temperature isT.

Equation (9.11) is plotted in Fig. 9.2 in terms of kT. The curve is not symmetrical

about the most probable energy because the lower limit to is 0 while there is,

in principle, no upper limit (although the likelihood of energies many times greater

than kTis small).

Average Molecular Energy

To find the average energy per molecule we begin by calculating the total internal

energy of the system. To do this we multiply n()dby the energy and then integrate

over all energies from 0 to :

E

0

n() d 

0

^3 ^2 ekTd

Making use of the definite integral



0

x^3 ^2 eaxdx 

we have

E (kT)^2 kT  NkT (9.12)

The average energy of an ideal-gas molecule is EN, or

 kT (9.13)

which is independent of the molecule’s mass: a light molecule has a greater average

speed at a given temperature than a heavy one. The value of at room temperature is

about 0.04 eV,  215 eV.

3



2

Average molecular

energy

3



2

3



4

2 N



(kT)^3 ^2

Total energy of N

gas molecules





a

3



4 a^2

2 N



(kT)^3 ^2

2 N



(kT)^3 ^2

Molecular energy

distribution

2 N



(kT)^3 ^2

C



2





a

1



2 a

302 Chapter Nine

0 kT 2 kT 3 kT

n(

e)

e

Figure 9.2 Maxwell-Boltzmann

energy distribution for the mole-

cules of an ideal gas. The average

molecular energy is   23 kT.

bei48482_Ch09.qxd 1/22/02 8:45 PM Page 302