bei48482_FM

The number of states g(p) dpwith momenta whose magnitudes are between pand

pdpis proportional to the volume of a spherical shell in momentum space pin

radius and dpthick, which is 4p^2 dp. Hence

g(p) dpBp^2 dp (9.5)

where Bis some constant. [The function g(p) here is not the same as the function g()

in Eq. (9.4).]

Since each momentum magnitude pcorresponds to a single energy , the number

of energy states g() dbetween and dis the same as the number of momentum

states g(p) dpbetween pand pdp, and so

g() dBp^2 dp (9.6)

Because

p^2  2 m and dp

Eq. (9.6) becomes

g() d 2 m^3 ^2 Bd (9.7)

The number of molecules with energies between and dis therefore

n() dCekTd (9.8)

where C( 2 m^3 ^2 AB) is a constant to be evaluated.

To find Cwe make use of the normalization condition that the total number of

molecules is N, so that

Normalization N

0

n() dC 

0

ekTd (9.9)

Number of energy

states

m d



 2 m

Number of

momentum states

Statistical Mechanics 301

Figure 9.1The coordinates in momentum space are px, py, pz. The number of momentum states avail-

able to a particle with a momentum whose magnitude is between pand p dpis proportional to the

volume of a spherical shell in momentum space of radius pand thickness dp.

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