Modern Thermodynamics

- Chapter 1

37

P(v)dvxdvydvz =

1

z

e−mv

2

/ 2k

B

Tdvxdvydvz (1.6.10)

in which:

v2

= vx

2

+ vy

2

+

vz2

Here, z is the normalization factor defined by:

−∞

∞

∫

−∞

∞

∫

e−mv

2

/ 2k

B

Tdvxdvydvz

−∞

∞

∫

= z (1.6.11)

so that a requirement of the very definition of a probability,

−∞

∞

∫

−∞

∞

∫

P(v)dvxdvydvz

−∞

∞

∫

=1,

is met. The

normalization factor, z, as defined in (1.6.11) can be calculated using the definite integral :

e−ax2

−∞

∞

∫

dx =

π1/

a

2

which gives

1

z

=

m

2πk

B

T

3 / 2

(1.6.12)

(Some integrals that are often used in kinetic theory are listed at the end of this chapter in Appendix 1.2.)

With the normalization factor thus determined, the probability distribution for the velocity can be written

explicitly as:

P(v)dvxdvydvz =

m

2πkBT

3 / 2

e−mv

2

/ 2k BTdvxdvydvz

. (1.6.13)

This is the Maxwell velocity distribution. Plots of this function shows well-known Gaussian or "bell

shaped" curves shown in Fig.10(a). It must be noted that this velocity distribution is that of a gas at

thermodynamic equilibrium. The width of the distribution is proportional to the temperature. A gas not

in thermodynamic equilibrium has a different velocity distribution and the very notion of a temperature

may not be well defined, but such cases are very rare. In most situations, even if the temperature changes

with location, locally the velocity distribution is very well approximated by (1.6.13). Indeed, in computer