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 =
π1/
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
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