**James Richard Fromm**

Comparison of the Ideal Gas Law, *pV* = *nRT*, with the kinetic-molecular
theory expression *pV* = 2*E*_{k}/3 derived in the previous section
shows that **the total kinetic energy of a collection of gas molecules is directly
proportional to the absolute temperature of the gas**. Equating the *pV* term
of both equations gives

*E*_{k} = 3*nRT*/2,

which rearranges to an explicit expression for temperature,

*T* = (2/3*R*)(*E*_{k}/*n*) = *Mv*^{2}/3*R*

We see that temperature is a function only of the mean kinetic energy *E*_{k},
the mean molecular velocity *v*, and the mean molar mass *M*.

As the absolute temperature decreases, the kinetic energy must decrease and thus the
mean velocity of the molecules must decrease also. At *T* = 0, the absolute zero of
temperature, all motion of gas molecules would cease and the pressure would then also be
zero. No molecules would be moving. Experimentally, the absolute zero of temperature has
never been attained, although modern experiments have extended to temperatures as low as
0.01 K.

It has been necessary to use the average velocity of the molecules of a gas because the
actual velocities are distributed over a very wide range. This distribution, the **Maxwell
law** or Maxwell's law of distribution of velocities, was developed by J.
Clerk-Maxwell (1860). The distribution is given by the equation:

*N*(*v*)/*N* = 4(pi)(*m*/2(pi)*kT*)^{3}/2*v*^{2}
exp(-*mv*^{2}/2*kT*)

where *N*(*v*) is the number of molecules moving with velocity *v*, *N*
is the total number of molecules, *m* is the mass of a molecule, *k* is the **Boltzmann
constant**, and *T* is the absolute temperature. Maxwell's law has been
experimentally demonstrated.

Maxwell's law is itself one particular example of the Maxwell-Boltzmann distribution which describes the distribution of any form of energy among interacting molecules as a function of temperature. The Maxwell-Boltzmann distribution, like Maxwell's law, is developed on the basis of probability arguments which are beyond our scope. Although these arguments preceded the development of quantum mechanics, the quantum-mechanical distributions of either Bose-Einstein statistics (even number of particles) or Fermi-Dirac statistics (odd number of particles) are essentially identical to the Maxwell-Boltzmann distribution for species of chemical interest at all but extremely low temperatures.

At any temperature, the distribution given by the Maxwell law has a single maximum due
to the interaction of two factors: the pre-exponential term 4(pi)(*m*/2*kT*)^{3}/2*v*^{2}
and the exponential term exp(-*mv*^{2}/2*kT*). The pre-exponential term
is directly proportional to the square of the velocity and so increases toward infinity,
while the exponential term declines from one toward zero, with increasing velocity. The
resulting Maxwell-Boltzmann distribution is one in which the height of the maximum, and
the velocity at which the maximum proportion of molecules are found, changes with
temperature alone.

It is not necessary to use the Maxwell-Boltzmann distribution of velocities to explain
either the nature of temperature or the Law of Charles,
although it is the correct expression of the distribution. The law of Charles will be
obtained for **any** distribution in which the velocities of the gas
molecules are a function of only the nature of the gas and the absolute temperature.

Copyright 1997 James R. Fromm