and 
,
were mixed. His conclusion was that, in equilibrium, the average
kinetic energy per molecule of both gases should be equal, that is,
one should have
The applet on this page was made in the summer of 2003 by Melinda Freeze thanks to an REU grant from the National Science Foundation, under my supervision.
In addition to deriving the equilibrium velocity distribution for
a gas consisting of identical molecules, Maxwell also considered what
should happen if two different gases, with molecular masses and ,
were mixed. His conclusion was that, in equilibrium, the average
kinetic energy per molecule of both gases should be equal, that is,
one should have
This is an extremely important result, for the following reason.
Consider two different gases, occupying equal volumes and at the same
pressure. Let 
be the number
of molecules of gas A and 
the number of
molecules of gas B. Then the kinetic model of pressure predicts
that
and
But if, in addition, the two gases are in thermal equilibrium
(that is, at the same temperature), then the product 
should have the same value for
both, and one concludes from the above two equations that 
; that is, the number of
molecules in equal volumes of any two gases, at the same pressure and
temperature, must be the same. This is known today as
Avogadro’s principle, which the Italian physicist Amedeo
Avogadro proposed around 1810 to explain, among other things,
Gay-Lussac’s results on combining gas volumes in chemical
reactions. Avogadro’s ideas were largely ignored for decades,
but they had just been resurrected and developed further by Stanislao
Cannizzaro, in 1858 (Avogadro had died in 1856), and more and more
chemists were recognizing their usefulness.
Since it was already known that the product PV, for a given
quantity of gas (that is to say, a given N) was proportional
to its temperature, Maxwell’s result made it natural to identify
the quantity 
with the
gas’ temperature (up to a proportionality constant); the ideal
gas law could then be written in the form PV=NkT, (k
would later come to be known as Boltzmann’s constant), and
Maxwell’s result for a mixture of gases in equilibrium could be
restated as saying that for any gas at temperature T,
This, in turn, can be recognized as a form of what would later be
known as the equipartition of energy theorem: in thermal
equilibrium, the average energy associated with every degree of
freedom (here, the three perpendicular directions of motion in
three-dimensional space) is equal to kT.
Feynman, in chapter 39 of Volume I of his Lectures in Physics,
going over Maxwell’s proof of this result (the equality of the
average kinetic energies for different gases in thermal equilibrium),
comments that there are some possible complications that the original
proof does not address in a fully satisfactory way, and also
acknowledges that he has been unable to find a simple (meaning,
presumably, simple enough) proof of it. You may, therefore, find it
satisfactory to see that the result indeed holds, from a direct
numerical calculation of the way two gases of rigid spheres interact
(in two dimensions, in this case), as provided by the applet on this
page.
In the applet you can set the ratio of the mass of the red particles
to that of the blue particles. The initial condition gives all the
particles the same speed (but random directions), so the more massive
particles will start out with more kinetic energy. The average
kinetic energy per molecule is calculated and plotted as a vertical
bar in the small box on the right. Assuming that Maxwell was right,
and that, in equilibrium, the total energy of the gas is shared
equally by all the molecules, regardless of their mass, it is easy to
predict where the bars should be at equlibrium, and this is indicated
by the horizontal black line. At equilibrium, you may think of the
bars as measuring the gas’ temperature, and you can see the
small fluctuations around the predicted equlibrium value. You can
also see that these fluctuations are proportionately smaller for the
gas that has the largest number of molecules; this makes sense,
physically, since, if you have a small number of molecules, a chance
collision giving a molecule a lot of energy can have a large effect
on the average.
At equilibrium, on the average, the heavier molecules should be
moving more slowly, and you can also check this, qualitatively, if
you watch the animation carefully (you may need to slow it down).
The applet can also be started with all the red molecules in the
upper left-hand corner (choose the “show diffusion”
option), to allow you to see how diffusion of a gas through another
one works. You may want to play with the numbers of molecules and see
how many collisions it takes until a good mixing has taken place, or
until some red molecules reach the farthest corner (remember how this
phenomenon is what prompted Clausius to introduce the “mean free
path” concept).
One last note about Avogadro’s principle. A problem with the
ideal gas law in the form PV = NkT is that there is no direct
way to measure or estimate N, the actual number of molecules.
This could be done (from measurements of the gas density) if one had
some idea of what the mass of a molecule is. Such estimates were to
follow shortly after Maxwell’s work, in fact, based on his
theoretical predictions for the ideal gas’ viscosity.
(Forward to "Viscosity of a gas")
(Back to "the Maxwell velocity
distribution")
(Back to "Irreversibility")