The pressure of a gas in a container is produced by the incessant drumming
of the gas molecules upon the walls. At each such wall collision - say,
against the right wall of a cubic box - the respective momentum of the
molecule
is reversed. The momentum transferred
to the wall is thus
. The force acting on the
unit area of the wall is then just the time average of the momentum transfer:

(2.35)

where is the number of wall impacts within the time .

To obtain a theoretical prediction for the value of the pressure we argue
as follows:

The number of particles having -velocity and impinging on the
right wall per unit time is obviously proportional to . The
momentum transfer from such a particle to the wall is .
Thus we find

(2.36)

Inserting the Boltzmann density for and performing the
integrations we have

Where do we stand now? Just by statistical reasoning we have actually
arrived at a prediction for the pressure - a thermodynamic quantity!

Having thus traversed the gap to macroscopic physics we take a few steps
further. From thermodynamics we know that the pressure of a dilute gas
at density
and temperature is .
Comparing this to the above formula we find for the mean energy of a
molecule

(2.38)

and the parameter (which was introduced in connection with
the equilibrium density in velocity space) turns out to be just
the inverse of .

The further explication of the dilute gas thermodynamics is easy.
Taking the formula for the internal energy,
, and using the First Law

(2.39)

we immediately find that amount of energy that is needed to raise the
temperature of a gas by degree - a.k.a. the heat capacity
: