Pressure is Confined Kinetic Energy

According the *kinetic theory of ideal gases*
[181], *air pressure* can be defined
as the *average momentum transfer per unit area per unit time*
due to molecular collisions between a confined gas and its boundary.
Using Newton's second law, this pressure can be shown to be given by
one third of the average kinetic energy of molecules in the gas.

Here, denotes the average squared particle velocity in the gas. (The constant comes from the fact that we are interested only in the kinetic energy directed along one dimension in 3D space.)

*Proof: *This is a classical result from the kinetic theory of gases
[181]. Let
be the total mass of a gas
confined to a rectangular volume
, where
is the area of
one side and
the distance to the opposite side. Let
denote the average molecule velocity in the
direction. Then the
total net molecular momentum in the
direction is given by
. Suppose the momentum
is directed
against a face of area
. A rigid-wall elastic collision by a mass
traveling into the wall at velocity
imparts a momentum of
magnitude
to the wall (because the momentum of the mass is
changed from
to
, and momentum is conserved).
The average momentum-transfer per unit area is therefore
at any instant in time. To obtain the definition of pressure, we need
only multiply by the average collision rate, which is given by
. That is, the average
-velocity divided by the
round-trip distance along the
dimension gives the collision rate
at either wall bounding the
dimension. Thus, we obtain

where is the density of the gas in mass per unit volume. The quantity is the average kinetic energy density of molecules in the gas along the dimension. The total kinetic energy density is , where is the average molecular velocity magnitude of the gas. Since the gas pressure must be the same in all directions, by symmetry, we must have , so that

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University