### Pressure is Confined Kinetic Energy

According the kinetic theory of ideal gases [180], 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 [180]. 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

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