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Wednesday, 6 July 2011



Pressure and Kinetic Energy

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L3. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:
\Delta p = p_{i,x} - p_{f,x} = 2 m v_x\,
where vx is the x-component of the initial velocity of the particle.
The particle impacts one specific side wall once every
\Delta t = \frac{2L}{v_x}
(where L is the distance between opposite walls).
The force due to this particle is:
F = \frac{\Delta p}{\Delta t} = \frac{m v_x^2}{L}.
The total force on the wall is
F = \frac{Nm \overline{v_x^2}}{L}
where the bar denotes an average over the N particles. Since the assumption of molecular chaos imposes  \overline{v_x^2} = \overline{v^2}/3 , we can rewrite the force as
F = \frac{Nm\overline{v^2}}{3L}.
This force is exerted on an area L2. Therefore the pressure of the gas is
P = \frac{F}{L^2} = \frac{Nm\overline{v^2}}{3V}
where V=L3 is the volume of the box. The fraction n=N/V is the number density of the gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic level). Using n, we can rewrite the pressure as
 P =  \frac{n m \overline{v^2}}{3}.
This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule {1 \over 2} m\overline{v^2} which is a microscopic property.

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