Kinetic theory

The kinetic theory of gases is a theory that explains the macroscopic properties of gases by consideration of their composition at a molecular level.

Pressure

Pressure is explained by the kinetic theory as arising from the force exerted by collisions of gas molecules with the walls of the container. The derivation of the mathematical expression for pressure is given below:

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Consider a gas with N molecules, each of mass m, enclosed in a cuboidal container of volume V.

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Suppose that a gas molecule collides with a wall of the container which is perpendicular to the x co-ordinate 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 given by

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:2mv_x

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where vx is the x-component of the initial velocity of the particle.

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Now, force is the rate of change of momentum. The particle under consideration impacts with the wall once every 2l/vx time units, where l is the length of the container. Therefore the force due to this particle is

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:mv_x cdot v_x over l

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and the total force on the wall is

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:msum_j v_{jx}^2 over l

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where the summation is over all the gas molecules in the container.

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Since the particles are moving randomly in all directions, and since

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: v^2 = v_x^2 + v_y^2 + v_z^2

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for each particle, the expression for the total force becomes

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:msum_j v_j^2 over 3l

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This can be written as

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:Nmv_{rms}^2 over 3l

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where vrms is the root mean square velocity of the gas.

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Therefore, pressure, the force per unit area, equals

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:Nmv_{rms}^2 over 3Al

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where A is the area of the wall.

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Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure

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:P = {Nmv_{rms}^2 over 3V}

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where V is the volume. Also, as Nm is the total mass of the gas, and mass divided by volume is density

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: P = {1 over 3} ho v_{rms}^2

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where ρ is the density of the gas.

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This result is interesting and significant because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule (1/2mvrms2), which is a microscopic property.

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Note that the product of pressure and volume is simply two-third of the total kinetic energy.

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