# Kinetic Theory

The theory for ideal gases makes the following assumptions:

The gas consists of very small particles. This smallness of their size is such that the total volume of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is large compared to their size. These particles have the same mass.

The number of molecules is so large that statistical treatment can be applied. These molecules are in constant, random, and rapid motion.

The rapidly moving particles constantly collide among themselves and with the walls of the container. All these collisions are perfectly elastic. This means, the molecules are considered to be perfectly spherical in shape, and elastic in nature. Except during collisions, the interactions among molecules are negligible. (That is, they exert no forces on one another.) This implies:

1. Relativistic effects are negligible.

2. Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects. 3. Because of the above two, their dynamics can be treated classically. This means, the equations of motion of the molecules are time-reversible. The average kinetic energy of the gas particles depends only on the temperature of the system. The time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions. More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.[1] In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number. [edit]Properties

[edit]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:

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

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