Consider a particle of mass *m* moving freely in a rectangular box, and undergoing perfectly elastic collisions with the walls of the box. Let us consider the motion in the *x* direction. Let the *x*-component of the particle’s velocity be *v*_{x}, and the length of the box in the *x* direction be *L*_{x}. The time it takes to travel from one of the *x*-bounding walls to the other is , and so the number of collisions with a particular *x*-bounding wall per unit time is . When it collides with the wall, the *x*-component of the velocity changes from *v*_{x} to –*v*_{x}, and so, due to conservation of momentum, it imparts a momentum of 2m|*v*_{x}|. The product of the momentum imparted per collision and the rate of collisions per unit time gives the (time-)average force exerted on the wall by the particle:

.

Now, let us replace the particle with *N* identical non-interacting particles (an ideal gas). Then the force on one of the *x*-bounding walls is

, where the bar represents the average over the *N* particles. Dividing this by the area *A* of the wall gives us pressure:

where is the volume of the box.

If our gas is isometric, then , where is the root-mean-square speed. This means that the pressure is the same on all sides of the box, and

.

Now, the Maxwell speed distribution tells us that for ideal gas molecule, the average kinetic energy is given by . This allows us to rewrite the above equation for pressure as

, or multiplying both sides by the volume,

.

This is the ideal gas law. Often, one is introduced to the law in terms of the number of moles, *n*, instead of the number of molecules *N*. As , where *N*_{A} is Avogadro’s number, this gives us the form more commonly seen in introductory courses:

,

where is the ideal gas constant.

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Tags: Boltzmann Constant, Friday Physics, Ideal Gas, Ideal Gas Constant, Ideal Gas Law, Maxwell Speed Distribution, physics, Statistical Mechanics

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