## Physics Friday 69

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 vx, and the length of the box in the x direction be Lx. 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 vx to –vx, and so, due to conservation of momentum, it imparts a momentum of 2m|vx|. 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 NA is Avogadro’s number, this gives us the form more commonly seen in introductory courses:
,
where  is the ideal gas constant.

### 4 Responses to “Physics Friday 69”

1. Physics Friday 71: The Isothermal Atmosphere « Twisted One 151’s Weblog Says:

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2. Physics Friday 72 « Twisted One 151’s Weblog Says:

[…] Friday 72 By twistedone151 When we previously derived the ideal gas law PV=NkT=nRT from a particle bouncing in a box, we used the Maxwell speed […]

3. Physics Friday 73 « Twisted One 151’s Weblog Says:

[…] freedom for each molecule (f≥3), we have equation of state and internal energy , as discussed here and here. Now, we can use this to find the heat capacity for a given quantity of an ideal gas. The […]

4. Physics Friday 88 « Twisted One 151’s Weblog Says:

[…] for a classical ideal gas whose molecules have only translational degrees of freedom available (see here and here). Next week, I intend to show that this reduces to the classical ideal gas for high […]