Molecular Pressure
The kinetic theory is a theory that a gas is made up of very small particles, like molecules. The theory simplifies things by assuming that the molecules do not interact with each other. This means they do not attract, repel or collide with each other. When these molecules hit the wall of a container, it gives pressure to the wall.
If the mass m or velocity v of the molecules are larger, the pressure should be larger. Let us see how to find a formula relating these quantities. We consider a cubic box with side L. Start with a single molecule parallel to one side of the box, and bouncing back and forth between two parallel walls.
When it bounces off one wall, it exerts a force for a very short time. The particle changes momentum from say +mv to -mv. So the change in momentum is mv - (-mv) = 2mv.
When the molecule goes to the opposite wall and bounces back, it travels a distance 2L. This takes a time of 2L/v. As it bounces back and forth, there is a momentum change at the wall of 2mv every 2L/v seconds.
According to Newton's second law, force = momentum change / time. So the molecule exerts an average force of 2mv / (2L/v) = mv2/L.
The pressure p = F/A, where F is the force and A the area. In this example, A would be L2. So the pressure due to the molecule is p = mv2/L / L2 = mv2/V, where the volume V is L3.
Suppose there are N molecules in the box, each with a different velocity. The sum of all the v2 would be N<v2>, where the symbol <v2> means the average over all v2. So the total pressure is p = Nm<v2>/V.
We have only considered motion parallel to one side of the cube. The molecules would really be moving randomly in all directions. Suppose a molecule has velocity c, with components u, v, w. They are related by a 3 dimension Pythagoras theorem: c2 = u2 + v2 + w2.
If we add up the c2> for all the molecules, we get <c2> = <u2> + <v2> + w<2>. <u2>, <v2> and <w2> are all equal, because there is no difference between any of the 3 directions.
S0 <c2> = 3<v2>. Substituting into p = Nm<v2>/3V, we get p = Nm<c2>/3V.
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 2 October 2010.