Derivation of pV = 1/3Nmc2

The Kinetic Theory of Gases and the Ideal Gas Equation

ASSUME:
Ideal gases are composed of:
    - Numerous
    - elastic molecules
    - of Negligible Size compared to Bulk Container
    - whose Thermal Motion is 'Random'

Consider a rectangular box length l, area of ends A, with a single molecule travelling  left and then right the length of the box because of a collision with the end wall.

The time between collisions with the left wall is the distance of travel between wall A and the other wall divided by the speed of the particle in the x-direction, u.

                       t   =    2 L       - (equation 1) 
     u

According to Newton, force is the rate of change of the momentum

 F   =    D (m u)
           Dt

The momentum change upon collision is the momentum after the collision minus the momentum before the collision To hit the left wall the initial velocity must have been -u, so:

change in momentum,  D (m u) = m u - (-m u)  = 2m u

The average force on the left end wall is the rate of change of momentum

F    =     2m u
             Dt

Combine equation (1) into the above equation and we get: 

  F    =     2m u
               (2 L /u)

The 2's cancel and the formula reduces to:

F     =    m  u2
             L

The pressure, p,  exerted by this single molecule constrained to move in one horizontal direction (one dimension) is the average force per unit area

            p  =   F   =    m  u2   =   m  u2 
                A         L . A          V
      where V = A . L is the volume of the rectangular box.

Now consider N gas molecules in the box   

p = Nm u2
   V

But they could be moving moving with velocities in ALL directions - not just horizontally. They could be moving in the:

  • x direction (ux)
  • y direction (uy)
  • z direction (uz)

Using the rule for adding vectors at right angles to each other - we have to use Pythagoras to add the three velocities.  (Square all the velocities and add them)

"mean square speed" of the gas molecules:  c2  =  ux2 +   uy2 +   uz2

but on average only a third of all molecules will be moving in any given direction, 
so  ux2 =   uy2 =  uz2

and so   c2 = 3 ux2          OR      ux2  = 1/3 c2

If the molecules are free to move in three dimensions, they will hit walls in one of the three dimensions one third as often. The pressure then of a gas sample of N molecules in 3-D is

  p        =       1      Nmc2
                 3        V

pV      =         1  Nmc2
                  3       

where p = gas pressure
           V = gas volume
           N = number of molecules
           m = mass of each molecule
           c2 = mean square speed of the gas molecules

You need to learn this derivation by heart

Good luck to you !