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Derivation of the kinetic theory formula

Remember that what follows applies to ideal gases only; the assumptions that we make certainly do not all apply to solids and liquids.

This proof was originally proposed by Maxwell in 1860. He considered a gas to be a collection of molecules and made the following assumptions about these molecules:

molecules behave as if they were hard, smooth, elastic spheres
molecules are in continuous random motion;
the average kinetic energy of the molecules is proportional to the absolute temperature of the gas;
the molecules do not exert any appreciable attraction on each other
the volume of the molecules is infinitesimal when compared with the volume of the gas
the time spent in collisions is small compared with the time between collisions.

Consider a volume of gas V enclosed by a cubical box of sides L. Let the box contain N molecules of gas each of mass m, and let the density of the gas be r. Let the velocities of the molecules be u1, u2, u3 . . . uN. (Figure 2)

Consider a molecule moving in the x-direction towards face A with velocity u1. On collision with face A the molecule will experience a change of momentum equal to 2mu1. (Figure 3)
It will then travel back across the box, collide with the opposite face and hit face A again after a time t, where t = 2L/u1.

The number of impacts per second on face A will therefore be 1/t = u1/2L.

Therefore rate of change of momentum = [mu12]/L = force on face A due to one molecule.

But the area of face A = L2, so pressure on face A = [mu12]/L3

But there are N molecules in the box and if they were all travelling along the x-direction then

total pressure on face A = [m/L3](u12 + u22 +...+ uN2)

But on average only one-third of the molecules will be travelling along the x-direction.

Therefore: pressure = 1/3 [m/L3](u12 + u22 +...+ uN2)

If we rewrite Nc2 = [u12 + u22 + + uN2 ] where c is the mean square velocity of the molecules:

pressure = 1/3 [m/L3]Nc2 But L3 is the volume of the gas and therefore:

and this is the kinetic theory equation.

Now the total mass of the gas M = mN, and since r = M/V we can write

The root mean square velocity or r.m.s. velocity is written as c r.m.s. and is given by the equation:
r.m.s. velocity = c r.m.s. = [c2]1/2 = [u12 + u22 + + uN2 ]1/2/N

We can use this equation to calculate the root mean square velocity of gas molecules at any given temperature and pressure.

Some further values of the root mean square velocity at s.t.p. for other gases are given below.
Hydrogen       18.39 x 102ms-1
Helium          13.10 x 102ms-1
Oxygen          4.61 x 102ms-1
Carbon dioxide    3.92 x 102ms-1
Bromine          2.06 X 102ms-1
© Keith Gibbs 2007