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 u
1,
u
2, u
3 . . . u
N. (Figure 2)
Consider a molecule moving in the x-direction towards face A with
velocity u
1. On collision with face A the molecule will experience a change of momentum
equal to 2mu
1. (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/u
1.
The
number of impacts per second on face A will therefore be 1/t =
u
1/2L.
Therefore rate of change of momentum = [mu
12]/L =
force on face A due to one molecule.
But the area of face A = L
2, so
pressure on face A = [mu
12]/L
3But there are N molecules in the box
and if they were all travelling along the x-direction then
total pressure on face A =
[m/L
3](u
12 + u
22 +...+ u
N2)
But on
average only one-third of the molecules will be travelling along the x-direction.
Therefore: pressure = 1/3 [m/L
3](u
12 +
u
22 +...+ u
N2)
If we rewrite Nc
2 =
[u
12 + u
22 + …+ u
N2 ] where c is the mean square velocity of
the molecules:
pressure = 1/3 [m/L
3]Nc
2 But
L
3 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.
= [c
2]
1/2 = [u
12 + u
22 + …+ u
N2
]
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 10
2ms
-1
Helium
13.10 x 10
2ms
-1
Oxygen
4.61 x 10
2ms
-1
Carbon dioxide
3.92 x 10
2ms
-1
Bromine
2.06 X 10
2ms
-1
