Review of the kinetic theory of gases

The kinetic theory of gases describes the connection between velocity of individual molecules and pressure.

The momentum of a particle is mu_x (in the x-direction).

For an elastic collision with the wall of the container the momentum of the particle is reversed and is now –mu_x. The change in momentum is D(mu_x) = mu_x – (-mu_x) = 2mu_x. If the distance between the two walls perpendicular to the x-direction is a, then the time elapsed between collisions is Dt = 2a/u_x because the molecules travels a distance of 2a to arrive back at the right-hand wall. The rate of change of momentum is equal to the force so

Since the area of wall is bc the pressure exerted on the wall is

since V = abc.

One molecule cannot really exert a pressure, but the extension of the above reasoning to a container with N molecules shows that

where

The choice of the x-direction is arbitrary and the results for the y- or z-direction must be equivalent

Since the mean square speed of any molecule is

we have

for the mean square speed in one direction.

The fundamental equation of the kinetic theory of gases is

This equation relates a macroscopic property, PV, on the left hand side with a microscopic (or molecular) property m<u²> on the left-hand side. The average kinetic energy is related to the temperature by

And therefore the equation also is equivalent to

the ideal gas law.

If we consider Avagadro's number of molecules then the molar mass M = N_Am. For one mole of gas molecules we have

The square root or the mean-square speed for an ideal gas is

This is the same as the root-mean-square speed u_rms = (áu²ñ - áuñ²)^1/2.

The distribution of molecular speeds is given by a Gaussian distribution

All molecules in a system do not have the same speed. This is important for simulations since we need to assign velocities to molecules to initiate the dynamics. The velocities are assigned according to a Maxwell-Boltzmann distribution, which is a Gaussian distribution of speeds. The derivation is due to Maxwell, and was formalized by Boltzmann.

Let h(u_x, u_y, u_z)du_xdu_ydu_z be the fraction of molecules that have velocity components between u_x and u_x + du_x etc. The key assumption is that the three components of velocity are statistically independent. Thus, we can write

h(u_x, u_y, u_z) = f(u_x) f(u_y) f(u_z)

where f(u_x), f(u_y), and f(u_z) are the probability distributions of the individual velocity components. Because the gas is isotropic, the function h(u_x, u_y, u_z) must depend only upon the magnitude of the velocity u.

h(u) = h(u_x, u_y, u_z) = f(u_x) f(u_y) f(u_z)

Taking the logarithm gives

ln h(u) = ln f(u_x) + ln f(u_y) + ln f(u_z)

Differentiating this equation with respect to ux gives

(¶ln h(u)/¶u_x)_uy,uz = ¶ln f(u_x)/¶u_x

Because the function h(u) depends on u we rewrite the derivative as a function of u.

(¶ln h(u)/¶u_x)_uy,uz = [d ln h(u)/du](¶u/¶u_x)_uy,uz= u_x/u [d ln h(u)/du]

Thus, we have

Because u_x, u_y, and u_z are independent of one another the above equation must be equal to a constant. We define this constant to be -g so that

Upon integration

Since f(u_j) is a probability distribution it must be normalized.

Substituting in for the Gaussian function we have obtained for the distribution. This is a Gaussian integral, so A = (g/p)^1/2.

Using the fact that áu_x²ñ = 1/3áu²ñ and áu²ñ = 3RT/M, it follows that áu_x²ñ = RT/M. We can use this information to determine the constant g in the probability distribution. The mean-square velocity is

Because the integral is an even integral it can be rewritten

This integral is discussed in the Math Review section on Gaussian integrals. We obtain

and g = M/2RT. The distribution function becomes

This is the formula used to assign velocities in the equilibration stage of a molecular dynamics simulation. Because of the independence of the x, y, and z dimensions, these can each be assigned velocities independently.

In three dimensions the expression becomes

and replacing the infinitesimal volume element du_xdu_ydu_z with 4pu²du we have