Maxwellian Distribution

the velocity (or momentum) distribution of the molecules of a system in thermodynamic equilibrium. It was established by J. C. Maxwell in 1859. According to the Maxwellian distribution, the probability Δw(v_x, v_y, v_z) that the components of the velocity of a molecule lie in the small ranges between v_x and v_x + Δv_x, v_y and v_y + Δv_y, v_z; and v_z + Δv_z is given by the formula

where m is the mass of the molecule, T is the absolute temperature of the system, and k is Boltzmann’s constant.

The probability that the absolute value of the velocity lies between v and v + Δv follows from (1) and has the form

This probability reaches a maximum when . The velocity vo is called the most probable molecular velocity. The lower the temperature of the system, the greater is the number of molecules that have velocities close to the most probable velocity (see Figure 1).

The average number of particles in 1 cm³ of gas having velocities between v and v + Δv is equal to Δn(v) = n₀Δw(v), where n₀ is the total number of particles in 1 cm³.

By using the Maxwellian distribution it is possible to calculate

Figure 1. Distribution over velocities v of nitrogen atoms at absolute temperatures T₁, and T₂; ΔW/ΔV is ratio of probability that absolute value of velocity lies between v and v + Δv to the velocity interval Δv

the mean molecular velocities and any functions of these velocities. In particular, the root mean square molecular velocity exceeds by a factor of the most probable molecular velocity. For example, for nitrogen when and v₀ ≈ 360 m/sec.

The Maxwellian distribution stems from the Gibbs canonical distribution for the case when the translational motion of the particles can be described in the classical approximation. The Maxwellian distribution does not depend on the nature of the interaction of the particles of the system or on external forces and therefore is valid both for the molecules of a gas and the molecules of liquids and solids. The Maxwellian distribution is also valid for Brownian particles suspended in a gas or liquid.

The Maxwellian distribution has been confirmed in experiments with molecular beams.