Maxwell Distribution of Speeds

We stipulated in the above discussion of the Maxwell-Boltzmann distribution that the medium defined by the particles is isotropic; that is, properties are not dependent on a direction (on internal or external gradients). Furthermore, though we wrote $\Omega({\nu})$ as a function of speed, $\nu$, there was no explicit dependence on speed, a non-vectorial quantity, in the final expression. With this in mind, we next consider the distribution of speeds that emerges from the velocity distribution we derived. Again, bear in mind that speed is not vectorial, velocity is.

Based on our earlier definition of the probability distribution based on speed, $\nu$, we can write now:

$$F(\nu) \ d \nu \ = \ f(v_x) \ f(v_y) \ f(v_z) \ dv_x \ dv_y \ dv_z \$$

$$\ = \ \left [ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right)^{1/2} e^{-m... ...rac{m}{2 \ \pi \ k_B \ T} \right)^{1/2} e^{-m \ v_y^2 \ /2 \ k_B \ T)} \right ]$$

$$\left [ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right)^{1/2} e^{-m \ v_z^2 \ /2 \ k_B \ T)} \right ] \ dv_x \ dv_y \ dv_z \$$

$$\ = \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right)^{3/2} e^{-m \ (v_x^2 + v_y^2 + v_z^2) \ /2 \ k_B \ T)} \ dv_x \ dv_y \ dv_z \$$

We are almost to the point of having our expression for the speed distribution, since we can see clearly that $v_x^2 + v_y^2 + v_z^2 = \nu^2$. However, the differential volume element $dv_x \ dv_y \ dv_z$ is the troubling part. Here we consider the idea of change of variables from Cartesian components of velocity to a spherical coordinate representation of speed.

Thus, the volume element becomes $4 \ \pi \ \nu^2 \ d \nu$ after integrating over the angular dimensions. The Maxwell speed distribution thus becomes:

$$F(\nu) \ d \nu \ = \ 4 \ \pi \ \left ( \frac{m}{2 \ \pi \ k_B \ T} \right)^{3/2} \ \nu^2 \ e^{-m \ \nu^2 \ /2 \ k_B \ T)} \ d \nu$$

$$F(\nu) \ d \nu \ = \ 4 \ \pi \ \left ( \frac{M}{2 \ \pi \ R \ T} \right)^{3/2} \ \nu^2 \ e^{-M \ \nu^2 \ /2 \ R \ T)} \ d \nu$$

Properties of this distribution include:

• At a particular temperature, there is a single maximum value of the speed
• Despite including a Gaussian factor, the distribution is not symmetric due to the $\nu^2$ term which introduces contributions from tail values at higher speeds
• At higher temperatures, the distribution becomes broader (more speeds, or translational energy levels accessible)
• At lower masses, the distribution becomes broader (more speeds are accessible)

• Comparison of distribution for gases of different mass At same temperature, kinetic theory says all gases have some energy (depends on T); thus, heavier masses have distributions that are narrower and peaked at lower speeds.