Distinguish between Maxwell speed and velocity distributions of a gas at a given temperature.
The Maxwell–Boltzmann distribution describes how the particles in a gas are distributed according to their velocities or speeds at a given temperature.
- Velocity Distribution: Probability distribution of velocity components (vx, vy, vz). It is a 3D Gaussian distribution.
- Speed Distribution: Probability distribution of the magnitude of velocity (v = |⃗v|). It is obtained by integrating over directions.
Mathematical Form
- Velocity Distribution: \[ f(v_x, v_y, v_z) = \left(\frac{m}{2\pi kT}\right)^{3/2} \exp\left(-\frac{m(v_x^2+v_y^2+v_z^2)}{2kT}\right) \]
- Speed Distribution: \[ f(v) = 4\pi \left(\frac{m}{2\pi kT}\right)^{3/2} v^2 \exp\left(-\frac{mv^2}{2kT}\right) \]
Comparison Table
| Aspect | Velocity Distribution | Speed Distribution |
|---|---|---|
| Quantity | Vector components (vx, vy, vz) | Magnitude of velocity (v) |
| Form | 3D Gaussian distribution | Chi distribution with 3 degrees of freedom |
| Expression | Exponential in vx² + vy² + vz² | Exponential in v² with factor v² |
| Use | Analyzing directional velocity components | Determining most probable, average, and RMS speeds |
Graphical Comparison
The following illustration compares the Maxwell speed distribution and the Maxwell velocity distribution side by side:
Right: Velocity distribution showing isotropic 3D Gaussian symmetry.
Key Notes
- Velocity distribution is symmetric in all directions (isotropic).
- Speed distribution peaks at the most probable speed, not at zero.
- Both depend on temperature and particle mass.
In short: Velocity distribution deals with vector components, while speed distribution deals with the scalar magnitude of velocity. Both are derived from the Maxwell–Boltzmann law but serve different purposes in kinetic theory.
Barasat Government College (West Bengal State University)