Understanding the Maxwell-Boltzmann Distribution: A Mathematical Derivation and Its Implications for Classical Gas Behavior

Derivation of the Maxwell-Boltzmann Distribution Law

The Maxwell-Boltzmann distribution describes the distribution of speeds (or energies) of particles in a classical ideal gas. It is based on the assumption that the particles in the gas follow classical mechanics and are non-interacting. The distribution provides a statistical description of the fraction of particles having a particular speed at thermal equilibrium.

1. Basic Assumptions:

The Maxwell-Boltzmann distribution law is derived for an ideal gas under the following assumptions:

• The gas consists of a large number of identical particles.
• The particles are point-like (i.e., their size is negligible compared to the average separation between particles).
• The particles do not interact with each other except through elastic collisions.
• The system is in thermal equilibrium at a temperature $T$.

2. Energy of a Particle:

The kinetic energy $E$ of a classical particle of mass $m$ moving with velocity $v$ is given by:

$$E = \frac{1}{2} m v^2$$

In three dimensions, the speed $v$ is related to the components of velocity along each axis $(v_x, v_y, v_z)$ by:

$$v^2 = v_x^2 + v_y^2 + v_z^2$$

Each component of the velocity $v_x, v_y, v_z$ is independently distributed according to a Gaussian (normal) distribution, as a consequence of the system's thermal equilibrium.

3. Phase Space and Probability:

The number of particles having velocities in the range $v_x$ to $v_x + dv_x$, $v_y$ to $v_y + dv_y$, and $v_z$ to $v_z + dv_z$ is proportional to the product of the Gaussian distributions in each direction:

$$f(v_x, v_y, v_z) dv_x dv_y dv_z = A e^{-\frac{m(v_x^2 + v_y^2 + v_z^2)}{2k_B T}} dv_x dv_y dv_z$$

Here:

• $A$ is a normalization constant.
• $k_B$ is the Boltzmann constant.
• $T$ is the absolute temperature of the system.

Rewriting this in terms of the speed $v$:

$$v^2 = v_x^2 + v_y^2 + v_z^2$$

Thus, the number of particles with speed between $v$ and $v + dv$ is:

$$f(v) dv = A v^2 e^{-\frac{mv^2}{2k_B T}} dv$$

The factor $v^2$ arises due to the transformation from Cartesian coordinates $(v_x, v_y, v_z)$ to spherical coordinates (speed $v$ and solid angle elements), accounting for the number of states with the same speed but different directions.

4. Normalization:

The normalization constant $A$ is determined by the condition that the total number of particles must equal $N$. That is, integrating the distribution over all possible speeds gives the total number of particles:

$$\int_0^\infty f(v) dv = N$$

Substituting $f(v) = A v^2 e^{-\frac{mv^2}{2k_B T}}$, we get:

$$A \int_0^\infty v^2 e^{-\frac{mv^2}{2k_B T}} dv = N$$

This integral can be evaluated using standard Gaussian integration techniques. The result is:

$$A = N \left( \frac{m}{2 \pi k_B T} \right)^{3/2}$$

5. Final Form of the Distribution:

Substituting the value of $A$, the Maxwell-Boltzmann distribution for the speed of particles in a gas is:

$$f(v) = N \left( \frac{m}{2 \pi k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}}$$

This is the Maxwell-Boltzmann speed distribution, which gives the probability density of finding particles with speed $v$ in a gas at temperature $T$.

Key Results:

• The distribution has a characteristic shape with a peak, indicating the most probable speed.
• The average speed, root mean square speed, and most probable speed can be derived from this distribution.

Interpretation:

• Most Probable Speed: The speed at which the distribution function $f(v)$ reaches its maximum.
• Average Speed: The mean speed of all particles.
• Root Mean Square Speed: The square root of the average of the square of the speeds.

These quantities describe different aspects of particle motion in the gas and are useful for understanding the thermodynamic behavior of gases.