Maxwell–Boltzmann distribution

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The Maxwell–Boltzmann distribution is a probability distribution used for describing the speeds of particles in a gas that is in thermal equilibrium. It is named after James Clerk Maxwell and Ludwig Boltzmann. The distribution is applicable to the classical ideal gas, where the particles do not interact with each other except for very brief collisions in which they exchange energy and momentum.

Derivation

The Maxwell–Boltzmann distribution can be derived from the principles of statistical mechanics. It describes the distribution of speeds among the molecules of a gas in thermal equilibrium. The probability density function for the speed \( v \) of a particle is given by: \[ f(v) = \left( \frac{m}{2 \pi k_B T} \right)^{3/2} 4 \pi v^2 \exp \left( - \frac{mv^2}{2k_B T} \right) \] where:

• \( m \) is the mass of a particle,
• \( k_B \) is the Boltzmann constant,
• \( T \) is the temperature in kelvin.

Properties

The Maxwell–Boltzmann distribution has several important properties:

• The distribution is skewed to the right, meaning that there are more particles with speeds lower than the most probable speed.
• The mean speed, median speed, and most probable speed are different.
• The distribution becomes broader and shifts to higher speeds as the temperature increases.

Applications

The Maxwell–Boltzmann distribution is fundamental in the field of kinetic theory of gases. It is used to predict the behavior of gases in various conditions, including:

• Effusion and diffusion of gases,
• Thermal conductivity,
• Viscosity of gases,
• Reaction rates in chemical kinetics.

Related Distributions

The Maxwell–Boltzmann distribution is related to other statistical distributions:

• The Boltzmann distribution describes the distribution of energy states in a system.
• The Maxwell distribution is a special case of the Maxwell–Boltzmann distribution for the speed of particles.
• The Rayleigh distribution is related to the Maxwell–Boltzmann distribution in two dimensions.

See Also

• Kinetic theory of gases
• Statistical mechanics
• Thermodynamics
• Ideal gas law
• Boltzmann constant
• James Clerk Maxwell
• Ludwig Boltzmann