Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Analytical Integration

Abstract

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. The Maxwell-Boltzmann speed distribution of mixed particles is based on kinetic theory; however, it has never been derived from a mechanical point of view. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles based on probability analysis and Newton’s law of motion. This paper requires the probability density function (PDF) ψ(ua; va, vb) of the speed ua of the particle with mass Ma after the collision of two particles with mass Ma in speed va and mass Mb in speed vb. The PDF ψ (ua; va, vb) in integral form has been obtained before. This paper further performs the exact integration from the integral form to obtain the PDF ψ(ua; va, vb) in an evaluated form, which is used in the following equation to get new distribution Pnew a (ua) from old distributions Pold a (va) and Pold b (vb). When Pold a (va) and Pold b (vb) are the Maxwell-Boltzmann speed distributions, the integration Pnew a (ua) obtained analytically is exactly the Maxwell-Boltzmann speed distribution. Pnew a (ua) = ∫ ∫ ψ (ua; va, vb)Pold a (va)Pold b (vb)dvadvb ∞