Math Problem Statement
Assume that the gas is homogeneous and isotropic. As the direction of the moving molecule is random, only the magnitude of the velocity, speed u, is significant. The Maxwell-Boltzmann (MB) distribution, F(u)du, is stated as the probability that a molecule has speed u ο u + du. It can be written as the product of the probabilities of the molecule having velocity components in the x, y, and z-directions: πΉπΉ(π’π’)πππ’π’ = ππ(π’π’ π₯π₯)πππ’π’ π₯π₯ ππ(π’π’ π¦π¦)πππ’π’ π¦π¦ ππ(π’π’ π§π§)πππ’π’ π§π§
Solution
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Statistical Mechanics
Maxwell-Boltzmann Distribution
Formulas
F(u) du = f(u_x) du_x f(u_y) du_y f(u_z) du_z
f(u_i) = (m / 2Ο k_B T)^(1/2) exp(-m u_i^2 / 2 k_B T)
F(u) = 4Ο (m / 2Ο k_B T)^(3/2) u^2 exp(-m u^2 / 2 k_B T)
Theorems
Maxwell-Boltzmann Speed Distribution
Gaussian Distribution
Suitable Grade Level
Undergraduate (Physics or Applied Mathematics)
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