The Maxwell-Boltzmann Speed Distribution:The Ideal Gas
For many systems, including the ideal gas, the Maxwell-Boltzmann distribution provides the probability of finding a component in a particular state with a particular amount of energy. In the case of the gas, many different states have the same speed; a particle could be going left, right, up, down, back, front, or at any angle, and still be traveling the same speed. To find the probability of a particle having a particular speed, we must compute how many states have the same speed, and add up the probabilities of being in any of these states. The kinetic energy of a particle with mass m and speed u is So the probability of finding the particle in any one state is The number of states with the same speed is given by the surface area of a spherical shell, (This is the same factor which enters when converting a three-dimensional Cartesian integral of a spherically-symmetric function to spherical coordinates.) Every state has a different velocity vector. All the states with the same speed have velocity vectors with the same length u, pointing in different directions. All these velocity vectors sweep out a spherical shell with radius u. Therefore, the probability of finding an atom with a particular speed is The next step is to evaluate the normalization constant C. Since P(u) is a continuous distribution, we must integrate over all values of u (from 0 to infinity). This involves some advanced mathematics, and gives (see the section on finding average properties of the gas for a step-by-step derivation):
Exercises using the Maxwell-Boltzmann Speed Distribution
The Big Picture
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