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The MaxwellBoltzmann distribution can be derived using Statistical Mechanics (see Derivation of the partition function). It corresponds to the most probable energy distribution in a system consisting of a large number of noninteracting particles. Since interactions between the molecules in a gas are generally quite small, the MaxwellBoltzmann distribution provides a very good approximation of the conditions in a gas (except at relatively high pressures and low temperatures, where intermolecular interactions become important).
The MaxwellBoltzmann distribution can be expressed as:
where N_{i} is the number of molecules at equilibrium temperature T, having energy level E_{i}, N is the total number of molecules in the system and k is Boltzmanns constant. Essentially Equation 1 provides a means for calculating the fraction of molecules (N_{i}/N) that have energy E_{i} at a given temperature, T. Because velocity and speed are related to energy, Equation 1 can be used to derive relationships between temperature and the speeds of molecules in a gas.
MaxwellBoltzmann Velocity Distribution
For the case of an "ideal gas" consisting of noninteracting atoms in the ground state, all energy is in the form of kinetic energy. From the Particle in a box problem in Quantum mechanics we know that the energy levels for a gas in a rectangular box with sides of lengths a_{x}, a_{y}, a_{z} are given by:
\left( \frac{n_x^2}{a_x^2} + \frac{n_y^2}{a_y^2} + \frac{n_z^2}{a_z^2} \right)
\left( \frac{h^2}{8m} \right) </math> (2)
where, n_{x}, n_{y}, and n_{z} are the quantum numbers for x,y, and z motion, respectively. However, for a macroscopic sized box, the energy levels are very closely spaced, so the energy levels can be considered continuous and we can replace the sum with an integral. Furthermore, we can recognize that (h^{2}n_{i}^{2}/4a_{i}^{2}) corresponds to the square of the ith component of momentum, p_{i}^{2} giving:
where q corresponds to the denominator in Equation 1. This distribution of N_{i}/N is proportional to the probability distribution function f_{p} for finding a molecule with these values of of momentum components, so:
The constant of proportionality, c, can be determined by recognizing that the probability of a molecule having any momentum must be 1. Therefore the integral of equation 4 over all p_{x}, p_{y}, and p_{z} must be 1.
It can be shown that:
Substituting Equation 6 into Equation 4 and using p_{i}=mv_{i} for each component of momentum gives:
Finally recognizing that the velocity probability distribution, f_{v} is proportional to the momentum probability distribution function as
we get:
Which is the MaxwellBoltzmann velocity distribution.
Velocity Distribution in One Direction
For the case of a single direction Equation 8 can be reduced to:
This distribution has the form of a Gaussian error curve. As expected for a gas at rest, the average velocity in any particular direction is zero.
Distribution of Speeds
Usually, we are more interested in the speed of molecules rather than the component velocities, where speed, v is defined such that:
The corresponding speed distribution is:
Average Speed
Although Equation 11 gives the distribution of speeds or in other words the fraction of molecules having a particular speed, we are often more interested in quantities such as the average speed of the particles rather than the actual distribution. In the following subsections we will define and derive the most probable speed, the mean speed and the rootmeansquare speed.
Most Probable Speed
The most probable speed, v_{p}, is the speed most likely to be possessed by any molecule in the system and corresponds to the maximum value of F(v). To find it, we calculate dF/dv, set it to zero and solve for v:
Mean Speed
The mean speed, <v>, or average speed can be calculated using the expression:
Substituting in Equation 11 and performing the integration gives:
Note that <v> and v_{p} differ by a constant factor (4/π)^{1/2}.
Rootmeansquare Speed
The root mean square speed, v_{rms} is given by
Thus, v_{p} < <v> < v_{rms}.
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