Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. It forms the basis of the kinetic theory of gases, which explains many fundamental gas properties, including pressure and diffusion. The Maxwell-Boltzmann distribution is also applied in electron transport and other phenomena. The Maxwell-Boltzmann distribution can apply to a number of related properties of the individual molecules in a gas. It is usually thought of as the distribution of molecular energies in a gas, but it can also refer to the distribution of velocities, speeds, momenta, and magnitude of the momenta of the molecules, each of which will have a different function describing the distribution, all of which are related. Also it may be expressed as a discrete distribution over a number of discrete energy levels, or as a continuous distribution over a continuum of energy levels.

The Maxwell-Boltzmann distribution

The distribution of the momentum vector

What follows is a derivation wildly different from the derivation described by James Clerk Maxwell and later described with fewer assumptions by Ludwig Boltzmann.

Related Topics:
James Clerk Maxwell - Ludwig Boltzmann

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For the case of an "ideal gas" consisting of non-interacting atoms in the ground state, all energy is in the form of kinetic energy. The relationship between kinetic energy and momentum for massive particles is

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:

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E=rac{p^2}{2m}

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where p2 is the square of the momentum vector

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p = . We may therefore rewrite Equation 1 as:

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:

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rac{N_i}{N} =

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rac{1}{Z}

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exp left[

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rac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT}

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ight]

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qquadqquad (3)

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where Z is the partition function, corresponding to the denominator in Equation 1. Here m is the molecular mass of the gas, T is the thermodynamic temperature and k is the Boltzmann constant. This distribution of Ni/N is proportional to the probability density function fp for finding a molecule with these values of momentum components, so:

Related Topics:
Boltzmann constant - Proportional - Probability density function

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:

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f_mathbf{p} (p_x, p_y, p_z) =

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rac{c}{Z}

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exp left[

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rac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT}

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ight]

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qquadqquad (4)

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The normalizing constant 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 px, py, and pz must be 1.

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It can be shown that:

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:

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int_{-infty}^infty int_{-infty}^infty int_{-infty}^infty

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rac{1}{Z}

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exp left[

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rac{-(p_x^2 + p_y^2 + p_z^2)}{2mkT}

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ight],dp_x , dp_y,dp_z

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