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.

Related Topics:
Probability distribution - Physics - Chemistry - Kinetic theory of gases - Gas - Pressure - Diffusion

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The Maxwell-Boltzmann distribution can be derived using statistical mechanics (see the derivation of the partition function). As an energy distribution, it corresponds to the most probable energy distribution, in a collisionally-dominated system consisting of a large number of non-interacting particles in which quantum effects are negligible. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas.

Related Topics:
Statistical mechanics - Derivation of the partition function

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In many other cases, however, the condition of elastic collisions dominating all other processes is not even approximately fulfilled. That is true, for instance, for the physics of the ionosphere and space plasmas where recombination and collisional excitation (i.e. radiative processes) are of far greater importance: in particular for the electrons. Not only would the assumption of a Maxwell distribution yield quantitatively wrong results, but even prevent a correct qualitative understanding of the physics involved. Also, in cases where the quantum thermal wavelength of the gas is not small compared to the distance between particles, there will be deviations from the Maxwell distribution due to quantum effects.

Related Topics:
Elastic collision - Ionosphere - Plasma - Thermal wavelength

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The Maxwell-Boltzmann energy distribution can be expressed as a discrete energy distribution by:

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:

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rac{N_i}{N} = rac{expleft(-E_i/kT ight) } { sum_{j}^{} {expleft(-E_j/kT ight)} }

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qquadqquad (1)

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where Ni is the number of molecules at equilibrium temperature T, in energy level i which has energy Ei, N is the total number of molecules in the system and k is the Boltzmann constant. (Note that sometimes the above equation is written with a factor gi denoting the degeneracy of energy states. In this case the sum will be over all energies, rather than all states.) 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. The denominator in this equation is known as the canonical partition function.

Related Topics:
Boltzmann constant - Partition function

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