Gas in A Box - Energy Distribution

Energy Distribution

Using the results derived from the previous sections of this article, some distributions for the "gas in a box" can now be determined. For a system of particles, the distribution for a variable is defined through the expression which is the fraction of particles that have values for between and

where

, number of particles which have values for between and
, number of states which have values for between and
, probability that a state which has the value is occupied by a particle
, total number of particles.

It follows that:

For a momentum distribution, the fraction of particles with magnitude of momentum between and is:

and for an energy distribution, the fraction of particles with energy between and is:

P_E~dE = P_p\frac{dp}{dE}~dE

For a particle in a box (and for a free particle as well), the relationship between energy and momentum is different for massive and massless particles. For massive particles,

while for massless particles,

where is the mass of the particle and is the speed of light. Using these relationships,

  • For massive particles
\begin{alignat}{2} dg_E & = \quad \ \left(\frac{Vf}{\Lambda^3}\right) \frac{2}{\sqrt{\pi}}~\beta^{3/2}E^{1/2}~dE \\ P_E~dE & = \frac{1}{N}\left(\frac{Vf}{\Lambda^3}\right) \frac{2}{\sqrt{\pi}}~\frac{\beta^{3/2}E^{1/2}}{\Phi(E)}~dE \\ \end{alignat}

where Λ is the thermal wavelength of the gas.

\Lambda =\sqrt{\frac{h^2 \beta }{2\pi m}}

This is an important quantity, since when Λ is on the order of the inter-particle distance 1/3, quantum effects begin to dominate and the gas can no longer be considered to be a Maxwell-Boltzmann gas.

  • For massless particles
\begin{alignat}{2} dg_E & = \quad \ \left(\frac{Vf}{\Lambda^3}\right) \frac{1}{2}~\beta^3E^2~dE \\ P_E~dE & = \frac{1}{N}\left(\frac{Vf}{\Lambda^3}\right) \frac{1}{2}~\frac{\beta^3E^2}{\Phi(E)}~dE \\ \end{alignat}

where Λ is now the thermal wavelength for massless particles.

Read more about this topic:  Gas In A Box

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