Question

Starting with the Maxwell Distribution of Energies, find an expression for Erms(root mean square energy). Use a gamma function or integration by parts.

Answer 1

The Maxwell-Boltzmann probability distribution function for the energies of the particles in an ideal gas is given by:

f(E) = dN/N = (2/(k*T)^(3/2))*sqrt(E/π)*exp(-E/(k*T)) dE

As is commonly done, let β = 1/(k*T)
f(E) = dN/N = (2/sqrt(π)) * β^(3/2) *sqrt(E)*exp(-β*E) dE

The mean squared energy is given by:
<E^2> = INTEGRAL from 0 to ∞ of {E^2 * f(E) dE}
<E^2> = (2/sqrt(π)) * β^(3/2)*INTEGRAL from 0 to ∞ of {E^(5/2) * exp(-β*E) dE}

Note that the definition of the Gamma function is:
Γ(x) = INTEGRAL from 0 to ∞ of {t^(x-1) * exp(-t) dt}

Letting t = β*E, (so E = t/β and dE = dt/β), we can write the integral above as:

<E^2> = (2/sqrt(π)) * β^(3/2)*β^(-7/2)*INTEGRAL from 0 to ∞ of {t^(5/2) * exp(-t) dt}

<E^2> = (2/sqrt(π)) * β^(-2) * Γ(7/2)

Γ(7/2) = 15π/8, so:

<E^2> = (2/sqrt(π)) * β^(-2) * 15π/8

<E^2> = 15*sqrt(π)*(k*T)^2/4

Erms = sqrt(15)*k*T/2