What fraction of molecules in a gas at a temperature `T` has the kinetic energy of translational motion exceeding `epsilon_0` if `epsilon_0 gt gt k T` ?

Explanation

`(Delta N)/(N) = (2)/(sqrt(pi)) (kT)^(-3//2) int_0^oo sqrt(epsilon) e^(-epsilon//kT) d epsilon`
`~~ (2)/(sqrt(pi)) (kT)^(-3//2) sqrt(epsilon_0) int_(epsilon_0)^oo e^(-epsilon//kT) d epsilon(epsilon_0 gt gt kT)`
=`(2)/(sqrt(pi)) (kT)^(-3//2) sqrt(epsilon_0) kT e^(-epsilon_0//kT) = 2 sqrt((epsilon_0)/(pi kT)) e^(-epsilon_0//kT)`
(In evaluating the integral, we have taken out `sqrt(epsilon)` as `sqrt(epsilon_0)` since the integral is dominated by the lower limit).