At what temperature of a gas will the number of molecules, whose velocities fall within the given interval from `v` to `v + dv`, be the greatest ? The mass of each molecule is equal to `m`.

Explanation

`dN (v) = (N 4)/(sqrt(pi)) (v^2 dv)/(v_p^3) e^(-v^2//v_p^2)`
For a given `v` to `v + dv` (i.e., given `v` and `dv`) this is maximum when
`(delta)/(delta v_p) (dN(v))/(N v^2 dv) = 0 = (-3 v_p^-4 + (2 v^2)/(v_p^6)) e ^(-v^2//v_p^2)`
or, `v^2 = (3)/(2) v_p^2 = (3 kT)/(m)`. Thus `T = (1)/(3) (mv^2)/(k)`.