From the conditions of the foregoing problem find :
(a) the number of molecules whose potential energy lies within the internal from `U` to `U + dU` ,
(b) the most probable value of the potential energy of a molecule , compare this value with the potential energy of a molecule located at its most probable distance from the centre of the field.

Explanation

Write `U = ar^2`or `r = sqrt((U)/(a))`, so `dr = sqrt((1)/(a)) (dU)/(2 sqrt(U)) = (dU)/(2 sqrt(a U))`
so `dN n_0 4 pi(U)/(a) (dU)/(2 sqrt(a U)) exp ((U)/(kT))`
=`2 pi n_0 a^(-3//2) U^(1//2) exp ((-U)/(kT)) dU`
The most probable value of `U` is given by
`(d)/(dU) ((dN)/(dU)) = 0 ((1)/(2 sqrt(u)) -U^(1//2)/(kT)) exp ((-U)/(kT))`or `U_(pr) = (1)/(2) kT`
From `2.111 (b)`, the potential energy at the most probable distance is `kT`.