The potential energy of gas molecules in a certain certral field depends on the distance `r` from the field's centre as `u( r) = ar^`, where `a` is a positive constant. The gas temperature is `T`, the concentration of molecules at the centre of the field is `n_0` Find :
(a) the number of molecules located at the distances between `r` and `r + dr` from the centre of the field ,
(b) the most probable distance separating the molecules from the centre of the field ,
( c) the fraction of molecules located in the spherical layer between `r` and `r + dr` ,
(d) how many times the concentration of molecules in the centre of the field will change if the temperature decreases `eta` times.

Explanation

Here `n(r) = n_0 exp (-(ar^2)/(kT))`
(a) The number of molecules located at the distance between `r` and `r + dr` is
`4 pi r^2 dr n(r) = 4 pi n_0 exp (-(ar^2)/(kT)) r^2 dr`
(b) `r_(pr)` is given by `(d)/(dr) r^2 n(r) = 0` or, `2r - (2 ar^3)/(kT) = 0` `r_(pr) = sqrt((kT)/(a))`
( c) The fraction of molecules lying between `r` and `r + dr` is
`(dN)/(N) = (4 pi r^2 dr n_0 exp (-ar^2//kT))/(int_0^oo 4 pi r^2 dr n_0 exp(-ar^2//kT))`
`int_0^oo 4 pi r^2 dr exp(-(ar^2)/(kT)) = ((kT)/(a))^(3//2) 4 pi int_0^oo x (dx)/(2 sqrt(x)) exp (-x)`
=`((kT)/(a))^(3//2) 2 pi Gamma ((3)/(2)) = ((pi kT)/(a))^(3//2)`
Thus `(dN)/(N) = ((a)/(pi kT))^(3//2) 4 pi r^2 dr exp ((-ar^2)/(kT))`
(d) `dN = N((a)/(pi kT))^(3//2) 4 pi r^2 dr exp ((-ar^2)/(kT))`,
So `n(r) = N((a)/(pi kT))^(1//2) exp ((-ar^2)/(kT))`
When `T` decrease `eta` times `n(0) = n_0` will increase `eta^(3//2)` times.