An ideal gas in under standard conditions. Find the diameter of the sphere within whose volume the relative fluctuation of the number of molecules is equal to `eta = 1.0.10^-3`. What is the average number of molecules inside such a sphere ?

Explanation

In a sphere of diameter `d` there are
`N = (pi d^3)/(6) n_0` molecules
where `n_0` = Loschmidt's number = No. of molecule per unit volume `(1 cc)` under NTP.
The relative flutuation in this number is
`(del N)/(N) = (sqrt(N))/(N) = (1)/(sqrt(N)) = eta`
or `(1)/(eta^2) = (pi)/(6) d^2 n_0` or `d^3 = (6)/(pi n_0eta^2)` or `d = ((6)/(pi eta^2 n_0))^(1//3) = 0.41 mu m`
The average number of molecules in this sphere is `(1)/(eta^2) = 10^6`.