What fraction of monatomic molecules of a gas a in a thermal equilibrium possesses kinetic energies differenting from the mean value by `delta eta = 1.0 %` and less ?

Explanation

The mean kinetic energy is
`lt epsilon gt = int_0^oo epsilon^(3//2) e^(- epsilon//kT)// int_0^oo epsilon^(1//2) e^(-epsilon//kT) d epsilon = kT (Gamma(5//2))/(Gamma(3//2)) = (3)/(2) kT`
Thus
`(delta N)/(N) = int_((3)/(2)kT(1 - delta eta))^((3)/(2)(1 + delta eta)kT) (2)/(sqrt(pi)) (kT)^(-3//2) e^(-epsilon//kT) epsilon^(1//2) epsilon`
=`(2)/(sqrt(pi)) e^(-3//2) ((3)/(2))^(3//2) 2 delta eta = 3 sqrt((6)/(pi)) e^(3//2) delta eta`
If `delta eta = 1 %` this gives `0.9 %`.