Find the fraction of gas molecules whose velocities differ by less than `delta eta = 1.00 %` from the value of
(a) the most probable velocity ,
(b) the root mean square velocity.

Explanation

(a) The formula is
`df (u) = (4)/(sqrt(pi)) u^2 e^(-u^2) du`, where `u = (v)/(v_p)`
Now prob `(|v - v_p|/(v_p) lt delta eta) = int_(1 - delta eta)^(1 + delta eta) df (u)`
=`(4)/(sqrt(pi)) e^(-1) xx 2 delta eta = (8)/(sqrt(pi) e) delta eta = 0.0166`
(b) `Prob (|(v - v_rms)/(v_(rms))| lt delta eta) = Prob(|(v)/(v_p) - v_(rms)/(v_p)|lt delta eta (v_(rms))/(v_p))`
=`Prob (|u - sqrt((3)/(2))| lt sqrt((3)/(2)) delta eta)`
`sqrt((3)/(2)) + sqrt((3)/(2)) delta eta`
=`int (4)/(sqrt(4)) u^2 e^(- u^2) du`
`sqrt((3)/(2)) - sqrt((3)/(2)) delta eta`
=`(4)/(sqrt(pi)) xx (3)/(2) e^(-3//2) xx 2 sqrt((3)/(2)) delta eta = (12 sqrt(3))/(sqrt(2 pi)) e^(-3//2) delta eta = 0.0185`.