The velocity distribution of molecules in a beam coming out of a hole in a vessel is described by the function `F(v) = A v^3 e^(- mv^2//2 k T)`, where `T` is the temperature of the gas in the vessel. Find the most probable values of
(a) the velocity of the molecules in the beam , compare the result obtained with the most probable velocity of the molecules in the vessel ,
(b) the kinetic energy of the molecules in the beam.

Explanation

(a) `F (v) = Av^3 e^(-mv^2//2kT)`
For the most probable value of the velocity
`(dF(v))/(dv) = 0` or `3A v^2 e^(-mv^2//2 kT) - A v^3 (2 mv)/(2 kT) e^(-mv^2//2 kT) = 0`
So, `v_(pr) = sqrt((3 kT)/(m))`
This should be compared with the value `v_(pr) = sqrt((2 kT)/(m))` for the Maxwellian distribution.
(b) `1 n` terms of energy, `epsilon = (1)/(2) mv^2`
`F (epsilon) = Av^3 e^(-mv^2//2 kT) (dv)/(d epsilon)`
=`A((2 epsilon)/(m))^(3//2) e^(-epsilon//kT) (1)/(sqrt(2 m epsilon)) = A (2 epsilon)/(m^2) e^(- epsilon//kT)`
From this the probable energy comes out as follows : `F' (epsilon) = 0` implies
`(2A)/(m^2) (e^(-epsilon//kT) - (epsilon)/(kT) e^(-epsilon//kT)) = 0`, or `epsilon_(pr) = kT`.