An ideal gas goes through a polytropic process with exponent `n`. Find the mean free path `lamda` and the number of collisions of each molecule per second `v` as a function of
(a) the volume `V` ,
(b) the pressure `p` ,
( c) the temperature `T`.

Explanation

In the polytropic process of index `n`
`pV^n =` constant, `TV^(n -1) =` constant and `p^(1 - n) T^n =` constant
(a) `lamda alpha V`
`v alpha (T^(1//2))/(V) = V^((1 - n)/(2)) V^-1 = V^((-n + 1)/(2))`
(b) `lamda alpha (T)/(p), T^n alpha p^(n -1)` or `T alpha p^(1 - (1)/(n)`
so `lamda alpha p^(-1//n)`
`v = (lt v gt)/(lamda) alpha (p)/(sqrt(T)) alpha p^(1 -(1)/(2) + (1)/(2n)) = p^((n + 1)/( 2n))`
( c) `lamda alpha (T)/(p), p alpha T^((n)/(n - 1))`
`lamda alpha T^(1 -(n)/(n 1)) = T^(-(1)/(n -1)) = T^((1)/(1 - n))`
`v alpha (p)/(sqrt(T)) alpha T^((n)/(n -1) -(1)/(2)) =T^((n + 1)/(2(n -1)))`.