An ideal gas consisting of rigid diatimic molecules goes through an adiabatic process process. How do the mean free path `lamda` and the number of collisions of each molecule per second `v` depend in this process on
(a) the volume`V` ,
(b) the pressure `p` ,
( c) the temperature `T` ?

Explanation

(a) `lamda alpha(1)/(n) = gt (1)/(N//V) = (V)/(N)`
Thus `lamda alpha V` and `v alpha (T^(1//2))/(V)`
But in an adiabatic process `(gamma = (7)/(5) "here")`
`TV^(gamma - 1) = "constant so" TV^(2//5) =` constant
or `T^(1//2) alpha V^(-1//5)` Thus `v alpha V^(-6//5)`
(b) `lamda alpha (T)/(p)`
But `p((T)/(p))^gamma =` constant or `(T)/(p) alpha p^(-1//gamma)` or `T alpha p^(1-1//gamma)`
Thus `lamda alpha p^(-1//gamma) = p^(-5//7)`
`v = (lt v gt )/(lamda) alpha(p)/(sqrt(T)) alpha p^(1//2 + (1)/(2 gamma)) =p^((gamma + 1)/(2 gamma) = p^(6//7))`
( c) `lamda alpha V`
But `TV^(2//5) =` constant or `V alpha T^(-5//2)`
Thus `lamda alpha T^(-5//2)`
`v alpha(T^(1//2))/(V) alpha T^3`.