Determine the molar heat capacity of a polytropic process through which an ideal gas consisting of rigid diatomic molecules goes and in which the number of collisions between the molecules remains constant
(a) in a unit volume ,
(b) in the total volume of the gas.

Explanation

(a) The number of collisions between the molecules in a unit volume is
`(1)/(2) n v= (1)/(sqrt(2)) pi d^2 n^2 lt v gt alpha (sqrt(T))/(V^2)`
This remains constant in the poly process `pV^-3 =` constant
Using `(2.122)` the molar specific heat for the polytropic process
`pV^(alpha) =` constant
is `C = R ((1)/(gamma - 1)-(1)/(alpha - 1))`
Thus `C = R((1)/(gamma - 1)+(1)/(4)) = R ((5)/(2) + (1)/(4)) = (11)/(4) R`
It can also be written as `(1)/(4) R(1 + 2 i)` where `i = 5`
(b) In this case `(sqrt(T))/(V) =` constant and so `pV^-1 =` constant
so `C =R ((1)/(gamma -1)+(1)/(2)) = R ((5)/(2) + (1)/(2)) = 3R`
It can also be written as `(R)/(2)(i + 1)`.