The volume of a diatomic gas `(gamma = 7//5)` is increased two times in a polytropic process with molar heat capacity ` C = R`. How many times will the rate of collision of molecules against the wall of the vessel be reduced as a result of this process?

Explanation

In a polytropic process `pV^n = constant`, where `n` is called the polytropic index. For this process
`PV^n = constant` or `TV^(n -1) = constant`
`(dT)/(T) + (n - 1) (dV)/(V) = 0`
Then `dQ = C d T = dU + pdV = C_V dT + pd V`
=`(i)/(2) R d T + (RT)/(V) dV = (i)/(2) R d T - (1)/(n - 1) R dT= ((i)/(2) - (1)/(n - 1)) R dT`
Now `C= R` so `(i)/(2) - (1)/( n - 1) = 1`
or, `(1)/(n - 1) = (i)/(2) - 1 = (i - 2)/(2)` or `n = (i)/(i - 2)`
Now `(v')/(v') = (n')/(n) (lt v' gt)/(lt v gt) = (1)/(eta) sqrt((T')/(T)) = (1)/(eta) ((V)/(V'))^((n - 1)/(2))`
=`(1)/(eta) ((1)/(eta))^((1)/(i - 2)) = ((1)/(eta))^((i - 1)/(i - 2)) = eta^((-i - 1)/(i - 2)) time s = (1)/(2.52)` times.