Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increase as `V^q`, where V is the volume of the gas. The value of q is : `(gamma=(C_p)/(C_v))`

Options

A. `(3(gamma) + 5)/6`
B. `(3(gamma) - 5)/6`
C. `((gamma) + 1)/2`
D. `((gamma) - 5)/2`

Answer

C

Explanation

Average time of collision, `tau = (lambda)/(upsilon_(rms))`
as, `lambda = (1)/(sqrt(2)pi d^(2)(N/V)) and upsilon_(rms) = sqrt((3RT)/(m))`
`:.tau = (1)/(sqrt(2)pi d^(2)(N/V) sqrt((3RT)/(m))) :. Tau prop (V)/(sqrt(T))`
In adiabatic expansion `TV^(gamma-1) = k or T prop V^(1-gamma)`
As `tau prop (V)/(sqrt(T)) prop (V)/(sqrt(V^(1-gamma))) or tau prop V (V^(1-gamma))^(-1/2)`
or, `tau = V^((1_gamma)/(2))`. On comparing with `V^(q)`, we get
`q = (1+gamma)/(2) prop (gamma +1)/(2)`.