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

Options

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

Answer

C

Explanation

The rms speed of the molecules , `c=sqrt(3RT//M)`
Mean free path , `lamda=1/(sigma^2n)=1/(sigma^2N/v)`
[N=Number of molecules , `sigma`= diameter of the molecules]
Average time of collision between the molecules
`t=lamda/c=V/(sigma^2N)=1/sqrt(3RT//M)`
or, `tpropV/sqrtTor,TpropV^2/t^2`
Again in the adiabatic process
`TV^(gamma-1)=constant or,T propV^(1-gamma)`
`thereforeV^2/t^2propV^(t-y)`
or,`t^2propv^(gamma+1)or,tpropV^((gamma+1)/2)`
Hence `q=(gamma+1)/2`