An ideal gas undergoes a process in which `PV^(-a)=` constant, where V is the volume occupied by the gas initially at pressure P. At the end of the process, 'rms' speed of gas molecules has become `a^(1//2)` times of its initial value. What will be the value of `C_(V)` so that energy transferred in the form of heat to the gas is 'a' times of the initial energy.

Options

A. `((a^(2)+1)R)/(a^(2)-1)`
B. `((a^(2)+1)R)/((a^(2)+1))`
C. `((a+1)R)/((a-1))`
D. `((a-1)R)/((a+1))`

Answer

D

Explanation

`DeltaQ=alphaU_(1)`
`nC(T_(2)-T_(1))=alpha(f)/(2)nRT_(1)`....(1)
but `v_(rms)alphasqrt(T)rArrT_(2)=alphaT_(1)`
(as rms speed became `sqrt(alpha)` times).
From (1) `therefore C(alpha-1)T_(1)=(alphaf)/(2)RT_(1)`
`C=(alphaf(R)/(2(alpha-1))("here",f=(2C_(v))/(R))`
`C=C_(v)+(R)/(1-n)` (for polytrophic process)
wherer n=-a