The molar specific heat of a gas is defined as `C=(Dq)/(ndT)` Where `dQ`=heat absorbed
`n` = mole number `dT` = change in temperature
`A` gas with adiabatic exponent `'gamma'` is expanded accrding to the law `p=alphaV` . The initial volume is `V_(0)` . The final volume is `eta V_(0)`. (`etagt1`). The molar heat capacity of the gas in the process is

Options

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

Answer

C

Explanation

`C=(R)/(gamma-1)+p(dV)/(dT)`
`=(R)/(gamma-1)+alphaV.(R)/(2alphaV)=(R)/(gamma-1)+(R)/(2)=(R(gamma+1))/(2(gamma-1))`
`Q=-DeltaU=DeltaU+W , or W=-2DeltaU`
`C=(R)/(gamma-1)+((-2RdT)/((gamma-1)dT))=(R)/(1-gamma)`
`pdV=(-2RdT)/(gamma-1)`
`(RT)/(V)dV=(-2)/(gamma-1)RdT(dT)/(T)=-((gamma-1))/(2)(dV)/(V)`
log `T=-((gamma-1))/(2)logV =logV^(-(gamma-1)/(2))+logK`
or log `TV^((gamma-1)/(2))=` constant