An ideal gas is made to undergo a process `T = T_(0)e^(alpha V)` where `T_(0)` and `alpha` are constants. Find the molar specific heat capacity of the gas in the process if its molar specific heat capacity at constant volume is `C_(v)`. Express your answer as a function of volume (V).

Explanation

By first law of thermodymanics
`Q=dU+W`
`impliesnCdT=nC_(V)dT+PdV`
`impliesC=C_(V)=(PdV)/(ndT)=C_(V)+(RT)/(V)(dV)/(dT)`
`impliesC=C_(V)+R((dV//V)/(dT//T))` . . . (i)
Process equation is given as
`T=T_(0)e^(alphaV)` . . . . (ii)
Tke log of equation (ii), we get
`log" "T=log" "T_(0)+alphaV` ltBrgt `implies(dT)/(T)=0+alpha" "dV` (Differentiating)
`implies(dT)/(T(dV))=alpha`
`impliesC=C_(V)+R(dV)/((V(dT)/(T)))=C_(V)=R(dV)/(V.alphadV)=C_(V)+(R)/(alphaV)`
`impliesC=C_(V)+(R)/(alphaV)`.