One mole of an ideal gas whose pressure changes with volume as `P=alphaV` , where `alpha` is a constant, is expanded so that its volume increase `eta` times. Find the change in internal energy and heat capacity of the gas.

Answer

A, B

Explanation

Let `V` be the initail bolume of the gas. It expanded to a volume `muV` . The work done in this process is given by
`W=int_(v)^(eta v)PdV=int_(v)^(eta v)alphaVdV=alpha[(V_(2))/(2)]_(v)^( eta v)`
`(alphaV_(2))/(2)eta^(2)-1]`
The pressure of the gas varies volume as `P=alphaV` So, the initial and final pressure will be `alphaV` and `eta alphaV` . The change in internal energy is given by
`dU=nC_(V)dT=(R(T_(f)-T))/(gamma-1)=(P_(f)V_(f)-P_(i)V_(i))/(gamma-1)`
`=(eta^(2)alphaV^(2)-alphaV^(2))/(gamma-1)=(alphaV^(2))/(gamma-1)(eta^(2)-1)`
The heat exchange in this process is given by
`Q=U+W`
`(alphaV^(2))/(gamma-1)[eta^(2)-1]+(alphaV^(2))/(2)[eta^(2)-1]=(alphaV^(2))/(2)[eta^(2)-1][(gamma+1)/(gamma-1)]`
Here `T_(i)=(P_(i)V_(i))/(nR)=(alphaV^(2))/(nR) and T_(f)=(P_(f)V_(f))/(nR)=(mu^(2)alphaV^(2))/(nR)`
Now heat capacity `C=(Q)/(T_(f)-T_(1))`
`C=(1)/(T_(f)-T_(1))[(alphaV^(2))/(2)(eta^(2)-1){(gamma+1)/(gamma-1)}]`
`=(nR)/(alphaV^(2)(mu^(2)-1))[(alphaV^(2))/(2)(eta^(2)-1){(gamma+1)/(gamma-1)}]`
`=(nR)/(2)[(gamma+1)/(gamma-1)]`
Here `n=1`