An ideal gas has molar heat capecity `C_(v)` at constant volume. The gas undergoes a process where in the temperature changes as `T=T_(0)(1+alphaV^(2))`, T and V are temperature and volume respectively, `T_(0) and alpha` are positive constant.The molar heat capecity C of the gas is given as `C=C_(v)+Rf(V),` where, f (V) is a funtion of volume. The expression for f (V) is

Options

A. `(alphaV^(2))/(1+alphaV^(2))`
B. `(1+ alphaV^(2))/(2alphaV^(2))`
C. `alphaV^(2)(1+alphaV^(2))`
D. `(1)/(2alphaV^(2)(1+alphaV^(2)))`

Answer

B

Explanation

Given,`T = T_(0)(1+alphaV^(2))`
`rArr (dT)/(dV)= T_(0). 2alphaV`
`rArr dV = (dT)/(T_(0).2alphaV)`
From first law of thermodynamics,
dQ = dU + dW
`rArr nC. dT = nC_(V).dT + pdV`
`rArr nC.dT + nC_(V).dT + p ((dT)/(T_(0).2alphaV))`
`rArr C = C_(V) + ((p)/(n))((1)/(T_(0)2alphaV))`
Now as gas is ideal,
pV = nRT
or `(p)/(n)=(RT)/(V)=(R[T_(0)(1+alphaV^(2))])/V`
Substitutting in Eq. (ii), we get,
`C = C_(V) + R.((1+alphaV^(2))/(2alphaV^(2)))`