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# Helium gas has undertaken a process in which the pressure and the volume are changed according to the equation $P=kV^2$. If $C_v$ is the molar specific heat at constant volume, then the molar hear capacity of the gas for the process is given by?

$(A) C = C_v + R/2$ $(B) C = C_v + R/4$ $(C) C = C_v + R/3$ $(D) C = C_v - R/2$

We know that $PV = n RT$ and are given $P = kV^2$.
$\Rightarrow PV = nRT = kV^3$
Differentiating, we get $nR dT = 3kV^2dV \rightarrow 3PdV = nRdT$
Now, $dQ = dU + pDV$
$\Rightarrow nCdT = nC_vdT +\large\frac{nRdT}{3}$
$\Rightarrow C = C_v + R/3$
Note: We can also substitute $C_v = 3R/2$ such that $C = 11R/6$
edited Jan 22, 2014