One mole of an ideal gas with heat capacity at constant pressure $C_{p}$ undergoes the process $T = T_{0} + α V$ , where $T_{0}$ and $α$ are constants, T is temperature and V is volume of the gas. The specific heat capacity of the gas as a function of its volume is: (if $C_{p}$ and $C_{V}$ denote specific heat at constant pressure and constant volume respectively)

C = C_{P} + \frac{R T_{0}}{α V}

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C = C_{P}

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C = C_{V} + \frac{R T_{0}}{α V}

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C = \frac{R T_{0}}{α V}

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Solution

The correct option is A $C = C_{P} + \frac{R T_{0}}{α V}$
The specific heat is given by
$C = C_{V} + \frac{P}{n} \frac{d V}{d T}$
We have,
$T = T_{0} + α V$
$\frac{d T}{d V} = 0 + α$
$\Rightarrow \frac{d V}{d T} = \frac{1}{α}$
$\frac{P}{n} = \frac{R T}{V} = \frac{R}{V} [T_{0} + α V]$
$= \frac{R T}{V} + R α$
$C = C_{V} + (\frac{R T_{0}}{V} + R α) \times \frac{1}{α}$
$= (C_{V} + R) + \frac{R T_{0}}{α V}$
$C = C_{P} + \frac{R T_{0}}{α V}$

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