One mole of an ideal diatomic gas undergoes a thermodynamic process, in which its molar heat capacity varies directly proportional to temperature as $C = α T,$ where $α$ is a positive constant. Work done by the gas when it is heated from initial temperature $T_{0}$ to a final temperature $3 T_{0}$ will be

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$(α T_{0} - R) \frac{3 T_{0}}{2}$

$(4 α T_{0} - 5 R) T_{0}$

$4 α T_{0}^{2}$

$(3 α T_{0} - 5 R) \frac{T_{0}}{2}$

answer is C.

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Detailed Solution

$C = C_{V} + \frac{d W}{n d T},$

$\int d W = \int_{T_{0}}^{3 T_{0}} α T d T - \int_{T_{0}}^{3 T_{0}} \frac{5 R}{2} d T$

for one mole $W = (4 α T_{0} - 5 R) T_{0}$

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