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Oxygen gas is made to undergo a process in which its molar heat capacity C depends on its absolute temperature T as C = $α$ T. Work done by it, when heated from an initial temperature $T_{o}$ to a final temperature $2 T_{o}$ will be

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$(3 {αT}_{0} - 5) \frac{T_{o}}{2}$

$({αT}_{o} - 1) \frac{3 T_{0}}{2}$

none of these

$4 {αT}_{0}^{2}$

answer is C.

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Detailed Solution

$C = C_{v} + W^{'}; where W^{'} is the work done by the gas per unit mole per unit rise in temperature$

so, $W^{'} = αT - C_{v} = αT - (\frac{5 R}{2}) dT$

$∴$ $W^{'} = \int W^{'} dT = \int_{T_{0}}^{2 T_{0}} (αT - \frac{5 R}{2}) dT$

= $(3 {αT}_{o} - 5) \frac{T_{0}}{2}$

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