One mole of an ideal gas undergoes a reversible process in which the entropy of the gas changes with

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One mole of an ideal gas undergoes a reversible process in which the entropy of the gas changes with absolute temperature T as S = aT + Cv,m lnT, where a is a positive constant. If T = To at V = Vo, then the volume dependence of the gas on temperature in this process is

Accepted Answer

$dS = C_{V,m}.\frac{dT}{T} + P.\frac{dV}{T} = a.dT + C_{V,m}.\frac{1}{T}.dT$

$\text{Or, }\int_{V_{o}}^{V}{\frac{R}{V}.dV} = a.\int_{T_{o}}^{T}{dT} \Rightarrow R.\ln\frac{V}{V_{0}} = a\left( T - T_{0} \right) $$$\therefore T = T_{0} + \frac{R}{a}.\ln\frac{V}{V_{0}}$$

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