Question

An ideal gas has a molar heat capacity $C_V$ at constant volume. Find the molar heat capacity of this gas as a function of volume, if the gas undergoes the process : $T=T_0{e}^{\alpha V}$.

Answer 1

The process is given as $T=T_o{e}^{\alpha V}$

Differentiating, we get $dT=T_o\alpha e^{\alpha V}dV$

$⟹$ $\frac{dT}{dV}=\frac{1}{\alpha T}$ ............(1)

Using 1st law of thermodynamics, $dQ=dU+dW$

where $\Delta U=C_V\Delta T$ and $dW=PdV$ and $\Delta Q=C\Delta T$

$\therefore$ $C\Delta T=C_V\Delta T+PdV$

OR $C=C_V+P\frac{dV}{dT}$

$⟹$ $C=C_V+P\times \frac{1}{\alpha T}$ ...........(1)

From ideal gas equation: $\frac{P}{T}=\frac{R}{V}$ in equation (1), we get

$C=C_V+\frac{R}{\alpha V}$