Question

For an ideal gas the molar heat capacity varies as $C=C_V+3a{T}^2$. Find the equation of the process in the variables (T,V) where $a$ is a constant.

• A. $V=e^{(\frac{2a{T}^2}{3R})}$
• B. $V=e^{(\frac{3a{T}^2}{2R})}$
• C. $V=e^{(\frac{5a{T}^2}{2R})}$
• D. $V=e^{(\frac{2a{T}^2}{5R})}$

Answer 1

The correct option is B $V=e^{(\frac{3a{T}^2}{2R})}$

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

$\therefore$ $nCdT=n{C}_{V}dT+PdV$

$⟹$ $C=C_V+\frac{P}{n}\frac{dV}{dT}$

Comparing with $C=C_V+3a{T}^2$ we get $\frac{P}{n}\frac{dV}{dT}=3a{T}^2$

$\therefore$ $\frac{RT}{V}\frac{dV}{dT}=3a{T}^2$ $(\because PV=nRT)$

OR $\int \frac{dV}{V}=\frac{3a}{R}\int TdT$

$⟹$ $lnV=\frac{3a}{R}\frac{T^2}{2}$

$⟹$ $V=e^{(\frac{3a{T}^2}{2R})}$