Question

For the case of an ideal gas , find the equation of the process (in the variables $T,V$) in which the molar heat capacity varies as $C=C_V+\frac{\alpha }{T}$, where $\alpha$ is constant.

• A. $V^3=\alpha T^3$
• B. $V^3=\alpha T^2$
• C. $V=\alpha T$
• D. $V^3=\alpha T$

Answer 1

The correct option is D $V^3=\alpha T$

Given that,
$C=C_V+\frac{\alpha }{T}.......\left(1\right)$
Also, from first law of thermodynamics we know that
$C=C_V+\frac{P}{n}\frac{dV}{dT}.......\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, we can say that
$\frac{\alpha }{T}=\frac{P}{n}\frac{dV}{dT}$
$\Rightarrow \frac{\alpha }{T}=\frac{nRT}{V\times n}\frac{dV}{dT}$
$\Rightarrow \frac{dV}{dT}=\frac{\alpha V}{R{T}^2}$ $\Rightarrow \int \frac{dV}{V}=\int \frac{\alpha }{R}{T}^{-2}dT\Rightarrow \ln V=\frac{-\alpha }{R}{T}^{-1}+\ln V_0$
$\Rightarrow \left(\frac{V}{V_0}\right)=e^{\frac{-\alpha }{RT}}$
Thus, option (d) is the correct answer.