Question

Consider an ideal gas having adiabatic exponent $\gamma$. In some process, its molar heat capacity varies with temperature as $C=\frac{\alpha }{T}$ where $\alpha$ is a constant. The work performed by one mole of the gas during its heating from the temperature of $T_0$ to $\eta T_0(\eta >1)$ is

• A. $\alpha ln\left[\eta \right]-\frac{R{T}_{\circ }(\eta -1)}{(\gamma -1)}$
• B. $2\alpha ln\left[\eta \right]-\frac{R{T}_{\circ }(\eta -1)}{(\gamma -1)}$
• C. $\alpha$ $ln\left[\eta \right]$
• D. $\frac{-R{T}_{\circ }(\eta -1)}{(\gamma -1)}$

Answer 1

The correct option is A $\alpha ln\left[\eta \right]-\frac{R{T}_{\circ }(\eta -1)}{(\gamma -1)}$

Heat required to raise the temperature of one mole of gas from $T_{\circ }$ to $\eta T_{\circ }$ is
$△Q=\int Cdt={\int }_{T_{\circ }}^{\eta T_{\circ }}\frac{\alpha }{T}dt=\alpha ln\left[\eta \right]$

Increase in the internal energy of one mole of gas is,
$△U=C_v[\eta T_{\circ }-T_{\circ }]=\frac{R{T}_{\circ }}{\gamma -1}[\eta -1]$
Work done, from first law of thermodynamics,
$△W=△Q-△U=\alpha ln\left(\eta \right)-\frac{R{T}_{\circ }[\eta -1]}{(\gamma -1)}$