Question

A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle, the volume of the gas increases from V to 32V, the efficiency of the engine is

• A. $0.75$
• B. $0.99$
• C. $0.25$
• D. $0.5$

Answer 1

Answer: (A)

Let $T_1$ be the temperature of diatomic gas at volume $V$ and $T_2$ is the temperature at volume $32V$. For an adiabatic process,
$TV^{γ−1} =$ constant
$\therefore T_{1}V_{1}^{\gamma-1}=T_{2}V_{2}^{\gamma-1}$
or $\frac{T_{2}}{T_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma-1}=\left(\frac{V}{32V}\right)^{\gamma-1}=\frac{1}{\left(32\right)^{\gamma-1}}$
For diatomic gas, $\gamma=\frac{7}{5}$
$\therefore \frac{T_{2}}{T_{1}}-\frac{1}{\left(32\right)^{\frac{7}{5}-1}}=\frac{1}{\left(2^{5}\right)^{\frac{7}{5}-1}}=\frac{1}{\left(2^{5}\right)^{2/5}}=\frac{1}{4}\,...\left(i\right)$
Efficiency of engine, $\eta=1-\frac{T_{2}}{T_{1}}$
$=1-\frac{1}{4}\,\left(Using \left(i\right)\right)$
$=\frac{3}{4}=0.75$