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,
$T{V}^{\gamma -1}=$ 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, $\eta =1-\frac{T_2}{T_1}$
$=1-\frac{1}{4}\left(Using\left(i\right)\right)$
$=\frac{3}{4}=0.75$