Q75 / 90 · JEE Main 2010 · Physics
A diatomic ideal gas is used in a Car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from $\mathrm{V}$ to $32 \mathrm{~V}$ the efficiency of the engine is
- A$0.5$
- Correct answerB$0.75$
- C$0.99$
- D$0.25$
Solution
The efficiency of cycle is
$$
\eta=1-\frac{T_2}{T_1}
$$
for adiabatic process
$$
\text { TV }^{r^{-1}}=\text { constant }
$$
For diatomic gas $\gamma=\frac{7}{5}$
$$
\begin{aligned}
& \mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1} \\
& \mathrm{~T}_1=\mathrm{T}_2\left(\frac{\mathrm{V}_2}{\mathrm{~V}_1}\right)^{\gamma-1} \\
& \mathrm{~T}_1=\mathrm{T}_2(32)^{\frac{7}{5}-1} \\
&=\mathrm{T}_2\left(2^5\right)^{2 / 5} \\
&=\mathrm{T}_2 \times 4 \\
& \mathrm{~T}_1=4 \mathrm{~T}_2 . \\
& \eta=\left(1-\frac{1}{4}\right)=\frac{3}{4}=0.75
\end{aligned}
$$
Hence, the correct answer is option: 'B'.