Question

A given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be $0.75$ of its initial pressure. The ratio of the specific heats of the gas is approximately

• A. 1.20
• B. 1.41
• C. 1.67
• D. 1.83

Answer 1

Answer: (B)

In isothermal process, temperature of the gas remains constant, so the gas obeys Boyle's law. That is,
$p \propto \frac{1}{V}$
$\Rightarrow \frac{p_{2}}{p_{1}}=\frac{V_{1}}{V_{2}}$
$\Rightarrow \frac{2 p}{p}=\frac{V_{1}}{V_{2}}$
$\therefore \frac{V_{1}}{V_{2}}=2$ ...(i)
Now, the gas is expanded adiabatically, so
$P^{V^{\gamma}}=$ constant
$\frac{p_{1}}{p_{2}}=\left(\frac{V_{2}}{V_{1}}\right)^{\gamma}$
$\Rightarrow \frac{2 p}{0.75 p}=\left(\frac{2}{1}\right)^{\gamma}$
(since volume is restored)
$\Rightarrow \log \left(\frac{8}{3}\right)=\gamma \log 2$
$\Rightarrow \log 8-\log 3=\gamma \log 2$
$\therefore \gamma=1.41$