Hydrogen is used in a Carnot cycle as a working substance. As a result of an adiabatic expansion the gas volume becomes 2 times. Let the efficiency of the cycle be $\mathrm{\eta }$. Find the value of $4\mathrm{\eta }$ to the nearest integer.

For ${\mathrm{H}}_2,\mathrm{\gamma }=7/5$
$\begin{array}{r}{\mathrm{p}}_1{\mathrm{V}}_1={\mathrm{p}}_2{\mathrm{V}}_2,{\mathrm{P}}_3{\mathrm{V}}_3={\mathrm{p}}_4{\mathrm{V}}_4\\ {\mathrm{p}}_2{\mathrm{V}}_2^{\mathrm{\gamma }}={\mathrm{p}}_3{\mathrm{V}}_3^{\mathrm{\gamma }},{\mathrm{P}}_1{\mathrm{V}}_1^{\mathrm{\gamma }}={\mathrm{p}}_4{\mathrm{V}}_4^{\mathrm{\gamma }}\end{array}$
Define n by ${\mathrm{V}}_3={\mathrm{nV}}_2$
Then ${\mathrm{p}}_3={\mathrm{p}}_2{\mathrm{n}}^{-\gamma },\mathrm{so}$
$\begin{array}{l}{\mathrm{p}}_4{\mathrm{V}}_4={\mathrm{p}}_3{\mathrm{V}}_3={\mathrm{p}}_2{\mathrm{V}}_2{\mathrm{n}}^{1-\mathrm{\gamma }}={\mathrm{p}}_1{\mathrm{V}}_1{\mathrm{n}}^{1-\mathrm{\gamma }}\\ {\mathrm{p}}_4{\mathrm{V}}_4^{\mathrm{\gamma }}={\mathrm{p}}_1{\mathrm{V}}_1^{\mathrm{\gamma }}\end{array}$
so, ${\mathrm{V}}_4^{1-\mathrm{\gamma }}={\mathrm{V}}_1^{1-\mathrm{\gamma }}{\mathrm{n}}^{1-\mathrm{\gamma }}$
or ${\mathrm{V}}_4={\mathrm{nV}}_1$
Also,
$\begin{array}{l}{\mathrm{Q}}_1={\mathrm{p}}_2{\mathrm{V}}_2\ln \frac{{\mathrm{V}}_2}{{\mathrm{V}}_1},{\mathrm{Q}}_2^{\mathrm{\prime }}={\mathrm{p}}_3{\mathrm{V}}_3\ln \frac{{\mathrm{V}}_3}{{\mathrm{V}}_4}={\mathrm{n}}^{1-\mathrm{\gamma }}{\mathrm{p}}_2{\mathrm{V}}_2\ln \frac{{\mathrm{V}}_3}{{\mathrm{V}}_4}\\ \mathrm{\eta }=1-\frac{{\mathrm{Q}}_2}{{\mathrm{Q}}_1}=1-{\mathrm{n}}^{1-\mathrm{\gamma }}=0.242\end{array}$