A cylinder contains \(1 \mathrm{~kg}\) carbon dioxide, and this is compressed adiabatically. Show the pressure, temperature and specific volume are related by the equation

\[\frac{1-\alpha}{\alpha} \sqrt{\frac{(2+\alpha)}{\alpha}}=K_{p_{\mathrm{r}}} \sqrt{\frac{p}{1.01325}}\]

and

\[p v=\frac{\Re(2+\alpha) T}{88},\]

where \(\alpha=\) degree of dissociation.

Note that the equilibrium constant for the reaction \(\mathrm{CO}+\frac{1}{2} \mathrm{O}_{2} \Leftrightarrow \mathrm{CO}_{2}\) is given by \(K_{p_{\mathrm{r}}}=\frac{\left(p_{\mathrm{CO}_{2}} / p_{0}\right)}{\left(p_{\mathrm{CO}} / p_{0}\right)\left(p_{\mathrm{O}_{2}} / p_{0}\right)^{1 / 2}}\), where \(p_{0}\) is the datum pressure of \(1 \mathrm{~atm}\).