For the reaction [ mathrm{CO}_{2}(mathrm{~g})+mathrm{H}_{2}(mathrm{~g}) ightarrow mathrm{CO}(mathrm{g})+mathrm{H}_{2} mathrm{O}(mathrm{g}) ] at (298 mathrm{~K}) is (8.685 times 10^{-6}).

For the reaction

\[ \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \]

at \(298 \mathrm{~K}\) is \(8.685 \times 10^{-6}\). Estimate the value of \(K_{a}\) at \(1000 \mathrm{~K}\), assuming that \(\Delta H^{0}\) is constant in the temperature range of \(298 \mathrm{~K}\) to \(1000 \mathrm{~K}\). The data given is:

\begin{tabular}{lc}

Component & \(\Delta H_{f}^{0}(\mathrm{~kJ} /\) mole \()\) \\

\(\mathrm{CO}(\mathrm{g})\) & 110.532 \\

\(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) & -241.997 \\

\(\mathrm{CO}_{2}(\mathrm{~g})\) & -393.978 \\

\(\mathrm{H}_{2}(\mathrm{~g})\) & 0

\end{tabular}

We are given that van't Hoff's equation

\[ \left(\frac{\partial \ln K_{a}}{\partial T}\right)=\frac{\Delta H^{0}}{R T^{2}} \]

where \(\Delta H^{0}\) is the standard heat of reaction.