$13\mathrm{.}12$ One of the current commercially important processes to produce acetone $(C_3{H}_{6}O)$ is the catalytic dehydrogenation of isopropanol $(C_3{H}_{8}O)$ in the vapor phase at $1$ bar according to the reaction

$C_3{H}_{8}O(g)=C_3{H}_{6}O(g)+H_2(g).$

$($a$)$ Using the procedure outlined in Section $13\mathrm{.}4\mathrm{.}3,$ estimate $tilde(G{)}_f^o$ for isopropanol and acetone at $400,450,500,550,600,$ and $650K.$

$($b$)$ Calculate $K_a$ at these temperatures.

$($c$)$ Plot $ln{K}_a$ versus $\frac{1}{T}$ and show that

$ln{K}_a=-\frac{6823\mathrm{.}7}{T}+14\mathrm{.}559.$

$($d$)$ Neglecting other reactions, estimate the minimum temperature for $95\%$ conversion of isopropanol to acetone.

$($Answer: $($d$)553\mathrm{.}3K)$