$15\mathrm{.}37$ A mixture of $0\mathrm{.}2000$mol of $C{O}_{2,0},1000$mol of $H_2$ and $0\mathrm{.}1600$mol of $H_{2}O$ is placed in a $2\mathrm{.}000-L$ vessel, The following equilibrium is established at $500K$ :

$C{O}_2(g)+H_2(g)CO(g)+H_{2}O(g)$

$($a$)$ Calculate the initial partial pressures of $C{O}_2,H_2,$ and $H_{2}O.$

$($b$)$ At equilibrium $P_{H_{2}O}=3\mathrm{.}51$atm. Calculate the equilibrium partial pressures of $C{O}_2,H_2,$ and $CO.($c$)$ Calculate $K_p$ for the reaction. $($d$)$ Calculate $K_c$ for the reaction.