P$5\mathrm{.}17$ To generate hydrogen for fuel cells, steam reforming of methane is to be conducted at in a steady$-$state continuous$-$flow reactor. For the reaction

$C{H}_4+H_{2}OCO+3H_2$

$K_a=5\mathrm{.}2$ at $600C,$ with pressure in atm. Assume that the reactor feed ratio is $5$:$1($moles:moles$)H_{2}O$:$C{H}_4$ and that the reactor operates at $2$atm. Derive an equation that relates $K_a$ to the fractional conversion of methane. Then determine whether the equilibrium conversion of methane is greater than or less than $90\%.$