Experimental equilibrium data for the adsorption of methane on activated carbon is follows Langmuir isotherm. The equilibrium constant for adsorption is measured to be $2\mathrm{.}045{10}^{-3}(kPa{)}^{-1}$ at $296K$ and $1\mathrm{.}888{10}^{-4}(kPa{)}^{-1}$ at $480K.$ The maximum amount of methane that can be adsorbed by $1kg$ of activated carbon at $373K$ is $4\mathrm{.}96$mol. Note that gas mixture behaves ideally and the ideal gas constant $R=1\mathrm{.}987ca\frac{l}{\text{m}\text{o}\text{l}*K}.$

$($a$)$ What is the heat of adsorption of methane on activated carbon $($in cal$/$mol$)?$

$($b$)$ What is the equilibrium constant at $373K?$

$($c$)$ Given that the feed gas contains $50$mol$\%$ of methane, determine the amount of methane $($in mol$)$ that can be adsorbed by $10kg$ of activated carbon at $373K$ and $20$ Bar $(2,000$kPa$).$