The catalytic reaction of carbon monoxide with hydrogen was conducted in a gas phase for the production of methanol according to following reaction:

$C{O}_{(g)}+2H_{2(g)}C{H}_{3}OH(s)$

The equilibrium conversion to methanol is large at $300K,$ but it decreases rapidly with increasing temperature. However, reaction rates become appreciable only at higher temperatures. For a feed mixture of carbon monoxide and hydrogen in the stoichiometric proportions, answer the followings:

$($a$)$ What is the equilibrium mole fractions of reactants and products at $1$ bar and $300K$ by assuming ideal gas?

$($b$)$ What is the equilibrium mole fractions of reactants and products at $1$ bar and $300K$ by assuming ideal solution of gases?

$($c$)$ At what temperature does the equilibrium mole fraction of methanol equal $0\mathrm{.}50$ for a pressure of $100$ bar, assuming the equilibrium mixture is an ideal gas?