The production of sulfur trioxide $($gas$)$ is carried out by the reaction of sulfur dioxide $($gas$)$ and oxygen gas.

The reactor is fed with $2$ moles of pure sulfur dioxide and a corresponding stoichiometric amount of air $($assume that air is $21\%$ mole oxygen gas and $79\%$ mole nitrogen gas$).$ The reaction is carried out at $P=1$ bar and at a temperature of $450C.$ Note that the heat of formation and Gibbs free energy of formation of elements are equal to zero.

a$.$ Determine the mole fractions of each component at the product stream at equilibrium. Show your complete solution, state assumptions and justifications of assumptions. $16$ points

b$.$ If vanadium oxide is added as a catalyst in the reactor, how does the equilibrium constant change $($comparing with the condition in $($a$))?$ Justify your answer. $2$ pts

c$.$ If the reaction is carried out at $P=5$ bar, how does the reaction coordinate at equilibrium change compared to the condition in $($a$)?$ Justify your answer. $2$ pts