Suppose the gas-phase isomerisation reactions $$A \rightleftharpoons B$$ ($$K_1$$); $$A \rightleftharpoons C$$ ($$K_2$$); $$B \rightleftharpoons C$$ ($$K_3$$) reach equilibrium at a fixed temperature. Express the equilibrium mole fractions of A, B and C in terms of equilibrium constants, $$K_1$$, $$K_2$$ and $$K_3$$.

PLACEHOLDER ANSWER - the answer is three expressions. Since the number of moles does not change, $$K_1 = \frac{x_B}{x_A}$$ and $$K_2 = \frac{x_C}{x_A}$$, with $$K_3 = \frac{x_C}{x_B} = \frac{K_2}{K_1}$$. Using $$x_A + x_B + x_C = 1$$, $$x_A = \frac{1}{1 + K_1 + K_2}$$, $$x_B = \frac{K_1}{1 + K_1 + K_2}$$ and $$x_C = \frac{K_2}{1 + K_1 + K_2}$$. In terms of $$K_1$$ and $$K_3$$ these are $$\frac{1}{1 + K_1 + K_1 K_3}$$, $$\frac{K_1}{1 + K_1 + K_1 K_3}$$ and $$\frac{K_1 K_3}{1 + K_1 + K_1 K_3}$$.