Answer
$$\frac{[SO_2]}{[SO_3]} = 0.516$$
Work Step by Step
1. Write the $K_c$ expression:
$$K_c = \frac{[SO_3]^2}{[SO_2]^2[O_2]} = 281$$
2. Calculate the molarity of oxygen gas:
$$\frac{0.00247 \space mol \space O_2}{0.185 \space L} = 0.0134 \space M \space O_2$$
3. Substitute into the equation and solve for $\frac{[SO_2]}{[SO_3]}$
$$281 = \frac{[SO_3]^2}{[SO_2]^2(0.0134)}$$ $$\frac{[SO_2]^2}{[SO_3]^2} = \frac{1}{281(0.0134)}$$ $$\frac{[SO_2]^2}{[SO_3]^2} = 0.266$$ $$\frac{[SO_2]}{[SO_3]} = \sqrt{0.266} = 0.516$$
