1. Given reaction and equilibrium constant

The given reaction is:

$SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2}O_2(g)$

with

$K_c = 4.9 \times 10^{-2}$

2. Required reaction

We need the value of $K_c$ for:

$2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$

3. Relate the required reaction to the given reaction

First, reverse the given reaction:

$SO_2(g) + \frac{1}{2}O_2(g) \rightleftharpoons SO_3(g)$

When a reaction is reversed, the equilibrium constant becomes the reciprocal:

$K_c' = \frac{1}{4.9 \times 10^{-2}}$

Now multiply the entire reaction by 2:

$2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$

When the coefficients are multiplied by 2, the equilibrium constant is squared:

$K_c'' = \left(\frac{1}{4.9 \times 10^{-2}}\right)^2$

4. Calculation

$\frac{1}{4.9 \times 10^{-2}} = \frac{1}{0.049} \approx 20.408$

So,

$K_c'' = (20.408)^2 \approx 416.5$

Hence,

$K_c \approx 416$

5. Option check

• A: $416$ ✅
• B: $9.8 \times 10^{-2}$ ❌
• C: $4.9 \times 10^{-2}$ ❌
• D: $2.40 \times 10^{-3}$ ❌

Therefore, the correct answer is A.