consider the following gaseous equilibrium with equilibrium
constant `K_(1)` and `K_(2)` respectively
`SO_(3)(g)hArrSO_(2)(g)+ 1//2O_(2)`
`2SO_(3)(g)hArr2SO_(2)(g)+O_(2)(g)`
The equilibrium constants are related as

Options

A. `K_(2)=K_(1)^(2)`
B. `K_(2)=(1)/(K_(1)^(2))`
C. `K_(2)^(2)=(1)/(K_(1))`
D. `K_(2)=(2)/(K_(1)^(2))`

Answer

To solve the problem, we need to analyze the two given reactions and their equilibrium constants, \( K_1 \) and \( K_2 \).

### Step-by-Step Solution:

1. **Identify the Reactions and Their Equilibrium Constants:**
- The first reaction is:
\[ SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \quad (K_1) \]
- The second reaction is:
\[ 2SO_3(g) \rightleftharpoons 2SO_2(g) + O_2(g) \quad (K_2) \]

2. **Write the Expression for the Equilibrium Constants:**
- For the first reaction:
\[ K_1 = \frac{[SO_2]^1 [O_2]^{1/2}}{[SO_3]^1} \]
- For the second reaction:
\[ K_2 = \frac{[SO_2]^2 [O_2]^1}{[SO_3]^2} \]

3. **Relate the Two Reactions:**
- Notice that the second reaction can be derived from the first reaction by multiplying the entire first reaction by 2. This means we can express \( K_2 \) in terms of \( K_1 \).
- When we multiply the first reaction by 2, the stoichiometric coefficients double, which affects the equilibrium constant as follows:
\[ K_1 \text{ becomes } K_1^2 \]

4. **Establish the Relationship Between \( K_1 \) and \( K_2 \):**
- Since the second reaction is essentially the first reaction multiplied by 2, we have:
\[ K_2 = K_1^2 \]

5. **Conclusion:**
- The relationship between the equilibrium constants is:
\[ K_2 = K_1^2 \]

### Final Answer:
The equilibrium constants are related as:
\[ K_2 = K_1^2 \]

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