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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Reactions and Their Equilibrium Constants:**
- The first reaction is:
\[ \text{SO}_2(g) + \frac{1}{2} \text{O}_2(g) \rightleftharpoons \text{SO}_3(g) \]
The equilibrium constant for this reaction is \( K_1 \).
- The second reaction is:
\[ 2 \text{SO}_3(g) \rightleftharpoons 2 \text{SO}_2(g) + \text{O}_2(g) \]
The equilibrium constant for this reaction is \( K_2 \).

2. **Write the Expression for \( K_1 \):**
- For the first reaction, the equilibrium constant \( K_1 \) can be expressed as:
\[ K_1 = \frac{[\text{SO}_3]}{[\text{SO}_2][\text{O}_2]^{1/2}} \]

3. **Write the Expression for \( K_2 \):**
- For the second reaction, the equilibrium constant \( K_2 \) can be expressed as:
\[ K_2 = \frac{[\text{SO}_2]^2[\text{O}_2]}{[\text{SO}_3]^2} \]

4. **Relate \( K_1 \) and \( K_2 \):**
- To relate \( K_1 \) and \( K_2 \), we can manipulate the expression for \( K_2 \):
- Rearranging \( K_2 \):
\[ K_2 = \frac{[\text{SO}_2]^2[\text{O}_2]}{[\text{SO}_3]^2} \]
- Now, notice that if we square \( K_1 \), we get:
\[ K_1^2 = \left(\frac{[\text{SO}_3]}{[\text{SO}_2][\text{O}_2]^{1/2}}\right)^2 = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2[\text{O}_2]} \]
- Taking the reciprocal of \( K_1^2 \):
\[ \frac{1}{K_1^2} = \frac{[\text{SO}_2]^2[\text{O}_2]}{[\text{SO}_3]^2} = K_2 \]

5. **Final Relationship:**
- From the above manipulations, we conclude that:
\[ K_2 = \frac{1}{K_1^2} \]

### Conclusion:
The relationship between the equilibrium constants \( K_1 \) and \( K_2 \) is:
\[ K_2 = \frac{1}{K_1^2} \]