If the equilibrium constant of the following equilibrium `SO_(2)+1//2O_(2) Leftrightarrow SO_(3) and 2SO_(3) Leftrightarrow 2SO_(2)+O_(2)` are given by `K_1 and K_2` respectively, which of the following relations is correct?

Options

A. `K_(2)=(1/K_(1))^2`
B. `K_(1)=(1/K_(2))^3`
C. `K_(2)=(1/K_(1))`
D. `K_2=(K_1)^2`

Answer

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

### Step-by-Step Solution:

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

2. **Reverse the First Reaction:**
- When we reverse the first reaction, we get:
\[ SO_3 \leftrightarrow SO_2 + \frac{1}{2} O_2 \]
- The equilibrium constant for the reversed reaction is given by:
\[ K' = \frac{1}{K_1} \]

3. **Adjust the Reversed Reaction:**
- Now, we need to multiply the reversed reaction by 2 to match the second reaction:
\[ 2SO_3 \leftrightarrow 2SO_2 + O_2 \]
- When we multiply the equilibrium expression by 2, the new equilibrium constant becomes:
\[ K_2 = (K')^2 = \left(\frac{1}{K_1}\right)^2 = \frac{1}{K_1^2} \]

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

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