For the following reactions `(1) , (2)` and `(3)` equilibrium constants are given
`(1) CO_(2)(g) +H_(2)O(g) hArr CO_(2)(g) +H_(2)(g), K_(1)`
`(2) CH_(4)(g) +H_(2)O(g) hArr CO(g)+3H_(2)(g),K_(2)`
`(3) CH_(4)(g)+2H_(2)O(g) hArr CO_(2)(g)+4H_(2)(g), K_(3)`
Which of the following relation is correct ?

Options

A. `K_(3)=K_(1) xx K_(2)`
B. `K_(3)K_(2)^(3)=K_(1)^(2)`
C. `K_(1) sqrt(K_(2))=K_(3)`
D. `K_(2)K_(3)=K_(1)`

Explanation

To solve the problem, we need to analyze the given reactions and their equilibrium constants.

### Step 1: Write the expressions for the equilibrium constants

1. For reaction (1):
\[ \text{CO}_2(g) + \text{H}_2O(g) \rightleftharpoons \text{CO}(g) + \text{H}_2(g) \]
The equilibrium constant \( K_1 \) is given by:
\[ K_1 = \frac{[\text{CO}][\text{H}_2]}{[\text{CO}_2][\text{H}_2O]} \]

2. For reaction (2):
\[ \text{CH}_4(g) + \text{H}_2O(g) \rightleftharpoons \text{CO}(g) + 3\text{H}_2(g) \]
The equilibrium constant \( K_2 \) is given by:
\[ K_2 = \frac{[\text{CO}][\text{H}_2]^3}{[\text{CH}_4][\text{H}_2O]} \]

3. For reaction (3):
\[ \text{CH}_4(g) + 2\text{H}_2O(g) \rightleftharpoons \text{CO}_2(g) + 4\text{H}_2(g) \]
The equilibrium constant \( K_3 \) is given by:
\[ K_3 = \frac{[\text{CO}_2][\text{H}_2]^4}{[\text{CH}_4][\text{H}_2O]^2} \]

### Step 2: Relate the reactions to find \( K_3 \)

To find the relationship between \( K_1 \), \( K_2 \), and \( K_3 \), we can express \( K_3 \) in terms of \( K_1 \) and \( K_2 \).

- We can combine reactions (1) and (2) to derive reaction (3):

1. **Reverse reaction (1)**:
\[ \text{CO}(g) + \text{H}_2(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2O(g) \]
The equilibrium constant for the reverse reaction is:
\[ K_1' = \frac{1}{K_1} \]

2. **Add the reversed reaction (1) to reaction (2)**:
\[ \text{CO}(g) + \text{H}_2(g) + \text{CH}_4(g) + \text{H}_2O(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2O(g) + 3\text{H}_2(g) \]
This simplifies to:
\[ \text{CH}_4(g) + 2\text{H}_2O(g) \rightleftharpoons \text{CO}_2(g) + 4\text{H}_2(g) \]
The equilibrium constant for this combined reaction is:
\[ K_3 = K_1' \cdot K_2 = \frac{1}{K_1} \cdot K_2 \]

### Step 3: Conclusion

Thus, we can express \( K_3 \) as:
\[ K_3 = \frac{K_2}{K_1} \]

### Final Relation
The correct relation is:
\[ K_3 = \frac{K_2}{K_1} \]