For the following three reaction 1, 2 and 3, equilibrium constants are given:
(1) `CO_((g)) + H_(2)O_((g))hArrCO_(2(g)) + H_(2(g)), K_(1)`
(2) `CH_(4(g)) + H_(2)O_((g))hArrCO_((g)) + 3H_(2(g)), K_(2)`
(3) `CH_(4(g)) + 2H_(2)O_((g))hArrCO_(2(g)) + 4H_(2(g)), K_(3)`
Which of the following relations is correct ?

Options

A. `K_(1)sqrt(K_2) = K_(3)`
B. `K_2K_3=K_1`
C. `K_3 = K_1K_2`
D. `K_3K_2^3 = K_1^2`

Answer

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

### Step-by-Step Solution:

1. **Identify the Reactions and Their Equilibrium Constants**:
- Reaction 1: \( CO_{(g)} + H_2O_{(g)} \rightleftharpoons CO_2_{(g)} + H_2_{(g)} \) with equilibrium constant \( K_1 \).
- Reaction 2: \( CH_4_{(g)} + H_2O_{(g)} \rightleftharpoons CO_{(g)} + 3H_2_{(g)} \) with equilibrium constant \( K_2 \).
- Reaction 3: \( CH_4_{(g)} + 2H_2O_{(g)} \rightleftharpoons CO_2_{(g)} + 4H_2_{(g)} \) with equilibrium constant \( K_3 \).

2. **Add Reactions 1 and 2**:
- When we add Reaction 1 and Reaction 2, we need to ensure that the products and reactants combine correctly.
- Adding Reaction 1 and Reaction 2:
\[ (CO_{(g)} + H_2O_{(g)} \rightleftharpoons CO_2_{(g)} + H_2_{(g)}) \\ + (CH_4_{(g)} + H_2O_{(g)} \rightleftharpoons CO_{(g)} + 3H_2_{(g)}) \]
- This results in:
\[ CH_4_{(g)} + 2H_2O_{(g)} \rightleftharpoons CO_2_{(g)} + 4H_2_{(g)} \]
- This is exactly Reaction 3.

3. **Relate the Equilibrium Constants**:
- According to the principle of equilibrium, when two reactions are added, the equilibrium constant of the resultant reaction is the product of the equilibrium constants of the individual reactions:
\[ K_3 = K_1 \times K_2 \]

4. **Conclusion**:
- Therefore, the correct relation among the equilibrium constants is:
\[ K_3 = K_1 \times K_2 \]

### Final Answer:
The correct relation is \( K_3 = K_1 \times K_2 \).

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