For the following three reactions (i),(ii) and (iii) equlibrium constant are given.
(i) ` CO(g) +H_2O (g) hArr CO_2( g)+ H_2(g) ,K_1`
(ii) ` CH_4(g) + H_2O (g) hArr CO(g) +3H_2(g) , K_2`
(iii)` CH_4(g) + 2H_2O(g) hArr CO_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_3.K_2^(3) = K_1^(2)`

Answer

To solve the problem, we need to analyze the given chemical reactions and their equilibrium constants. We will derive the relationship between \( K_1 \), \( K_2 \), and \( K_3 \) based on the reactions provided.

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

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

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

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

### Step 2: Relate \( K_3 \) to \( K_1 \) and \( K_2 \)

We need to express \( K_3 \) in terms of \( K_1 \) and \( K_2 \). We can assume that:
\[ K_3 = K_1^a \cdot K_2^b \]

### Step 3: Substitute the expressions for \( K_1 \) and \( K_2 \)

Substituting the expressions for \( K_1 \) and \( K_2 \) into the equation:
\[ K_3 = \left(\frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]}\right)^a \cdot \left(\frac{[\text{CO}][\text{H}_2]^3}{[\text{CH}_4][\text{H}_2\text{O}]}\right)^b \]

### Step 4: Rearranging the equation

Rearranging gives:
\[ K_3 = \frac{[\text{CO}_2]^a [\text{H}_2]^{a + 3b}}{[\text{CO}]^b [\text{H}_2\text{O}]^{a + b} [\text{CH}_4]^b} \]

### Step 5: Compare the coefficients

Now we compare the coefficients of \( K_3 \) with the expression derived from the equilibrium constant:
\[ K_3 = \frac{[\text{CO}_2][\text{H}_2]^4}{[\text{CH}_4][\text{H}_2\text{O}]^2} \]

From this, we can set up equations based on the exponents of each species:

1. For \( \text{CO}_2 \): \( a = 1 \)
2. For \( \text{H}_2 \): \( a + 3b = 4 \)
3. For \( \text{CH}_4 \): \( b = 1 \)
4. For \( \text{H}_2\text{O} \): \( a + b = 2 \)

### Step 6: Solve the equations

From \( b = 1 \):
- Substituting into \( a + 3b = 4 \):
\[ a + 3(1) = 4 \implies a = 1 \]
- Check \( a + b = 2 \):
\[ 1 + 1 = 2 \quad \text{(True)} \]

### Conclusion

Thus, we find that:
\[ K_3 = K_1 \cdot K_2 \]

### Final Answer
The correct relationship is:
\[ K_3 = K_1 \cdot K_2 \]