Equilibrium constant for the reactions,
`2 NO+O_(2)hArr2 NO_(2) "is" K_(c_(1)`,
`NO_(2)+SO_(2)hArrSO_(3)+NO "is" L_(C_(2)` and
`2 SO_(3)hArr2 SO_(2)+O_(2) "is" K_(c_(3)` then correct reaction is:

Options

A. `K_(C_(3))=K_(C_(1))xxK_(C)(2)`
B. `K_(C_(3))xxK_(C_1)xxK_(C_(2))^(2)=1`
C. `K_(C_(3))xxK_(C_(1))xxK_(C_2)=1`
D. `K_(C_(3))xxK_(C_(1))^(2)xxK_(C_(2)=1`

Answer

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

### Step 1: Write down the reactions and their equilibrium constants.
1. **Reaction 1:**
\[ 2 \text{NO} + \text{O}_2 \rightleftharpoons 2 \text{NO}_2 \quad (K_{c1}) \]
The equilibrium constant \( K_{c1} \) is given by:
\[ K_{c1} = \frac{[\text{NO}_2]^2}{[\text{NO}]^2 [\text{O}_2]} \]

2. **Reaction 2:**
\[ \text{NO}_2 + \text{SO}_2 \rightleftharpoons \text{SO}_3 + \text{NO} \quad (K_{c2}) \]
The equilibrium constant \( K_{c2} \) is given by:
\[ K_{c2} = \frac{[\text{SO}_3][\text{NO}]}{[\text{NO}_2][\text{SO}_2]} \]

3. **Reaction 3:**
\[ 2 \text{SO}_3 \rightleftharpoons 2 \text{SO}_2 + \text{O}_2 \quad (K_{c3}) \]
The equilibrium constant \( K_{c3} \) is given by:
\[ K_{c3} = \frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2} \]

### Step 2: Analyze the options.
We need to check which of the given options is correct based on the relationships between the equilibrium constants.

#### Option A: \( K_{c3} = K_{c1} \times K_{c2} \)
- Calculate \( K_{c1} \times K_{c2} \):
\[ K_{c1} \times K_{c2} = \left(\frac{[\text{NO}_2]^2}{[\text{NO}]^2 [\text{O}_2]}\right) \times \left(\frac{[\text{SO}_3][\text{NO}]}{[\text{NO}_2][\text{SO}_2]}\right) \]
After simplification, we find that this does not equal \( K_{c3} \).

#### Option B: \( K_{c3} \times K_{c1} \times K_{c2}^2 = 1 \)
- Calculate \( K_{c3} \times K_{c1} \times K_{c2}^2 \):
\[ K_{c3} \times K_{c1} \times K_{c2}^2 = \left(\frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2}\right) \times \left(\frac{[\text{NO}_2]^2}{[\text{NO}]^2 [\text{O}_2]}\right) \times \left(\frac{[\text{SO}_3]^2 [\text{NO}]^2}{[\text{NO}_2]^2 [\text{SO}_2]}\right) \]
After simplification, all terms cancel out, resulting in 1. Therefore, Option B is correct.

#### Option C: \( K_{c3} \times K_{c1} \times K_{c2} = 1 \)
- Calculate \( K_{c3} \times K_{c1} \times K_{c2} \):
\[ K_{c3} \times K_{c1} \times K_{c2} = \left(\frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2}\right) \times \left(\frac{[\text{NO}_2]^2}{[\text{NO}]^2 [\text{O}_2]}\right) \times \left(\frac{[\text{SO}_3][\text{NO}]}{[\text{NO}_2][\text{SO}_2]}\right) \]
After simplification, this does not equal 1.

#### Option D: \( K_{c3} \times K_{c1}^2 \times K_{c2} = 1 \)
- Calculate \( K_{c3} \times K_{c1}^2 \times K_{c2} \):
\[ K_{c3} \times K_{c1}^2 \times K_{c2} = \left(\frac{[\text{SO}_2]^2 [\text{O}_2]}{[\text{SO}_3]^2}\right) \times \left(\frac{[\text{NO}_2]^4}{[\text{NO}]^4 [\text{O}_2]^2}\right) \times \left(\frac{[\text{SO}_3][\text{NO}]}{[\text{NO}_2][\text{SO}_2]}\right) \]
After simplification, this does not equal 1.

### Conclusion
The only correct option is **B**: \( K_{c3} \times K_{c1} \times K_{c2}^2 = 1 \).