2 moles each of `SO_(3)`, CO, `SO_(2)` and `CO_(2)` is taken in a 1 L vessel. If `K_(C)` for `SO_(3) + COhArrSO_(2) + CO_(2)` is `1//9` then:

Options

A. (A) total no. of moles at equilibrium are less than `8`
B. (B) `n(SO_(3))+n(CO_(2))=4`
C. (C) `[n(SO_(2))//n(CO)]lt1`
D. (D) both (B) and (C).

Explanation

To solve the problem step by step, we will analyze the equilibrium reaction and the given conditions:

### Step 1: Write the balanced chemical equation
The equilibrium reaction given is:
\[ SO_3(g) + CO(g) \rightleftharpoons SO_2(g) + CO_2(g) \]

### Step 2: Identify initial moles
According to the problem, we start with 2 moles of each gas in a 1 L vessel:
- \( SO_3 = 2 \, \text{moles} \)
- \( CO = 2 \, \text{moles} \)
- \( SO_2 = 2 \, \text{moles} \)
- \( CO_2 = 2 \, \text{moles} \)

### Step 3: Set up the expression for equilibrium constant \( K_C \)
The equilibrium constant \( K_C \) is given by:
\[ K_C = \frac{[SO_2][CO_2]}{[SO_3][CO]} \]
Given \( K_C = \frac{1}{9} \).

### Step 4: Define changes in concentration at equilibrium
Let \( x \) be the change in moles of \( SO_3 \) and \( CO \) that react to form \( SO_2 \) and \( CO_2 \). At equilibrium, the moles will be:
- \( [SO_3] = 2 - x \)
- \( [CO] = 2 - x \)
- \( [SO_2] = 2 + x \)
- \( [CO_2] = 2 + x \)

### Step 5: Substitute equilibrium concentrations into the \( K_C \) expression
Substituting the equilibrium concentrations into the \( K_C \) expression:
\[ K_C = \frac{(2 + x)(2 + x)}{(2 - x)(2 - x)} = \frac{(2 + x)^2}{(2 - x)^2} \]
Setting this equal to \( \frac{1}{9} \):
\[ \frac{(2 + x)^2}{(2 - x)^2} = \frac{1}{9} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ 9(2 + x)^2 = (2 - x)^2 \]
Expanding both sides:
\[ 9(4 + 4x + x^2) = 4 - 4x + x^2 \]
This simplifies to:
\[ 36 + 36x + 9x^2 = 4 - 4x + x^2 \]
Rearranging gives:
\[ 8x^2 + 40x + 32 = 0 \]

### Step 7: Solve the quadratic equation
Dividing through by 8:
\[ x^2 + 5x + 4 = 0 \]
Factoring gives:
\[ (x + 4)(x + 1) = 0 \]
Thus, \( x = -4 \) or \( x = -1 \). Since \( x \) cannot be negative in this context, we take \( x = -1 \).

### Step 8: Calculate equilibrium moles
Substituting \( x = -1 \) back into the expressions for equilibrium moles:
- \( [SO_3] = 2 - (-1) = 3 \)
- \( [CO] = 2 - (-1) = 3 \)
- \( [SO_2] = 2 + (-1) = 1 \)
- \( [CO_2] = 2 + (-1) = 1 \)

### Step 9: Total moles at equilibrium
Total moles at equilibrium:
\[ 3 + 3 + 1 + 1 = 8 \]

### Step 10: Check the options
1. Total number of moles at equilibrium are less than 8: **Incorrect** (Total = 8)
2. Number of moles of \( SO_3 + CO_2 = 3 + 1 = 4 \): **Correct**
3. Number of moles of \( SO_2 + CO < 1 \):
\[ \frac{1}{3} < 1 \text{ (Correct)} \]
4. Both B and C are correct: **Correct**

### Final Answer
The correct options are B and C.