A mixture of 3 moles of `SO_(2)`, 4 moles of `NO_(2)`, 1 mole of `SO_(3)` and 4 moles of NO is placed in a 2.0L vessel. `SO_(2)(g)+NO_(2)(g) iff SO_(3)(g)+NO(g)`.
At equilibrium, the vessel is found to contain 1 mole of `SO_(2)`. Calculate the value of `K_(C)`.

Answer

To solve the problem step by step, we will follow these procedures:

### Step 1: Write the balanced chemical equation
The reaction given is:
\[ \text{SO}_2(g) + \text{NO}_2(g) \rightleftharpoons \text{SO}_3(g) + \text{NO}(g) \]

### Step 2: Determine initial moles and concentrations
We are given the initial moles of each substance:
- SO₂: 3 moles
- NO₂: 4 moles
- SO₃: 1 mole
- NO: 4 moles

The volume of the vessel is 2.0 L. Therefore, we can calculate the initial concentrations:
- \[ [\text{SO}_2] = \frac{3 \text{ moles}}{2 \text{ L}} = 1.5 \, \text{M} \]
- \[ [\text{NO}_2] = \frac{4 \text{ moles}}{2 \text{ L}} = 2.0 \, \text{M} \]
- \[ [\text{SO}_3] = \frac{1 \text{ mole}}{2 \text{ L}} = 0.5 \, \text{M} \]
- \[ [\text{NO}] = \frac{4 \text{ moles}}{2 \text{ L}} = 2.0 \, \text{M} \]

### Step 3: Set up the equilibrium expression
At equilibrium, we denote the change in concentration of SO₂ and NO₂ as \( -x \) (since they are reactants) and for SO₃ and NO as \( +x \) (since they are products).

The equilibrium concentrations will be:
- \[ [\text{SO}_2] = 1.5 - x \]
- \[ [\text{NO}_2] = 2.0 - x \]
- \[ [\text{SO}_3] = 0.5 + x \]
- \[ [\text{NO}] = 2.0 + x \]

### Step 4: Use the information given to find \( x \)
We know that at equilibrium, the concentration of SO₂ is 1 mole in a 2 L vessel, which means:
\[ [\text{SO}_2] = \frac{1 \text{ mole}}{2 \text{ L}} = 0.5 \, \text{M} \]

Setting up the equation:
\[ 1.5 - x = 0.5 \]

Solving for \( x \):
\[ x = 1.5 - 0.5 = 1 \]

### Step 5: Calculate equilibrium concentrations
Now substitute \( x = 1 \) back into the expressions for equilibrium concentrations:
- \[ [\text{SO}_2] = 1.5 - 1 = 0.5 \, \text{M} \]
- \[ [\text{NO}_2] = 2.0 - 1 = 1.0 \, \text{M} \]
- \[ [\text{SO}_3] = 0.5 + 1 = 1.5 \, \text{M} \]
- \[ [\text{NO}] = 2.0 + 1 = 3.0 \, \text{M} \]

### Step 6: Write the expression for \( K_c \)
The equilibrium constant \( K_c \) is given by:
\[ K_c = \frac{[\text{SO}_3][\text{NO}]}{[\text{SO}_2][\text{NO}_2]} \]

Substituting the equilibrium concentrations:
\[ K_c = \frac{(1.5)(3.0)}{(0.5)(1.0)} \]

### Step 7: Calculate \( K_c \)
Calculating the above expression:
\[ K_c = \frac{4.5}{0.5} = 9 \]

### Final Answer
The value of \( K_c \) is:
\[ K_c = 9 \]

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