In a 500mL falsk, the degree of dissociation of `PCI_(5)` at equilibrium is `40%` and the initial amount is 5 moles. The value of equilibrium constant in mol `L^(-1)` for the decomposition of `PCI_(5)` is

Options

A. `2.33`
B. `2.66`
C. `5.32`
D. `4.66`

Explanation

To find the equilibrium constant \( K_c \) for the decomposition of \( PCl_5 \) at equilibrium, we will follow these steps:

### Step 1: Write the balanced chemical equation
The decomposition of \( PCl_5 \) can be represented as:
\[ PCl_5 \rightleftharpoons PCl_3 + Cl_2 \]

### Step 2: Determine the initial moles and degree of dissociation
Given:
- Initial moles of \( PCl_5 = 5 \) moles
- Degree of dissociation \( \alpha = 40\% = 0.4 \)

### Step 3: Calculate moles at equilibrium
At equilibrium, the moles of each species can be calculated as follows:
- Moles of \( PCl_5 \) remaining:
\[ \text{Moles of } PCl_5 = 5(1 - \alpha) = 5(1 - 0.4) = 5 \times 0.6 = 3 \text{ moles} \]

- Moles of \( PCl_3 \) formed:
\[ \text{Moles of } PCl_3 = 5\alpha = 5 \times 0.4 = 2 \text{ moles} \]

- Moles of \( Cl_2 \) formed:
\[ \text{Moles of } Cl_2 = 5\alpha = 5 \times 0.4 = 2 \text{ moles} \]

### Step 4: Calculate concentrations at equilibrium
Since the volume of the flask is 500 mL (or 0.5 L), we can calculate the concentrations:
- Concentration of \( PCl_5 \):
\[ \text{Concentration of } PCl_5 = \frac{3 \text{ moles}}{0.5 \text{ L}} = 6 \text{ mol/L} \]

- Concentration of \( PCl_3 \):
\[ \text{Concentration of } PCl_3 = \frac{2 \text{ moles}}{0.5 \text{ L}} = 4 \text{ mol/L} \]

- Concentration of \( Cl_2 \):
\[ \text{Concentration of } Cl_2 = \frac{2 \text{ moles}}{0.5 \text{ L}} = 4 \text{ mol/L} \]

### Step 5: Write the expression for the equilibrium constant \( K_c \)
The expression for the equilibrium constant \( K_c \) for the reaction is:
\[ K_c = \frac{[PCl_3][Cl_2]}{[PCl_5]} \]

### Step 6: Substitute the equilibrium concentrations into the expression
Substituting the concentrations we calculated:
\[ K_c = \frac{(4)(4)}{6} = \frac{16}{6} = \frac{8}{3} \]

### Step 7: Calculate the numerical value of \( K_c \)
Calculating \( \frac{8}{3} \):
\[ K_c \approx 2.67 \text{ mol/L} \]

Thus, the equilibrium constant \( K_c \) for the decomposition of \( PCl_5 \) is approximately \( 2.67 \text{ mol/L} \).

### Final Answer
The value of the equilibrium constant \( K_c \) for the decomposition of \( PCl_5 \) is \( 2.67 \text{ mol/L} \).

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