1.6 mol of `PCl_(5)(g)` is placed in 4 `dm^(-3)` closed vessel. When the temperature is raised to 500 K, it decompses and at equilibrium, 1.2 mol of `PCl_(5)(g)` remains. What is `K_(c)` value for the decomposition of `PCl_(5)(g)` to `PCl_(3)(g)` and `Cl_(2)(g)` at 500K.

Options

A. 0.013
B. 0.05
C. 0.033
D. 0.067

Explanation

To find the equilibrium constant \( K_c \) for the decomposition of \( PCl_5(g) \) into \( PCl_3(g) \) and \( Cl_2(g) \), we can follow these steps:

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

### Step 2: Determine initial moles and equilibrium moles
Initially, we have:
- Moles of \( PCl_5 \) = 1.6 mol
- Moles of \( PCl_3 \) = 0 mol
- Moles of \( Cl_2 \) = 0 mol

At equilibrium, we are given that:
- Moles of \( PCl_5 \) remaining = 1.2 mol

Let \( x \) be the moles of \( PCl_5 \) that decomposed. Thus:
\[ 1.6 - x = 1.2 \implies x = 1.6 - 1.2 = 0.4 \text{ mol} \]

At equilibrium, the moles are:
- Moles of \( PCl_5 \) = 1.2 mol
- Moles of \( PCl_3 \) = \( x = 0.4 \) mol
- Moles of \( Cl_2 \) = \( x = 0.4 \) mol

### Step 3: Calculate concentrations at equilibrium
The volume of the closed vessel is 4 dm³ (or 4 L). Therefore, we can calculate the concentrations:
\[ \text{Concentration of } PCl_5 = \frac{1.2 \text{ mol}}{4 \text{ L}} = 0.3 \text{ mol/L} \]
\[ \text{Concentration of } PCl_3 = \frac{0.4 \text{ mol}}{4 \text{ L}} = 0.1 \text{ mol/L} \]
\[ \text{Concentration of } Cl_2 = \frac{0.4 \text{ mol}}{4 \text{ L}} = 0.1 \text{ mol/L} \]

### Step 4: Write the expression for \( K_c \)
The equilibrium constant \( K_c \) is given by the expression:
\[ K_c = \frac{[PCl_3][Cl_2]}{[PCl_5]} \]
Substituting the concentrations we calculated:
\[ K_c = \frac{(0.1)(0.1)}{0.3} \]

### Step 5: Calculate \( K_c \)
Now, we can perform the calculation:
\[ K_c = \frac{0.01}{0.3} = 0.0333 \]

### Step 6: Round to appropriate significant figures
Thus, rounding to two decimal places, we find:
\[ K_c \approx 0.033 \]

### Final Answer
The value of \( K_c \) for the decomposition of \( PCl_5(g) \) at 500 K is approximately \( 0.033 \).

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