The equilibrium constant Kc for the decomposition of `PCl_5` is 0.625 mol/litre at 300 K. Then the value of Kp is-

Answer

To find the value of \( K_p \) for the decomposition of \( PCl_5 \) given that \( K_c = 0.625 \, \text{mol/L} \) at 300 K, we can follow these steps:

### Step 1: Write the balanced equation for the decomposition of \( PCl_5 \)
The decomposition of phosphorus pentachloride can be represented by the following balanced equation:
\[ PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \]

### Step 2: Identify the change in the number of moles of gas (\( \Delta n_g \))
To calculate \( \Delta n_g \), we need to find the difference between the number of moles of gaseous products and the number of moles of gaseous reactants.

- **Products**: \( PCl_3 \) (1 mole) + \( Cl_2 \) (1 mole) = 2 moles
- **Reactants**: \( PCl_5 \) (1 mole) = 1 mole

Thus,
\[ \Delta n_g = \text{moles of products} - \text{moles of reactants} = 2 - 1 = 1 \]

### Step 3: Use the relationship between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \times R^T \times \Delta n_g \]
Where:
- \( R \) is the ideal gas constant. In this case, we will use \( R = 0.0821 \, \text{L atm/(K mol)} \).
- \( T \) is the temperature in Kelvin (300 K).
- \( K_c \) is given as 0.625.

### Step 4: Substitute the values into the equation
Now we can substitute the known values into the equation:
\[ K_p = 0.625 \times (0.0821 \, \text{L atm/(K mol)})^{300} \times 1 \]

### Step 5: Calculate \( K_p \)
Calculating \( R^T \):
\[ R^T = 0.0821 \times 300 = 24.63 \]
Now substituting back into the equation:
\[ K_p = 0.625 \times 24.63 \approx 15.38 \]

### Final Answer
Thus, the value of \( K_p \) is approximately:
\[ K_p \approx 15.38 \]

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