Phosphorous pentachloride when heated in a sealed tube at `700 K` it undergoes decomposition as
`PCl_(5)(g) hArr PCl_(3)(g)+Cl_(2)(g), K_(p)=38` atm
Vapour density of the mixture is `74.25`.
Equilibrium constant `K_(c)` for the reaction will be

Options

A. `0.66 M`
B. `0.56 M`
C. `0.46 M`
D. `0.36 M`

Explanation

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

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

### Step 2: Identify the change in moles (Δn)
To find \( \Delta n \), we need to count the number of moles of gaseous products and reactants:
- Products: \( PCl_3 \) (1 mole) + \( Cl_2 \) (1 mole) = 2 moles
- Reactants: \( PCl_5 \) (1 mole) = 1 mole

Now, we calculate \( \Delta n \):
\[ \Delta n = \text{moles of products} - \text{moles of reactants} = 2 - 1 = 1 \]

### Step 3: Use the relationship between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \cdot R^n \cdot T^{\Delta n} \]
where:
- \( R \) is the ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) is the temperature in Kelvin (700 K)
- \( \Delta n \) is the change in moles (which we found to be 1)

### Step 4: Rearranging the equation to find \( K_c \)
From the equation, we can rearrange to find \( K_c \):
\[ K_c = \frac{K_p}{R^n \cdot T^{\Delta n}} \]
Substituting \( K_p = 38 \) atm, \( R = 0.0821 \) L·atm/(K·mol), \( T = 700 \) K, and \( \Delta n = 1 \):
\[ K_c = \frac{38}{(0.0821)^{1} \cdot (700)^{1}} \]

### Step 5: Calculate \( K_c \)
Now we compute \( K_c \):
\[ K_c = \frac{38}{0.0821 \cdot 700} \]
Calculating the denominator:
\[ 0.0821 \cdot 700 = 57.47 \]
Now, substituting back:
\[ K_c = \frac{38}{57.47} \approx 0.66 \, \text{mol/L} \]

### Final Answer
Thus, the equilibrium constant \( K_c \) for the reaction is approximately:
\[ K_c \approx 0.66 \, \text{mol/L} \]

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