`PCl_(5)hArrPCl_(3)+Cl_(2)` in the reversible reaction the moles of `PCl_(5).PCl_(3)` and `Cl_(2)` are a,b and c respectively and total pressure is `P` then value of `K_(p)` will be

Options

A. (A) `(bc)/(a).RT`
B. (B) `(b)/((a+b+c).P`
C. `(bc.P)/(a(a+b+c))`
D. (D) `(c)/((a+b+c)).P`

Answer

To find the value of \( K_p \) for the reaction \( PCl_5 \rightleftharpoons PCl_3 + Cl_2 \), we can follow these steps:

### Step 1: Write the Reaction
The reaction is:
\[ PCl_5 \rightleftharpoons PCl_3 + Cl_2 \]

### Step 2: Define Moles
Let:
- Moles of \( PCl_5 = a \)
- Moles of \( PCl_3 = b \)
- Moles of \( Cl_2 = c \)

### Step 3: Calculate Total Moles
The total number of moles in the system is:
\[ \text{Total moles} = a + b + c \]

### Step 4: Calculate Mole Fractions
The mole fractions for each component are:
- Mole fraction of \( PCl_5 \):
\[ \chi_{PCl_5} = \frac{a}{a + b + c} \]
- Mole fraction of \( PCl_3 \):
\[ \chi_{PCl_3} = \frac{b}{a + b + c} \]
- Mole fraction of \( Cl_2 \):
\[ \chi_{Cl_2} = \frac{c}{a + b + c} \]

### Step 5: Calculate Partial Pressures
Using the total pressure \( P \), the partial pressures can be calculated as:
- Partial pressure of \( PCl_5 \):
\[ P_{PCl_5} = \chi_{PCl_5} \cdot P = \frac{a}{a + b + c} \cdot P \]
- Partial pressure of \( PCl_3 \):
\[ P_{PCl_3} = \chi_{PCl_3} \cdot P = \frac{b}{a + b + c} \cdot P \]
- Partial pressure of \( Cl_2 \):
\[ P_{Cl_2} = \chi_{Cl_2} \cdot P = \frac{c}{a + b + c} \cdot P \]

### Step 6: Write the Expression for \( K_p \)
The equilibrium constant \( K_p \) for the reaction is given by:
\[ K_p = \frac{P_{PCl_3} \cdot P_{Cl_2}}{P_{PCl_5}} \]

### Step 7: Substitute the Partial Pressures
Substituting the expressions for the partial pressures into the \( K_p \) expression:
\[ K_p = \frac{\left(\frac{b}{a + b + c} \cdot P\right) \cdot \left(\frac{c}{a + b + c} \cdot P\right)}{\frac{a}{a + b + c} \cdot P} \]

### Step 8: Simplify the Expression
This simplifies to:
\[ K_p = \frac{bc \cdot P^2}{a \cdot (a + b + c)} \]
After canceling out common terms, we arrive at:
\[ K_p = \frac{bc \cdot P}{a + b + c} \]

### Final Answer
Thus, the value of \( K_p \) is:
\[ K_p = \frac{bc \cdot P}{a + b + c} \]

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