`n` mole of `PCl_(3)` and `n` mole of `Cl_(2)` are allowed to react at constant temperature `T` to have a total equilibrium pressure `P` , as `:`
`PCl_(3)(g)+Cl_(2)(g) hArr PCl_(5)(g)`
If `y` mole of `PCl_(5)` are formed at equilibrium , find `K_(p)` for the given reaction .

Options

A. `((2n-y)y)/((n-y)^(2)P)`
B. `(y)/((n-y)^(2)(2n-y)P)`
C. `((n-y)^(2).P)/((2n-y)y)`
D. `((n-y)^(2)(2n-y)P)/(y)`

Explanation

To find the equilibrium constant \( K_p \) for the reaction

\[ \text{PCl}_3(g) + \text{Cl}_2(g) \rightleftharpoons \text{PCl}_5(g) \]

given that \( n \) moles of \( \text{PCl}_3 \) and \( n \) moles of \( \text{Cl}_2 \) are allowed to react and \( y \) moles of \( \text{PCl}_5 \) are formed at equilibrium, we can follow these steps:

### Step 1: Write the initial and equilibrium conditions

At the start (initial conditions):
- Moles of \( \text{PCl}_3 = n \)
- Moles of \( \text{Cl}_2 = n \)
- Moles of \( \text{PCl}_5 = 0 \)

At equilibrium:
- Moles of \( \text{PCl}_3 = n - y \)
- Moles of \( \text{Cl}_2 = n - y \)
- Moles of \( \text{PCl}_5 = y \)

### Step 2: Calculate the total number of moles at equilibrium

The total number of moles at equilibrium is given by:

\[ \text{Total moles} = (n - y) + (n - y) + y = 2n - y \]

### Step 3: Determine the partial pressures

The partial pressure of each component can be calculated using the formula:

\[ P_i = \left( \frac{\text{moles of } i}{\text{total moles}} \right) \times P \]

where \( P \) is the total pressure.

1. **Partial pressure of \( \text{PCl}_3 \)**:

\[ P_{\text{PCl}_3} = \left( \frac{n - y}{2n - y} \right) P \]

2. **Partial pressure of \( \text{Cl}_2 \)**:

\[ P_{\text{Cl}_2} = \left( \frac{n - y}{2n - y} \right) P \]

3. **Partial pressure of \( \text{PCl}_5 \)**:

\[ P_{\text{PCl}_5} = \left( \frac{y}{2n - y} \right) P \]

### Step 4: Write the expression for \( K_p \)

The equilibrium constant \( K_p \) for the reaction is given by:

\[ K_p = \frac{P_{\text{PCl}_5}}{P_{\text{PCl}_3} \cdot P_{\text{Cl}_2}} \]

Substituting the expressions for the partial pressures:

\[ K_p = \frac{\left( \frac{y}{2n - y} \right) P}{\left( \frac{n - y}{2n - y} \right) P \cdot \left( \frac{n - y}{2n - y} \right) P} \]

### Step 5: Simplify the expression

Cancelling \( P \) from the numerator and denominator:

\[ K_p = \frac{y}{2n - y} \cdot \frac{(2n - y)^2}{(n - y)^2} \]

This simplifies to:

\[ K_p = \frac{y \cdot (2n - y)}{(n - y)^2} \]

### Final Expression

Thus, the expression for \( K_p \) is:

\[ K_p = \frac{2n - y}{(n - y)^2} \cdot y \]