Which of the following expression is true regarding formation of `PCl_(5)` by equation given
`PCl_(3)(g)+Cl_(2)(g)hArr PCl_(5)(g)`

Options

A. `(K_(p))/(K_(c))lt1`
B. `(K_(p))/(K_(c))gt1`
C. `(K_(p))/(K_(c))=1`
D. None of these

Answer

To determine which expression is true regarding the formation of \( PCl_5 \) from the reaction:

\[ PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g) \]

we need to analyze the relationship between the equilibrium constants \( K_p \) and \( K_c \).

### Step 1: Write the expressions for \( K_p \) and \( K_c \)

1. **Equilibrium Constant \( K_c \)**:
The equilibrium constant in terms of concentration (molarity) is given by:
\[ K_c = \frac{[PCl_5]}{[PCl_3][Cl_2]} \]

2. **Equilibrium Constant \( K_p \)**:
The equilibrium constant in terms of partial pressures is given by:
\[ K_p = \frac{P_{PCl_5}}{P_{PCl_3} \cdot P_{Cl_2}} \]

### Step 2: Determine the change in moles (\( \Delta n \))

To relate \( K_p \) and \( K_c \), we need to calculate the change in moles of gas (\( \Delta n \)):
- On the **reactant side**, we have:
- \( 1 \) mole of \( PCl_3 \)
- \( 1 \) mole of \( Cl_2 \)
- Total = \( 1 + 1 = 2 \) moles

- On the **product side**, we have:
- \( 1 \) mole of \( PCl_5 \)
- Total = \( 1 \) mole

Thus, the change in moles (\( \Delta n \)) is:
\[ \Delta n = \text{moles of products} - \text{moles of reactants} = 1 - 2 = -1 \]

### Step 3: Relate \( K_p \) and \( K_c \)

The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c (RT)^{\Delta n} \]
where \( R \) is the ideal gas constant and \( T \) is the temperature in Kelvin.

Substituting \( \Delta n = -1 \):
\[ K_p = K_c (RT)^{-1} = \frac{K_c}{RT} \]

### Step 4: Analyze the relationship

Since \( R \) and \( T \) are both positive constants, it follows that:
\[ K_p < K_c \]
This implies:
\[ \frac{K_p}{K_c} < 1 \]

### Conclusion

The correct expression regarding the formation of \( PCl_5 \) is:
\[ K_p < K_c \quad \text{or} \quad \frac{K_p}{K_c} < 1 \]