Write the relation between `K_(p) " and "K_(c)` for the following reactions :
`"(i)"" " PCI_(5) (g) hArr PCI_(3)(g) +CI_(2) (g)`
`"(ii)"" "N_(2)(g) +3H_(2)(g) hArr 2NH_(3)(g)`
`"(iii)"" "2H_(2)O(g) hArr 2H_(2)(g) +O_(2)(g)`
`"(iv)"" "H_(2)(g) + I_(2)(g) hArr 2HI(g)`

Answer

To find the relation between \( K_p \) and \( K_c \) for the given reactions, we will use the formula:

\[ K_p = K_c \cdot R^{\Delta N} \cdot T^{\Delta N} \]

where:
- \( R \) is the universal gas constant (0.0821 L·atm/(K·mol)),
- \( T \) is the temperature in Kelvin,
- \( \Delta N \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

Now, let's analyze each reaction step by step.

### (i) Reaction: \( PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \)

1. **Identify gaseous species**:
- Reactants: 1 mole of \( PCl_5 \)
- Products: 1 mole of \( PCl_3 \) + 1 mole of \( Cl_2 \) = 2 moles

2. **Calculate \( \Delta N \)**:
\[ \Delta N = \text{moles of products} - \text{moles of reactants} = 2 - 1 = 1 \]

3. **Write the relation**:
\[ K_p = K_c \cdot R^{1} \cdot T^{1} \implies K_p = K_c \cdot RT \]

### (ii) Reaction: \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \)

1. **Identify gaseous species**:
- Reactants: 1 mole of \( N_2 \) + 3 moles of \( H_2 \) = 4 moles
- Products: 2 moles of \( NH_3 \)

2. **Calculate \( \Delta N \)**:
\[ \Delta N = 2 - 4 = -2 \]

3. **Write the relation**:
\[ K_p = K_c \cdot R^{-2} \implies K_p = \frac{K_c}{R^2} \]

### (iii) Reaction: \( 2H_2O(g) \rightleftharpoons 2H_2(g) + O_2(g) \)

1. **Identify gaseous species**:
- Reactants: 2 moles of \( H_2O \)
- Products: 2 moles of \( H_2 \) + 1 mole of \( O_2 \) = 3 moles

2. **Calculate \( \Delta N \)**:
\[ \Delta N = 3 - 2 = 1 \]

3. **Write the relation**:
\[ K_p = K_c \cdot R^{1} \implies K_p = K_c \cdot RT \]

### (iv) Reaction: \( H_2(g) + I_2(g) \rightleftharpoons 2HI(g) \)

1. **Identify gaseous species**:
- Reactants: 1 mole of \( H_2 \) + 1 mole of \( I_2 \) = 2 moles
- Products: 2 moles of \( HI \)

2. **Calculate \( \Delta N \)**:
\[ \Delta N = 2 - 2 = 0 \]

3. **Write the relation**:
\[ K_p = K_c \cdot R^{0} \implies K_p = K_c \]

### Summary of Relations

1. For \( PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \):
\[ K_p = K_c \cdot RT \]

2. For \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \):
\[ K_p = \frac{K_c}{R^2} \]

3. For \( 2H_2O(g) \rightleftharpoons 2H_2(g) + O_2(g) \):
\[ K_p = K_c \cdot RT \]

4. For \( H_2(g) + I_2(g) \rightleftharpoons 2HI(g) \):
\[ K_p = K_c \]