In which one of the followng gaseous equilibria, `K_(p)` is less than `K_(c)`?

Options

A. `N_(2)O_(4)hArr 2NO_(2)`
B. `2SO_(2)+O_(2)hArr 2SO_(3)`
C. `2HIhArr H_(2)+I_(2)`
D. `N_(2)+O_(2)hArr 2NO`

Answer

To determine in which gaseous equilibrium \( K_p \) is less than \( K_c \), we need to understand the relationship between these two equilibrium constants. The relationship is given by the formula:

\[ K_p = K_c \cdot (RT)^{\Delta n} \]

where:
- \( R \) is the ideal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta n \) is the change in the number of moles of gas, calculated as the number of moles of products minus the number of moles of reactants.

### Step-by-Step Solution:

1. **Identify the Relationship**:
We know that \( K_p \) is related to \( K_c \) through the equation \( K_p = K_c \cdot (RT)^{\Delta n} \).

2. **Understand \( \Delta n \)**:
- \( \Delta n = n_{\text{products}} - n_{\text{reactants}} \)
- If \( \Delta n \) is positive, \( K_p \) will be greater than \( K_c \).
- If \( \Delta n \) is zero, \( K_p \) will be equal to \( K_c \).
- If \( \Delta n \) is negative, \( K_p \) will be less than \( K_c \).

3. **Determine Conditions for \( K_p < K_c \)**:
For \( K_p \) to be less than \( K_c \), we require \( \Delta n < 0 \). This means that there are more moles of gas on the reactant side than on the product side.

4. **Analyze Given Equilibria**:
We need to analyze the provided equilibria to find which one has a negative \( \Delta n \).

For example:
- If we have a reaction like \( A \rightleftharpoons 2B \), then \( \Delta n = 2 - 1 = 1 \) (positive).
- If we have a reaction like \( 2A \rightleftharpoons B \), then \( \Delta n = 1 - 2 = -1 \) (negative).

5. **Select the Correct Option**:
Based on the analysis of the reactions, we can conclude that the equilibrium where \( \Delta n < 0 \) will be the one where \( K_p < K_c \).

### Conclusion:
The correct option is the one where the number of gaseous moles on the reactant side exceeds that on the product side, leading to a negative \( \Delta n \).