In which of the following equilibrium, `(K_c = K_p)` ?

Options

A. `[N_2(g) + O_2(g) ltimplies 2NO(g)]`
B. `[4NH_3(g) + 5O_2(g) ltimplies 4NO(g) + 6H_2O(g)]`
C. `[N_2(g) + 3H_2(g)ltimplies 2NH_3(g)]`
D. `[2NO(g) + O_2(g) ltimplies 2NO_2(g)]`

Answer

To determine in which of the following equilibria \( K_c = K_p \), we need to understand the relationship between these two equilibrium constants. The key equation to remember is:

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

Where:
- \( K_p \) is the equilibrium constant in terms of partial pressures.
- \( K_c \) is the equilibrium constant in terms of concentrations.
- \( R \) is the universal gas constant.
- \( T \) is the temperature in Kelvin.
- \( \Delta N_g \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

### Step-by-Step Solution:

1. **Identify the Equilibrium Reactions**:
We need to look at the given equilibrium reactions (not provided in the question) and identify the number of moles of gaseous reactants and products for each.

2. **Calculate \( \Delta N_g \)**:
For each reaction, calculate \( \Delta N_g \) using the formula:
\[ \Delta N_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants}) \]

3. **Check the Condition for \( K_c = K_p \)**:
According to the equation \( K_p = K_c (RT)^{\Delta N_g} \), for \( K_c \) to equal \( K_p \), we need:
\[ \Delta N_g = 0 \]
This means that the number of moles of gaseous products must equal the number of moles of gaseous reactants.

4. **Evaluate Each Option**:
For each equilibrium reaction, evaluate the calculated \( \Delta N_g \):
- If \( \Delta N_g = 0 \), then \( K_c = K_p \) for that reaction.
- If \( \Delta N_g \neq 0 \), then \( K_c \neq K_p \).

5. **Select the Correct Option**:
Based on the evaluations, select the option(s) where \( \Delta N_g = 0 \).