For which of the following reactions, `K_p` may be equal to 0.5 atm?

Options

A. `2HI Leftrightarrow H_2+I_2`
B. `PCl_5 Leftrightarrow PCl_3+Cl_2`
C. `N_2+3H_2 Leftrightarrow 2NH_3`
D. `2NO_2 Leftrightarrow N_2O_4`

Answer

To determine for which reaction \( K_p \) may be equal to 0.5 atm, we need to analyze the reactions provided in the options based on the relationship between \( K_p \) and \( K_c \), and the change in the number of moles of gas (\( \Delta N_g \)).

### Step-by-Step Solution:

1. **Understanding \( K_p \) and \( K_c \)**:
- The equilibrium constant \( K_p \) is related to \( K_c \) by the equation:
\[ K_p = K_c (RT)^{\Delta N_g} \]
- Here, \( R \) is the ideal gas constant and \( T \) is the temperature in Kelvin. \( \Delta N_g \) is the change in the number of moles of gas during the reaction.

2. **Identifying \( \Delta N_g \)**:
- For a reaction, \( \Delta N_g \) is calculated as:
\[ \Delta N_g = \text{moles of gaseous products} - \text{moles of gaseous reactants} \]
- If \( \Delta N_g = 0 \), then \( K_p = K_c \).
- If \( \Delta N_g > 0 \), \( K_p > K_c \).
- If \( \Delta N_g < 0 \), \( K_p < K_c \).

3. **Analyzing the Given Reactions**:
- For each reaction provided in the options, calculate \( \Delta N_g \) and determine the potential value of \( K_p \).

4. **Calculating \( K_p \) for Each Reaction**:
- **Reaction 1**: If \( \Delta N_g = 0 \), then \( K_p \) can be equal to \( K_c \) which can be 0.5 atm.
- **Reaction 2**: If \( PCl_5 \) dissociates to form \( PCl_3 \) and \( Cl_2 \), then:
\[ \Delta N_g = 2 - 1 = 1 \quad (\text{since 1 mole of } PCl_5 \text{ gives 2 moles of products}) \]
- Here, \( K_p \) will be greater than \( K_c \).
- **Reaction 3**: If \( \Delta N_g = -1 \), then \( K_p < K_c \), and it cannot be equal to 0.5 atm.
- **Reaction 4**: Similar to Reaction 3, if \( \Delta N_g = -1 \), then \( K_p < K_c \), and it cannot be equal to 0.5 atm.

5. **Conclusion**:
- The only reaction where \( K_p \) can potentially be equal to 0.5 atm is the first reaction where \( \Delta N_g = 0 \).

### Final Answer:
The reaction for which \( K_p \) may be equal to 0.5 atm is **Reaction 1**.