To determine for which reaction \( K_p = K_c \), we need to analyze the relationship between the two equilibrium constants. The relationship is given by the equation:
\[
K_p = K_c (RT)^{\Delta n}
\]
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 gaseous products minus the number of moles of gaseous reactants.
### Step-by-Step Solution:
1. **Understand the Relationship**:
- Recall that \( K_p \) is the equilibrium constant based on partial pressures, while \( K_c \) is based on molar concentrations. The relationship between them involves \( \Delta n \).
2. **Identify the Condition for Equality**:
- For \( K_p \) to equal \( K_c \), the term \( (RT)^{\Delta n} \) must equal 1. This occurs when \( \Delta n = 0 \).
3. **Calculate \( \Delta n \)**:
- \( \Delta n = n_{products} - n_{reactants} \)
- Count the number of moles of gaseous products and subtract the number of moles of gaseous reactants.
4. **Analyze Each Reaction**:
- For each provided reaction, calculate \( \Delta n \):
- If \( \Delta n = 0 \), then \( K_p = K_c \).
- If \( \Delta n > 0 \), then \( K_p > K_c \).
- If \( \Delta n < 0 \), then \( K_p < K_c \).
5. **Select the Correct Reaction**:
- From the options given, identify the reaction where \( \Delta n = 0 \).
### Example Analysis:
- **Reaction 1**: \( A \rightleftharpoons 2B \)
- \( n_{products} = 2 \), \( n_{reactants} = 1 \) → \( \Delta n = 2 - 1 = 1 \) (So, \( K_p > K_c \))
- **Reaction 2**: \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \)
- \( n_{products} = 2 \), \( n_{reactants} = 2 \) → \( \Delta n = 2 - 2 = 0 \) (So, \( K_p = K_c \))
- **Reaction 3**: \( 2NO(g) \rightleftharpoons N_2(g) + O_2(g) \)
- \( n_{products} = 2 \), \( n_{reactants} = 3 \) → \( \Delta n = 2 - 3 = -1 \) (So, \( K_p < K_c \))
- **Reaction 4**: \( 2A(g) \rightleftharpoons B(g) + C(g) \)
- \( n_{products} = 2 \), \( n_{reactants} = 2 \) → \( \Delta n = 2 - 2 = 0 \) (So, \( K_p = K_c \))
### Conclusion:
The reactions for which \( K_p = K_c \) are those where \( \Delta n = 0 \). In this case, the correct options are the reactions where the total number of gaseous products equals the total number of gaseous reactants.
