To solve the problem, we need to analyze the relationship between the heat of reaction at constant volume and constant pressure, and how it relates to the equilibrium constants \( K_p \) and \( K_c \).
### Step-by-Step Solution:
1. **Understanding the Given Information**:
- The heat of reaction at constant volume (\( \Delta E \)) is 1200 cal more than at constant pressure (\( \Delta H \)).
- The temperature is given as \( T = 300 \, K \).
2. **Using the Relationship Between \( \Delta H \) and \( \Delta E \)**:
- The relationship between change in enthalpy (\( \Delta H \)) and change in internal energy (\( \Delta E \)) is given by:
\[
\Delta H = \Delta E + P \Delta V
\]
- Rearranging this gives:
\[
\Delta E = \Delta H - P \Delta V
\]
3. **Substituting the Given Heat of Reaction**:
- We know that:
\[
\Delta H = \Delta E + 1200 \, \text{cal}
\]
- Substituting \( \Delta E \) into this equation:
\[
\Delta H = (\Delta H - P \Delta V) + 1200 \, \text{cal}
\]
- This simplifies to:
\[
P \Delta V = 1200 \, \text{cal}
\]
4. **Converting Calories to Joules**:
- To convert calories to joules, we use the conversion factor \( 1 \, \text{cal} = 4.18 \, \text{J} \):
\[
P \Delta V = 1200 \times 4.18 \, \text{J} = 5016 \, \text{J}
\]
5. **Using the Ideal Gas Law**:
- From the ideal gas law, we know:
\[
P \Delta V = nRT
\]
- Rearranging gives:
\[
\Delta n = \frac{P \Delta V}{RT}
\]
6. **Substituting Known Values**:
- We know \( R = 8.314 \, \text{J/(mol K)} \) and \( T = 300 \, K \):
\[
\Delta n = \frac{5016 \, \text{J}}{8.314 \, \text{J/(mol K)} \times 300 \, K}
\]
- Calculating \( \Delta n \):
\[
\Delta n = \frac{5016}{2494.2} \approx 2.01
\]
7. **Determining the Sign of \( \Delta n \)**:
- Since the heat of reaction is given as 1200 cal more at constant volume than at constant pressure, this indicates that the number of moles of products is less than that of reactants, thus:
\[
\Delta n = -2
\]
8. **Finding the Relationship Between \( K_p \) and \( K_c \)**:
- The relationship between \( K_p \) and \( K_c \) is given by:
\[
K_p = K_c (RT)^{\Delta n}
\]
- Substituting \( \Delta n = -2 \):
\[
K_p = K_c \left(8.314 \times 300\right)^{-2}
\]
- Since \( \Delta n \) is negative, it implies:
\[
K_p < K_c
\]
### Conclusion:
The correct expression is:
\[
K_p < K_c
\]
