To solve the question regarding the effect of temperature on the equilibrium constant (K) in terms of change in entropy (ΔS), we can follow these steps:
### Step 1: Understanding the Relationship Between K and Temperature
The equilibrium constant (K) for a reaction is temperature-dependent. The Van 't Hoff equation describes how K changes with temperature. It is given by:
\[
\log \left( \frac{K_2}{K_1} \right) = -\frac{\Delta H}{2.303R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)
\]
where:
- \( K_1 \) and \( K_2 \) are the equilibrium constants at temperatures \( T_1 \) and \( T_2 \) respectively,
- \( \Delta H \) is the change in enthalpy,
- \( R \) is the universal gas constant.
### Step 2: Analyzing Exothermic Reactions
For exothermic reactions, the change in enthalpy (\( \Delta H \)) is negative. According to the Van 't Hoff equation:
- If the temperature increases (\( T_2 > T_1 \)), the term \( \frac{1}{T_2} - \frac{1}{T_1} \) becomes negative, making \( \log \left( \frac{K_2}{K_1} \right) \) positive.
- This implies that \( K_2 < K_1 \), meaning that the equilibrium constant decreases with an increase in temperature.
### Step 3: Analyzing Endothermic Reactions
For endothermic reactions, the change in enthalpy (\( \Delta H \)) is positive. In this case:
- If the temperature increases, the term \( \frac{1}{T_2} - \frac{1}{T_1} \) remains positive, making \( \log \left( \frac{K_2}{K_1} \right) \) negative.
- This implies that \( K_2 > K_1 \), meaning that the equilibrium constant increases with an increase in temperature.
### Step 4: Relating ΔS to K
The change in entropy (\( \Delta S \)) is related to the change in enthalpy and the change in Gibbs free energy (\( \Delta G \)) by the equation:
\[
\Delta G = \Delta H - T\Delta S
\]
At equilibrium, \( \Delta G = 0 \), thus:
\[
0 = \Delta H - T\Delta S \implies \Delta S = \frac{\Delta H}{T}
\]
### Step 5: Conclusion
- For exothermic reactions, as temperature increases, \( K \) decreases, and the entropy change of the system is negative while that of the surroundings is positive.
- For endothermic reactions, as temperature increases, \( K \) increases, and the entropy change of the system is positive while that of the surroundings is negative.
### Summary of Key Points
1. **Exothermic Reactions**: \( K \) decreases with increasing temperature.
2. **Endothermic Reactions**: \( K \) increases with increasing temperature.
3. **Entropy Changes**: The sign of \( \Delta S \) for the system and surroundings varies based on whether the reaction is exothermic or endothermic.
