Variation of equilibrium constan K with temperature is given by van't Hoff equation
`InK=(Delta_(r)S^(@))/R-(Delta_(r)H^(@))/(RT)`
for this equation, `(Delta_(r)H^(@))` can be evaluated if equilibrium constans `K_(1)` and `K_(2)` at two temperature `T_(1)` and `T_(2)` are known.
`log(K_(2)/K_(1))=(Delta_(r)H^(@))/(2.303R)[1/T_(1)-1/T_(2)]`
Select the correct statement :

Options

A. Value of `K_(eq)` always increases with increasing temperature
B. For expthermic reaction of value of `K_(eq)` increases with decreasing in temperature
C. For endothermic reaction value of `K_(eq)` increases with decreasihng in temperature
D. For exothermic reactionslope is `(logK Vs.1//T)` negative

Answer

To solve the question regarding the variation of the equilibrium constant \( K \) with temperature using the van't Hoff equation, we will analyze the statements provided and determine which one is correct.

### Step-by-Step Solution:

1. **Understanding the van't Hoff Equation**:
The van't Hoff equation is given by:
\[ \ln K = \frac{\Delta_r S^\circ}{R} - \frac{\Delta_r H^\circ}{RT} \]
This equation relates the equilibrium constant \( K \) to the standard entropy change \( \Delta_r S^\circ \) and the standard enthalpy change \( \Delta_r H^\circ \) at a given temperature \( T \).

2. **Analyzing Statement A**:
"The value of \( K \) always increases with temperature."
- This statement is **incorrect**. The effect of temperature on \( K \) depends on whether the reaction is exothermic or endothermic. For exothermic reactions, increasing temperature decreases \( K \), while for endothermic reactions, increasing temperature increases \( K \).

3. **Analyzing Statement B**:
"For exothermic reactions, the value of \( K \) always increases with a decrease in temperature."
- This statement is **correct**. For exothermic reactions, \( \Delta_r H^\circ \) is negative. According to the van't Hoff equation, decreasing temperature (lowering \( T \)) will lead to an increase in \( K \) because the negative term becomes less negative, thus increasing \( K \).

4. **Analyzing Statement C**:
"For endothermic reactions, the value of \( K \) increases with decreasing temperature."
- This statement is **incorrect**. For endothermic reactions, \( \Delta_r H^\circ \) is positive. Decreasing temperature will lead to a decrease in \( K \) because the positive term becomes less significant, thus decreasing \( K \).

5. **Analyzing Statement D**:
"For exothermic reactions, the slope of \( \log K \) versus \( \frac{1}{T} \) is negative."
- This statement is **incorrect**. For exothermic reactions, since \( \Delta_r H^\circ \) is negative, the slope of the graph of \( \log K \) versus \( \frac{1}{T} \) will be positive, not negative.

### Conclusion:
The correct statement is **B**: "For exothermic reactions, the value of \( K \) always increases with a decrease in temperature."