According to van 't Hoff equation, K varies with temperature as:

Options

A. `log ""K_(2)/K_(1)=(triangleH)/(2.303R) [1/T_(1)-1/T_(2)]`
B. `log ""K_(1)/K_(2)=+ (triangleH)/(2.303R) [1/T_(2)+1/T]`
C. `log ""K_(1)/K_(2)=-(triangleH)/(2.303R) [1/T_(2)+1/T_2]`
D. `log ""K_(2)/K_(1)=+ (triangleH)/(2.303R) [1/T_(1)+1/T_2]`

Answer

To solve the question regarding how the equilibrium constant (K) varies with temperature according to van 't Hoff's equation, we can follow these steps:

### Step 1: Understand the van 't Hoff Equation
The van 't Hoff equation relates the change in the equilibrium constant (K) with temperature (T). It is derived from the Gibbs free energy equation and can be expressed as:

\[ \Delta G^\circ = -RT \ln K \]

where:
- \(\Delta G^\circ\) is the standard Gibbs free energy change,
- \(R\) is the universal gas constant,
- \(T\) is the temperature in Kelvin,
- \(K\) is the equilibrium constant.

### Step 2: Relate Gibbs Free Energy to Enthalpy and Entropy
We can also express the Gibbs free energy change in terms of enthalpy (\(\Delta H^\circ\)) and entropy (\(\Delta S^\circ\)):

\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]

### Step 3: Combine the Two Equations
By equating the two expressions for \(\Delta G^\circ\):

\[ -RT \ln K = \Delta H^\circ - T \Delta S^\circ \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:

\[ \ln K = \frac{\Delta H^\circ}{RT} - \frac{\Delta S^\circ}{R} \]

### Step 5: Deriving the van 't Hoff Equation
To find how \(K\) changes with temperature, we can differentiate the equation with respect to temperature. This leads to the van 't Hoff equation in its integrated form:

\[ \log \frac{K_2}{K_1} = \frac{\Delta H^\circ}{2.303 R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]

where:
- \(K_1\) is the equilibrium constant at temperature \(T_1\),
- \(K_2\) is the equilibrium constant at temperature \(T_2\).

### Step 6: Analyze the Equation
From the derived equation, we can see that:
- If \(\Delta H^\circ\) is positive (endothermic reaction), increasing temperature increases \(K\).
- If \(\Delta H^\circ\) is negative (exothermic reaction), increasing temperature decreases \(K\).

### Conclusion
Thus, according to van 't Hoff's equation, the equilibrium constant \(K\) varies with temperature based on the enthalpy change of the reaction.