In which reaction the equilibrium constant value increases with increase in temperature ?

Options

A. a. `H_(2)O_((l)) hArr H_(2)O_((g))`
B. b. `CaCO_(3)(s) hArr CaO(s) + CO_(2)(g)`
C. c. `H_(2) O_((aq)) hArr H_((aq))^(+) + OH^(-) (aq)`
D. d. `N_(2(g)) + 3H_(2(g)) hArr 2 NH_(3(g))`

Answer

To determine in which reaction the equilibrium constant (Kc) increases with an increase in temperature, we need to understand the relationship between temperature and the equilibrium constant for different types of reactions.

### Step-by-Step Solution:

1. **Understanding the Equilibrium Constant (Kc)**:
The equilibrium constant (Kc) is defined as the ratio of the concentrations of products to the concentrations of reactants at equilibrium. For a general reaction:
\[ aA + bB \rightleftharpoons cC + dD \]
The equilibrium constant is given by:
\[ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]

2. **Le Chatelier's Principle**:
According to Le Chatelier's principle, if a system at equilibrium is subjected to a change in temperature, pressure, or concentration, the system will adjust to counteract the change and restore a new equilibrium.

3. **Endothermic vs. Exothermic Reactions**:
- **Endothermic Reactions**: These reactions absorb heat. For endothermic reactions, increasing the temperature shifts the equilibrium to the right (towards the products), thus increasing Kc.
- **Exothermic Reactions**: These reactions release heat. For exothermic reactions, increasing the temperature shifts the equilibrium to the left (towards the reactants), thus decreasing Kc.

4. **Analyzing Each Reaction**:
- **Option A**: \( H_2O (l) \rightleftharpoons H_2O (g) \)
- This is an endothermic process (vaporization). Increasing temperature will increase the concentration of water vapor, thus increasing Kc.
- **Option B**: \( CaCO_3 (s) \rightleftharpoons CaO (s) + CO_2 (g) \)
- This is also an endothermic reaction (decomposition). Increasing temperature will favor the formation of products, thus increasing Kc.
- **Option C**: \( H_2O (l) \rightleftharpoons H^+ (aq) + OH^- (aq) \)
- This dissociation is endothermic. Increasing temperature will increase the concentration of ions, thus increasing Kc.
- **Option D**: \( N_2 (g) + 3H_2 (g) \rightleftharpoons 2NH_3 (g) \)
- This is an exothermic reaction (Haber process). Increasing temperature will shift the equilibrium to the left, thus decreasing Kc.

5. **Conclusion**:
The reactions where the equilibrium constant increases with an increase in temperature are:
- Option A: \( H_2O (l) \rightleftharpoons H_2O (g) \)
- Option B: \( CaCO_3 (s) \rightleftharpoons CaO (s) + CO_2 (g) \)
- Option C: \( H_2O (l) \rightleftharpoons H^+ (aq) + OH^- (aq) \)

Thus, the correct options are A, B, and C.