The equilibrium constant of the following isomerisation reaction at 400 K and 298 K are 2.07 and 3.42 respectively. `cis-" butene"overset(k_(1))underset(k_(-1))hArr "trans - butene"`
Which of the following is/are correct?
I. The reaction is exothermic
II. The reaction is endothermic
III. At 400 K 50% of cis - butene and `50%` of trans - butene are present of equilibrium
IV. Both at 298 K and 400 K, `k_(1)=k_(-1)`.

Options

A. I and IV
B. II and IV
C. I and III
D. I only

Answer

To solve the problem, we need to analyze the isomerization reaction of cis-butene to trans-butene and the given equilibrium constants at two different temperatures (400 K and 298 K).

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The reaction is:
\[ \text{cis-butene} \overset{k_1}{\rightleftharpoons} \text{trans-butene} \]
The equilibrium constants are given as:
- \( K_{298} = 3.42 \) at 298 K
- \( K_{400} = 2.07 \) at 400 K

2. **Analyzing the Equilibrium Constants**:
The equilibrium constant \( K \) is defined as:
\[ K = \frac{[\text{trans-butene}]}{[\text{cis-butene}]} \]
A higher \( K \) value at a lower temperature suggests that the reaction favors the formation of trans-butene at lower temperatures.

3. **Determining the Nature of the Reaction**:
To determine if the reaction is exothermic or endothermic, we can use the van 't Hoff equation:
\[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]
Here, \( K_1 = 3.42 \) at \( T_1 = 298 \, K \) and \( K_2 = 2.07 \) at \( T_2 = 400 \, K \).

4. **Calculating the Change in Enthalpy (\( \Delta H \))**:
- Calculate \( \frac{K_2}{K_1} = \frac{2.07}{3.42} \).
- Calculate \( \ln\left(\frac{2.07}{3.42}\right) \). Since \( \frac{2.07}{3.42} < 1 \), \( \ln \) will be negative.
- The term \( \left(\frac{1}{400} - \frac{1}{298}\right) \) is positive because \( \frac{1}{298} > \frac{1}{400} \).

Since \( \ln\left(\frac{K_2}{K_1}\right) < 0 \) and \( \left(\frac{1}{400} - \frac{1}{298}\right) > 0 \), it follows that \( \Delta H < 0 \). Thus, the reaction is exothermic.

5. **Evaluating the Statements**:
- **I. The reaction is exothermic**: **True** (as shown above).
- **II. The reaction is endothermic**: **False** (the reaction is exothermic).
- **III. At 400 K, 50% of cis-butene and 50% of trans-butene are present at equilibrium**:
- Calculate the concentrations using \( K = \frac{[\text{trans-butene}]}{[\text{cis-butene}]} = 2.07 \).
- If we assume \( [\text{cis-butene}] = a \) and \( [\text{trans-butene}] = x \), then \( K = \frac{x}{a} \) implies \( x = 2.07a \).
- The total concentration \( a + x = a + 2.07a = 3.07a \).
- Therefore, \( \frac{x}{a + x} = \frac{2.07a}{3.07a} = \frac{2.07}{3.07} \approx 0.674 \), which is not 50%. Thus, this statement is **False**.
- **IV. Both at 298 K and 400 K, \( k_1 = k_{-1} \)**: This statement is **False** because the equilibrium constants change with temperature.

### Summary of Correct Answers:
- **I. True**
- **II. False**
- **III. False**
- **IV. False**