Two different first order reaction have rate constants `k_1 and k_2` at `T_1(k_1 gt k_2)` > If temperature is increased from `T_1 " to " T_2`. Then new constants become `k_3 and k_4` respectively. Which among the following relations is correct /

Options

A. `k_1 gt k_2 = k_3 = k_4`
B. `k_1 lt k_3 and k_2 lt k_4`
C. `k_1 = k_3 = k_4`
D. `k_1 gt k_2 gt k_3 gt k_4`

Answer

To solve the problem, we need to analyze the effects of temperature on the rate constants of two different first-order reactions. Let's break it down step by step.

### Step 1: Understand the Initial Conditions
We have two first-order reactions with rate constants \( k_1 \) and \( k_2 \) at temperature \( T_1 \). It is given that \( k_1 > k_2 \).

### Step 2: Effect of Temperature on Rate Constants
When the temperature increases from \( T_1 \) to \( T_2 \), the rate constants change to \( k_3 \) and \( k_4 \) for the two reactions, respectively. According to the principles of chemical kinetics, for first-order reactions, an increase in temperature generally leads to an increase in the rate constants.

### Step 3: Establish Relationships
Since the temperature is increased:
- The new rate constant \( k_3 \) for the first reaction must be greater than \( k_1 \) because increasing temperature increases the rate constant.
- Similarly, the new rate constant \( k_4 \) for the second reaction must be greater than \( k_2 \).

Thus, we can establish the following relationships:
- \( k_3 > k_1 \)
- \( k_4 > k_2 \)

### Step 4: Compare the Rate Constants
Since \( k_1 > k_2 \), and we know that \( k_3 > k_1 \) and \( k_4 > k_2 \), we can infer:
- \( k_3 \) is greater than both \( k_1 \) and \( k_2 \).
- \( k_4 \) is greater than \( k_2 \) but we cannot directly compare \( k_3 \) and \( k_4 \) without additional information.

### Step 5: Conclusion
Given the relationships we have established, the most logical conclusion is:
- \( k_1 > k_2 \)
- \( k_3 > k_1 \)
- \( k_4 > k_2 \)

Thus, the correct relation is:
- \( k_1 > k_2 < k_4 \) and \( k_3 > k_1 \)

Based on the options provided, the correct option is:
- \( k_1 > k_2 < k_3 < k_4 \)

### Final Answer
The correct relation is \( k_1 > k_2 < k_3 < k_4 \).

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