For three reactions of first, second and third-order, the numerical value of the rate constant is the same. Which of the following is correct?
Given , [A] = the concentration of the reactant & `r_1 , r_2 and r_3` are the rates of first, second and third-order reaction respectively

Options

A. If `[A] = 1`, then `r_1 = r_2 = r_3`
B. If `[A] lt 1`, then , `r_1 gt r_2 gt r_3`
C. If `[A] gt 1`, then `r_3 gt r_2 gt r_1`
D. All of the above

Answer

To solve the question, we need to analyze the rate expressions for first, second, and third-order reactions given that the rate constant \( k \) is the same for all three reactions.

### Step 1: Write the rate expressions for each order of reaction.

1. **First-order reaction:**
\[ R_1 = k[A]^1 = k[A] \]

2. **Second-order reaction:**
\[ R_2 = k[A]^2 \]

3. **Third-order reaction:**
\[ R_3 = k[A]^3 \]

### Step 2: Analyze the rates when the concentration \( [A] \) is equal to 1.

If we set \( [A] = 1 \):
- For the first-order reaction:
\[ R_1 = k \cdot 1 = k \]

- For the second-order reaction:
\[ R_2 = k \cdot 1^2 = k \]

- For the third-order reaction:
\[ R_3 = k \cdot 1^3 = k \]

Thus, when \( [A] = 1 \):
\[ R_1 = R_2 = R_3 = k \]

### Step 3: Analyze the rates when the concentration \( [A] \) is less than 1.

Let’s assume \( [A] < 1 \):
- For the first-order reaction:
\[ R_1 = k[A] \quad \text{(will be less than } k\text{)} \]

- For the second-order reaction:
\[ R_2 = k[A]^2 \quad \text{(will be less than } k\text{ since } [A]^2 < [A]\text{)} \]

- For the third-order reaction:
\[ R_3 = k[A]^3 \quad \text{(will be the least since } [A]^3 < [A]^2\text{)} \]

Thus, the order of rates when \( [A] < 1 \) will be:
\[ R_1 > R_2 > R_3 \]

### Step 4: Analyze the rates when the concentration \( [A] \) is greater than 1.

Let’s assume \( [A] > 1 \):
- For the first-order reaction:
\[ R_1 = k[A] \quad \text{(will be greater than } k\text{)} \]

- For the second-order reaction:
\[ R_2 = k[A]^2 \quad \text{(will be greater than } R_1\text{ since } [A]^2 > [A]\text{)} \]

- For the third-order reaction:
\[ R_3 = k[A]^3 \quad \text{(will be the greatest since } [A]^3 > [A]^2\text{)} \]

Thus, the order of rates when \( [A] > 1 \) will be:
\[ R_3 > R_2 > R_1 \]

### Conclusion

Based on the analysis:
1. When \( [A] = 1 \), \( R_1 = R_2 = R_3 \).
2. When \( [A] < 1 \), \( R_1 > R_2 > R_3 \).
3. When \( [A] > 1 \), \( R_3 > R_2 > R_1 \).

Thus, the correct answer is that all the statements about the rates are correct, leading us to choose option 4: "All of the above."