The rate constant is numerically the same for three reactions of first, second and third order respectively. Which one of the following is true for the rate of these reactions if concentration of the reactant is same and greater than 1 M ?

Options

A. `r_(1)=r_(2)=r_(3)`
B. `r_(1)gt r_(2)gt r_(3)`
C. `r_(1) lt r_(2) lt r_(3)`
D. There can be no definite order

Explanation

To solve the problem, we need to analyze the rates of first, second, and third-order reactions given that the rate constant (K) is the same and the concentration of the reactant (A) is greater than 1 M.

### Step-by-Step Solution:

1. **Understand the Rate Laws**:
- For a first-order reaction:
\[ \text{Rate} (R_1) = k[A]^1 = k[A] \]
- For a second-order reaction:
\[ \text{Rate} (R_2) = k[A]^2 \]
- For a third-order reaction:
\[ \text{Rate} (R_3) = k[A]^3 \]

2. **Substituting the Same Concentration**:
Let’s denote the concentration of the reactant A as [A] = C, where C > 1 M.

3. **Express the Rates**:
- For first order:
\[ R_1 = kC \]
- For second order:
\[ R_2 = kC^2 \]
- For third order:
\[ R_3 = kC^3 \]

4. **Compare the Rates**:
Since k is the same for all three reactions, we can compare the rates directly based on the powers of C:
- \( R_1 = kC \)
- \( R_2 = kC^2 \)
- \( R_3 = kC^3 \)

5. **Analyzing the Values**:
Given that C > 1:
- \( C^2 > C \) (This means \( R_2 > R_1 \))
- \( C^3 > C^2 \) (This means \( R_3 > R_2 \))

6. **Final Conclusion**:
Therefore, we can conclude that:
\[ R_1 < R_2 < R_3 \]
This means the rate of the first-order reaction is less than the rate of the second-order reaction, which is less than the rate of the third-order reaction.

### Answer:
The correct relationship is:
\[ R_1 < R_2 < R_3 \]