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

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. All of these

Answer

To solve the problem, we need to analyze the rates of three reactions of different orders (first, second, and third) given that their rate constants are numerically the same and the concentration of the reactant is greater than 1 M.

### Step-by-Step Solution:

1. **Define the Rate Expressions**:
- For a first-order reaction, the rate \( R_1 \) is given by:
\[ R_1 = k[A] \]
- For a second-order reaction, the rate \( R_2 \) is given by:
\[ R_2 = k[A]^2 \]
- For a third-order reaction, the rate \( R_3 \) is given by:
\[ R_3 = k[A]^3 \]

2. **Substitute the Concentration**:
- We know that the concentration \( [A] \) is greater than 1 M. Let's denote \( [A] = a \), where \( a > 1 \).
- Thus, we can rewrite the rates as:
\[ R_1 = k \cdot a \]
\[ R_2 = k \cdot a^2 \]
\[ R_3 = k \cdot a^3 \]

3. **Compare the Rates**:
- Since \( a > 1 \), we can analyze how the rates compare:
- \( R_1 = k \cdot a \) will be the smallest since it is linear with respect to \( a \).
- \( R_2 = k \cdot a^2 \) will be larger than \( R_1 \) because \( a^2 > a \) when \( a > 1 \).
- \( R_3 = k \cdot a^3 \) will be the largest since \( a^3 > a^2 \) when \( a > 1 \).

4. **Conclusion**:
- Therefore, the order of the rates is:
\[ R_3 > R_2 > R_1 \]
- This means that the correct answer to the question is that the rate of the third-order reaction is greater than the second-order reaction, which in turn is greater than the first-order reaction.

### Final Answer:
The correct order of the rates is:
\[ R_3 > R_2 > R_1 \]