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.

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The rate law expression for the given reactions will be as follows:

$\begin{matrix} r_{1} = k [A] & (I^{st} order) \\ r_{2} = k {[A]}^{2} & ({II}^{nd} order) \\ r_{3} = k {[A]}^{3} & ({III}^{rd} order) \end{matrix}} k = same$

Thus, the following conclusions can be drawn based on the mathematical analysis of the rate law expressions:

1) $r_{1} = r_{2} = r_{3};$ if $[A] = 1$

2) $r_{1} < r_{2} < r_{3};$ if $[A] > 1$

For example, if $[A] = 2 M, r_{1} = 2 k, r_{2} = 4 k, r_{3} = 8 k$

3) $r_{1} > r_{2} > r_{3};$ if $[A] < 1$

For example, if $[A] = \frac{1}{2} M, r_{1} = \frac{k}{2}, r_{2} = \frac{k}{4}, r_{3} = \frac{k}{8}$

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