Consider the reaction aX arrow bY , for which the rate constant at...

The unit of the rate constant $k$ is mol$^{-1}$ L s$^{-1}$.

The general unit of the rate constant for an $n$-th order reaction is (mol L$^{-1}$)$^{1-n}$ s$^{-1}$.

Equating the powers of the concentration units, we get $1-n = -1 \Rightarrow n = 2$.

Thus, the reaction is a second-order reaction. Statement B is correct.

For a second-order reaction, Rate $= k[X]^2$. When the concentration of $X$ is increased to $4$ times, the new rate is $k(4[X])^2 = 16k[X]^2$, which is $16$ times the original rate. Statement A is correct.

The half-life of a second-order reaction is $t_{1/2} = \dfrac{1}{k[X]_0}$, which depends inversely on the initial concentration. Statement C is incorrect.

The decomposition of $N_2O_5$ is a well-known first-order reaction. Statement D is incorrect.

The equation $\ln\dfrac{[R_o]}{[R]} = kt$ represents the integrated rate law for a first-order reaction, not a second-order reaction. Statement E is incorrect.

Therefore, only statements A and B are correct.

Answer: A and B Only