Determination of Reaction Order from Concentration Change

To determine the order of the reaction with respect to $\text{M}$, we write the general rate law for the disappearance of reactant $\text{M}$:

$r = -\frac{d[\text{M}]}{dt} = k [\text{M}]^n$

where:

• $r$ is the rate of disappearance of $\text{M}$,
• $k$ is the rate constant,
• $[\text{M}]$ is the concentration of $\text{M}$,
• $n$ is the order of the reaction with respect to $\text{M}$.

According to the given information, when the concentration of $\text{M}$ is doubled, i.e., $[\text{M}]' = 2[\text{M}]$, the new rate $r'$ increases by a factor of $8$:

$r' = 8r$

Using the rate law for the new concentration:

$r' = k [\text{M}']^n = k (2[\text{M}])^n = 2^n \cdot k [\text{M}]^n = 2^n \cdot r$

Equating the two expressions for $r'$:

$2^n \cdot r = 8r$

$2^n = 8$

Since $8 = 2^3$:

$2^n = 2^3 \implies n = 3$

Therefore, the order of the reaction with respect to $\text{M}$ is $3$.

Correct Option: (B)