Step-by-step Derivation:

1. Define the Rate Law: Let the order of the reaction with respect to the reactant M be 'n'. The rate law for the reaction can be expressed as: $\text{Rate} = k[\text{M}]^n$ where Rate is the rate of the reaction, k is the rate constant, and [M] is the concentration of M. The rate of disappearance of M is directly proportional to the rate of the reaction, so we can use this expression.

2. Set up Initial Conditions: Let the initial rate of disappearance of M be $R_1$ when the concentration of M is $[\text{M}]_1$. According to the rate law: $R_1 = k[\text{M}]_1^n$

3. Set up Final Conditions: The problem states that the concentration of M is doubled. So, the new concentration, $[\text{M}]_2$, is: $[\text{M}]_2 = 2[\text{M}]_1$ Upon doubling the concentration, the rate of disappearance of M increases by a factor of 8. So, the new rate, $R_2$, is: $R_2 = 8R_1$ The new rate can also be expressed using the rate law with the new concentration: $R_2 = k[\text{M}]_2^n$

4. Solve for the Order (n): Now we have a system of two equations: (i) $R_1 = k[\text{M}]_1^n$ (ii) $R_2 = k[\text{M}]_2^n$

   Substitute the given relationships into equation (ii): $8R_1 = k(2[\text{M}]_1)^n$

   Divide the modified equation (ii) by equation (i): $\frac{8R_1}{R_1} = \frac{k(2[\text{M}]_1)^n}{k[\text{M}]_1^n}$

   Simplify the expression: $8 = \frac{2^n [\text{M}]_1^n}{[\text{M}]_1^n}$ $8 = 2^n$

   To find the value of n, we can express 8 as a power of 2: $2^3 = 2^n$

   Therefore, the order of the reaction with respect to M is: $n = 3$

5. Conclusion: The order of the reaction with respect to M is 3. This corresponds to option B.

   Note: The question describes the reaction M $\to$ N as "elementary". For an elementary reaction, the order is typically equal to the stoichiometric coefficient of the reactant, which is 1 in this case. However, the experimental data provided (rate increases 8-fold when concentration doubles) leads to an order of 3. In chemical kinetics, the order of a reaction is an experimentally determined quantity. When experimental data contradicts the stoichiometry of a supposedly elementary reaction, the experimental data is used to determine the rate law. Therefore, we base our answer on the given rate change.