If the rate of a reaction increases by a factor of 64 when the concentration of reactant increases by a factor of 4, what is the order of the reaction with respect to this reactant?

The reaction is of order three with respect to the reactant.

The rate of a reaction of order n about a certain reactant is proportion to the concentration of that reactant raised to the n-th power. This is true only if  concentrations of any other reactants stay constant in the whole process.

In other words, Rate = constant × [Reactant]ⁿ, Rate ∝ [Reactant]ⁿ. (The symbol "∝" reads "proportional to".)

In this question,

[4 × Reactant]ⁿ ÷ [Reactant]ⁿ = 64.

In other words, 4ⁿ = 64, where n is the order of the reaction with respect to this reactant.

It might take some guesswork to find the value of n. Alternatively, n can be solved directly with a calculator using logarithms. Taking natural log of both sides:

.

Evaluating on Google or on a calculator with support for ln (the natural log) will give the value of n- no guesswork required.

n = 3. Therefore, the reaction is of order three with respect to this reactant.

every chemical equation adheres to the law of conservation of mass, which states that matter cannot be created or destroyed. therefore, there must be the same number of atoms of each element on each side of a chemical equation.