Short Answer 
The reaction order with respect to the reactant is determined to be 2. This conclusion is reached by observing that increasing the reactant concentration by four times increases the reaction rate by sixteen times, leading to the equation 4^n = 16, which simplifies to n = 2.
Step 1: Understanding the Given Data
To find the order of the reaction with respect to the reactant, we start by analyzing the provided information. The problem states that the rate of reaction increases by sixteen times when the concentration of the reactant is increased by four times. To represent this mathematically, we can use the rate law equation: r = k[A]^n, where r is the rate, k is the rate constant, [A] is the reactant concentration, and n is the order of the reaction.
Step 2: Setting Up the Equations
We will define the initial concentration and rate for easier calculations. Let the initial concentration be [A] = [A]_0 and the rate at this concentration be r_0. We can express this as:
- Initial equation: r_0 = k[A]_0^n
- New concentration when increased: r_1 = k(4[A]_0)^n = k ¬¨‚à ë 4^n ¬¨‚à ë [A]_0^n
This sets the stage to relate the original and new reaction rates based on the concentration change.
Step 3: Using the Rate Increase to Solve for n
According to the problem, the new rate, r_1, is sixteen times the original rate, r_0. Therefore, we can establish the equation: r_1 = 16r_0. By substituting our earlier equations:
- k ¬¨‚à ë 4^n ¬¨‚à ë [A]_0^n = 16(k[A]_0^n)
- This simplifies to: 4^n = 16
Recognizing that 16 is equivalent to 4^2, we can equate the exponents and solve for n: n = 2. Thus, the order of the reaction is 2.