(a) Determining the order of the reaction

To determine the order of the reaction, we can use the method of initial rates. This involves measuring the initial rate of reaction for different initial concentrations of the reactant. If the initial rate of reaction changes with changes in the initial concentration, then the reaction is not a zero-order reaction. If the initial rate changes with the square of the initial concentration, then the reaction is second-order. If the initial rate changes with the initial concentration, then the reaction is first-order.

Using the given concentrations of species A at different times, we can calculate the initial rates of reaction as follows:

At t = 0 s: initial rate = k[A]0 = k(1.0) = k
At t = 1 s: initial rate = (k/2)[A]1 = (k/2)(0.5) = 0.25k
At t = 2 s: initial rate = (k/4)[A]2 = (k/4)(0.33) = 0.0825k
At t = 3 s: initial rate = (k/8)[A]3 = (k/8)(0.25) = 0.03125k

Since the initial rate of reaction changes with changes in the initial concentration of A, the reaction is not zero-order. Furthermore, the initial rate changes with the time, so we can rule out third-order kinetics. Comparing the initial rates at different concentrations, we see that the initial rate changes with the initial concentration to the power of 1. Therefore, the reaction is first-order kinetics.

(b) The order of the reaction is one.

(c) Explanation

From the initial rates calculated above, we can see that the initial rate of the reaction changes with changes in the initial concentration of A. This indicates that the reaction is not zero-order. Furthermore, the initial rate changes with time, so we can rule out third-order kinetics. Comparing the initial rates at different concentrations, we see that the initial rate changes with the initial concentration to the power of 1. Therefore, the reaction is first-order kinetics.