Consider two reactions. Reaction (1) has a constant half-life, whereas reaction (2) has a half-life that gets longer as the reaction proceeds. What can you conclude about the rate laws of these reactions from these observations?

Half-life is the time it takes for the concentration of a reactant to decrease by half. The rate law is an equation that describes the relationship between the reaction rate and the concentrations of the reactants. The order of the reaction is determined by the exponents of the concentration terms in the rate law equation. For first-order reactions, the rate law equation is: \(rate = k[A]\), where k is the rate constant, and [A] is the concentration of the reactant A. For first-order reactions, the half-life is constant and independent of the initial reactant concentration. In other words, it takes the same time to go from 100% to 50% of the reactant as it does to go from 50% to 25%. For other reaction orders, the half-life depends on the initial concentration of the reactants.

Since reaction (1) has a constant half-life, it is indicative of a first-order reaction. Therefore, the rate law for reaction (1) can be represented as: \(rate = k_1[A_1]\), where k_1 is the rate constant for reaction (1) and [A_1] is the concentration of reactant A for reaction (1).

As the half-life for reaction (2) gets longer as the reaction proceeds, it suggests that the reaction is not first-order; otherwise, the half-life would be constant. Therefore, we can conclude that the rate law for reaction (2) is not of the form \(rate = k_2[A_2]\), where k_2 is the rate constant for reaction (2) and [A_2] is the concentration of reactant A for reaction (2). This reaction could be a second-order or a higher-order reaction, depending on the experimental data. In general, a zeroth-order reaction and a second-order reaction have a half-life that depends on the initial concentration of the reactants, but given the information we have, we cannot say for sure which order it is. We can only conclude that reaction (2) is not a first-order reaction.

From the given information about the half-lives of the reactions, we can conclude that: 1. Reaction (1) is a first-order reaction with a rate law of the form \(rate = k_1[A_1]\). 2. Reaction (2) is not a first-order reaction, and its rate law must be of a different form, which could be either zeroth-order, second-order, or a higher-order. The exact form depends on the experimental data.