This reaction was monitored as a function of time: A plot of ln[A] versus time yields a straight line with slope -0.0045>s. a. What is the value of the rate constant (k) for this reaction at this temperature? b. Write the rate law for the reaction. c. What is the half-life? d. If the initial concentration of A is 0.250 M, what is the concentration after 225 s?

Understanding the rate constant is fundamental in chemical kinetics, particularly in studying the speed at which a chemical reaction occurs. For a first-order reaction, the rate constant, denoted as 'k', is a proportionality factor that relates the rate of the reaction to the concentration of the reactants. In the given exercise, the value of the rate constant is deduced from the slope of the plot of ln[A] against time, which is -0.0045 s^-1. Since the slope is negative, it signifies a decrease in the concentration of reactant A over time, typical of a reaction proceeding forward.

Determining the rate constant is a critical step as it not only helps to predict how fast a reaction proceeds but also assists in calculating other essential parameters such as the half-life of the reaction and the concentration of reactants at any given time. Beyond academic exercises, the rate constant reveals much about the reaction conditions, such as temperature and the presence of a catalyst, making it a valuable metric in both laboratory studies and large-scale industrial processes.

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of the reactants. For the reaction described in the exercise, which is of first-order, the rate law takes a simple form: Rate = k[A]. This equation states that the rate of the reaction is directly proportional to the concentration of reactant A, with 'k' being the rate constant.

Writing the rate law provides a predictive power to chemists; it allows them to estimate the speed of a reaction under various conditions by just knowing the concentrations of the reactants involved. One important aspect of rate laws is that they cannot be determined solely by stoichiometry of the reaction, but must be established experimentally, as was done in the step-by-step solution provided.

The half-life of a reaction, represented as t_1/2, is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, this concept exceptionally simplifies, as the half-life is independent of the initial concentration and only depends on the rate constant. The formula t_1/2 = ln(2)/k illustrates this relationship.

In the context of the exercise, knowing that the rate constant k is 0.0045 s^-1, the half-life can be straightforwardly calculated. The beauty of first-order kinetics is reflected here – regardless of the starting amount of reactant A, the time it takes to reduce it to half will always remain constant. This property can be quite advantageous in fields such as pharmacology, where it is crucial to understand how quickly a drug's active ingredient will decrease to half its initial concentration in the body.

Calculating the concentration of reactants at any given time is a pivotal aspect of chemical kinetics, and in first-order reactions, it involves the integrated rate law. This mathematical model relates the concentration of a reactant at time 't' to its initial concentration and the elapsed time. The integrated rate law for a first-order reaction is given by ln([A]_t/[A]₀) = -kt.

In practice, as demonstrated in the exercise, to find the concentration of A after 225 s, you would rearrange the integrated rate law to solve for [A]_t and apply the known values of k, the initial concentration [A]₀, and time 't'. This exercise is not just an academic drill but a fundamental technique used in labs and industries to track the progress of reactions over time. The ability to predict concentrations at future times helps chemists control and optimize reactions for desired outcomes.