Consider the hypothetical reaction $$ \mathrm{A}+\mathrm{B}+2 \mathrm{C} \longrightarrow 2 \mathrm{D}+3 \mathrm{E} $$ where the rate law is $$ \text { Rate }=-\frac{\Delta[\mathrm{A}]}{\Delta t}=k[\mathrm{~A}][\mathrm{B}]^{2} $$ An experiment is carried out where \([\mathrm{A}]_{0}=1.0 \times 10^{-2} M\), \([\mathrm{B}]_{0}=3.0 \mathrm{M}\), and \([\mathrm{C}]_{0}=2.0 \mathrm{M} .\) The reaction is started, and after \(8.0\) seconds, the concentration of \(\mathrm{A}\) is \(3.8 \times 10^{-3} \mathrm{M}\). a. Calculate the value of \(k\) for this reaction. b. Calculate the half-life for this experiment. c. Calculate the concentration of A after \(13.0\) seconds. d. Calculate the concentration of \(\mathrm{C}\) after \(13.0\) seconds.

Understanding the speed of chemical reactions is essential in the study of chemical kinetics. The reaction rate refers to the rate at which the reactants are consumed or the products are formed over time. In the given hypothetical reaction \[ \mathrm{A}+\mathrm{B}+2 \mathrm{C} \longrightarrow 2 \mathrm{D}+3 \mathrm{E} \], the reaction rate can be calculated by measuring the change in concentration of a reactant or product over time. The rate law for this reaction follows \[\text{Rate} = k[\mathrm{~A}][\mathrm{B}]^{2}\]where \(k\) is the rate constant, and the concentrations of \(\mathrm{A}\) and \(\mathrm{B}\) are raised to the powers corresponding to their stoichiometric coefficients in the rate-determining step. By plugging the concentrations of the reactants into the rate law equation and measuring the change in concentration over time, we can calculate the reaction rate.

For instance, to find the initial rate, we would use the initial concentrations and the rate law equation while to find the average rate over 8 seconds, we would use the change in concentration of \(\mathrm{A}\). These calculations provide a basis to characterize the kinetics of the reaction and determine the rate constant \(k\), which is crucial for predicting the reaction's behavior under different conditions.

The reaction order is a critical concept in kinetics as it provides insight into the relationship between the concentration of reactants and the rate of reaction. It is often not apparent from the chemical equation and must be determined experimentally. The order of reaction with respect to each reactant shows the power to which its concentration term is raised in the rate law. In our example, the rate law is given by \[\text{Rate}=k[\mathrm{A}][\mathrm{B}]^{2}\]which suggests that the reaction is first-order with respect to [\(\mathrm{A}\)] and second-order with respect to [\(\mathrm{B}\)]. The overall reaction order is the sum of these individual orders, thus making it a third-order reaction.

To confirm these orders experimentally, one would typically perform a series of reactions varying the concentration of one reactant while holding others constant and observing the effect on the reaction rate. The data obtained helps establish the rate law's exponents empirically and define the kinetics of the reaction.

Integrated rate laws are algebraic expressions that relate concentrations of reactants to time and are derived from the differential rate laws for a given order of reaction. For a second-order reaction, the integrated rate law expression is \[\frac{1}{[A]_{t}} = kt + \frac{1}{[A]_{0}}\]where \([A]_{t}\) is the concentration of reactant A at time \(t\), \([A]_{0}\) is the initial concentration, \(k\) is the rate constant, and \(t\) is time. By applying this formula, we can predict future concentrations of reactants or determine how the concentration has changed over a given time.

In solving the exercise problem, we can calculate the concentration of A after 13.0 seconds by first finding the rate constant \(k\) using initial rate and average rate calculations. Once \(k\) is determined, we can manipulate the integrated rate law to find \([A]_{t}\) at any point in time, providing valuable insight into how concentrations vary throughout the course of the reaction.

Half-life, symbolized as \(t_{1/2}\), is the amount of time required for half of the reactant to be consumed in a chemical reaction. For different orders of reactions, the half-life formulas vary. In a second-order reaction, the half-life is directly dependent on the initial concentration and inversely proportional to the rate constant. The half-life expression for a second-order reaction is given by \[t_{1/2} = \frac{1}{k[A]_0}\]where \(k\) is the rate constant, and \([A]_0\) is the initial concentration of the reactant.

Calculating half-life is crucial in understanding how quickly a reaction proceeds and can be especially important in pharmaceuticals, environmental engineering, and other fields where reaction timing is critical. In the case of our particular problem, knowing \(k\) and \([A]_0\), we can determine the half-life of the reaction which essentially provides a timescale for the reaction's progress under the given conditions.

A second-order reaction is characterized by a rate that is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. In our textbook case, we deal with the second-order kinetics since the rate law is \[\text{Rate}= -\frac{\Delta[\mathrm{A}]}{\Delta t} = k[\mathrm{A}][\mathrm{B}]^{2}\]suggesting the reaction rate is proportional to the concentration of A and the square of the concentration of B. For second-order reactions, the integrated rate law and half-life expressions are unique as previously discussed.

These reactions exemplify kinetic behavior where interactions between two reactant molecules or two reactive sites within a single molecule are necessary for the reaction to proceed. The nuances of second-order reactions are important to master as they often appear in biochemical processes, such as enzyme kinetics, and have different characteristics and mathematical treatments compared to zero or first-order kinetics.