A certain first-order reaction is 35.5 percent complete in 4.90 min at \(25^{\circ} \mathrm{C}\). What is its rate constant?

Reaction kinetics is the study of rates of chemical processes and the factors that affect these rates. It is a sub-discipline of physical chemistry that examines how the speed of a chemical reaction is influenced by various variables such as temperature, concentration of reactants, surface area, and the presence of catalysts. Understanding the speed at which reactants are transformed into products is crucial for controlling industrial chemical processes and for gaining insight into the mechanism of the reaction.

For example, consider the exercise where we are asked to determine the rate constant for a reaction that is 35.5 percent complete after 4.90 minutes. To solve such problems, we apply the principles of reaction kinetics, specifically the mathematics of first-order reactions, to find how quickly the reaction is progressing towards completion.

Rate laws are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentrations of reactants. According to the law of mass action, the rate of a chemical reaction at a fixed temperature depends on the concentrations of the substances involved in the reaction. This expression takes the form of an equation known as the rate law.

For first-order reactions, the rate law is expressed as 'rate = k[A]', where 'k' is the rate constant and '[A]' is the concentration of the reactant A. The rate law tells us that the reaction rate is directly proportional to the concentration of the reactant. In the solution to our exercise, we use the rate law for first-order kinetics to establish the value of the rate constant 'k', which is a measure of how quickly a reaction proceeds.

Chemical kinetics involves the experimental and theoretical determinations of how different experimental conditions can change the speed of a chemical reaction and yield information about the reaction's mechanism and transition states. It is an essential field for understanding how reactions occur. In the context of our exercise, we use a formula derived from the concepts of chemical kinetics to calculate the rate constant 'k' from the given reaction completion percentage and time.

This incorporation is a practical example of how knowledge in chemical kinetics can be applied to real-world problems to find the rate at which a reaction is proceeding. By understanding the rate constant and other kinetic parameters, chemists can design and optimize reactions for industrial production, pharmaceuticals, and environmental processes.

The half-life of a reaction is the time required for the concentration of a reactant to reduce to half its initial value. It is a vital concept in both chemistry and physics with wide applications ranging from radioactive decay to the metabolism of drugs in the body. For first-order reactions, the half-life is particularly important because it is constant regardless of the initial concentration of the reactant.

In the first-order reaction, the half-life can be derived from the rate constant 'k' using the equation \( t_{1/2} = \frac{\ln(2)}{k} \). Our previous example doesn't directly ask for the half-life, but knowing the rate constant, one could easily calculate it. The concept of half-life was not needed explicitly in this exercise, but it's a fundamental property to understand when interpreting reaction rates.

Integrated rate equations are mathematical expressions that relate the concentrations of reactants or products to time. They are obtained by integrating the corresponding differential rate laws. Each order of reaction has its own form of the integrated rate equation, which can be used to determine reaction kinetics parameters such as the rate constant 'k'.

For our given problem, a first-order reaction's integrated rate equation is \( \ln[A] = -kt + \ln[A]_{0} \) or rearranged to solve for 'k', \( k = \frac{\ln([A]_{0}/[A])}{t} \). This formula takes into account the initial concentration \( [A]_{0} \) and the concentration at time 't', [A]. This integrated rate equation is crucial for solving the exercise as it allows us to plug in the given completion percentage and time to find the rate constant \( k \) for the chemical reaction in question.