What is the half-life for the decomposition of \(\mathrm{O}_{3}\) when the concentration of \(\mathrm{O}_{3}\) is \(2.35 \times 10^{-6} \mathrm{M}\) ? The rate constant for this second-order reaction is \(50.4 \mathrm{L} \mathrm{mol}^{-1} \mathrm{h}^{-1}\).

Chemical kinetics is a branch of physical chemistry that concerns the rates at which chemical reactions occur. Its study involves understanding how different variables—such as temperature, pressure, reactant concentration, and the presence of a catalyst—affect the speed of a reaction. The rate of a chemical reaction is usually expressed in terms of the change in concentration of reactants or products per unit of time. For students tackling kinetic problems, grasping the fundamental concept of reaction rates is crucial. It’s this rate that governs how fast a product is formed or a reactant is consumed.

In the case of second-order reactions, the rate is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. Differentiating between reaction orders is vital because it determines the mathematical relationships that allow us to predict reaction behavior and calculate essential parameters like half-life—a concept we explore in the context of the given exercise.

The reaction rate constant, denoted as 'k', is a proportionality factor that provides a quantitative measure of the speed of a chemical reaction. For a given reaction at a particular temperature, the rate constant ties the reaction rate to the concentration(s) of the reactant(s) as described by the rate law. It is affected by environmental conditions, notably temperature, and in the case of heterogeneous reactions, surface area.

When solving kinetics problems, understanding the implications of the rate constant is essential. For instance, in a second-order reaction, a larger 'k' value means a faster reaction, as you can see from the inverse relationship in the second-order half-life equation. As demonstrated in the methodology provided for calculating the half-life of ozone decomposition, the rate constant is indispensable in predicting how long it takes for half of the reactant to react.

Reactant concentration, often signified by square brackets \( [A] \), plays a pivotal role in the kinetics of a reaction. In simple terms, it's the measure of how much of a substance is present in a given volume of solution. The change in this concentration over time is what chemists refer to as the reaction rate. As you advance in chemistry, you'll often come across concentration expressed in molarity (M), or moles per liter (mol/L).

For second-order reactions specifically, the reaction rate depends on the concentration of one reactant raised to the second power, or the product of two reactant concentrations. This distinctive feature is reflected in the unique form of the half-life equation for these reactions. In practice, as the reactant concentration decreases, the rate of reaction slows down - a concept vividly illustrated by the exercise in question. Students should note that keeping track of concentration changes is fundamental for accurate reaction rate analysis and predictions.