Question

Define the half-life of a reaction. Write the equation relating the half-life of a first-order reaction to the rate constant.

Short answer

The half-life of a reaction is the time it takes for the concentration of a reactant to decrease to half of its initial value. In a first-order reaction, the half-life (\(t_{1/2}\)) is related to the rate constant (\(k\)) by the equation \(t_{1/2} = 0.693/k\).

Step-by-step solution

Define half-life of a reaction

The half-life of a reaction, often denoted as \(t_{1/2}\), is the time it takes for the concentration of a reactant to decrease to half of its initial value. It is a measure of how quickly a reaction occurs.

Understand first-order reaction characteristics

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. This means the reaction rate decreases as the concentration of the reactant decreases.

Derive the equation for half-life in first-order reactions

In a first-order reaction, the rate constant (\(k\)) is given by the equation \(k = 2.303/t \cdot \log(C_0/C)\), where \(C_0\) is the initial concentration, \(C\) is the concentration at time \(t\). When \(t = t_{1/2}\), \(C = C_0/2\). Substituting these values into the equation, we get: \(k = 2.303/t_{1/2} \cdot \log(2)\), which simplifies to: \(t_{1/2} = 0.693/k\).

Key concepts

First-Order Reaction

Understanding a first-order reaction is central to grasping the nuances of chemical kinetics, a field that studies the speed of chemical processes. In this type of reaction, the rate at which the reaction occurs is directly proportional to the concentration of a single reactant. This proportionality means that if you were to double the concentration of the reactant, the rate of the reaction would also double.

Moreover, the distinctive feature of a first-order reaction lies in its exponential decay behavior—over equal time intervals, a constant fraction of the reactant is consumed rather than a constant amount. This trait leads to a linear relationship when plotting the natural logarithm of the reactant concentration versus time, making it a handy tool for identifying such reactions experimentally.

A practical aspect of this concept is seen in pharmaceuticals, where the degradation of certain drugs follows first-order kinetics, and knowing this helps in determining the drug's shelf life and proper dosage.

Rate Constant

The rate constant, symbolized by the letter 'k,' is a pivotal parameter in the realm of reaction kinetics. It quantifies the speed of a reaction, linking the rate of a reaction to the concentrations of reactants according to the rate law. For a first-order reaction, the rate law can be expressed as:
\[\text{Rate} = k [A]\]
with '[A]' representing the concentration of the reactant.

The magnitude of 'k' not only provides insights into how fast a reaction proceeds but also is influenced by external factors such as temperature and the presence of catalysts. In thermal processes, for example, the rate constant typically increases with temperature, following the Arrhenius equation. A higher rate constant implies a faster reaction, conversely, a low rate constant indicates a slowly proceeding chemical process. The units of 'k' for a first-order reaction are reciprocal seconds (s^-1), emphasizing the time component in reaction rates.

Reaction Kinetics

Delving into reaction kinetics allows us to comprehend the speed (rate) and mechanism of a chemical reaction. One of the core aspects of reaction kinetics is determining how different variables such as concentration, temperature, and catalysts affect the reaction rate.

It is through studying kinetics that scientists and engineers are able to optimize conditions for industrial reactions, control pollution, and even understand metabolic pathways in biochemistry. In an educational context, mastering reaction kinetics equips students with the skills to predict how changes in conditions can alter the outcome of a chemical reaction, making it a fundamental topic in chemistry.

The chemical rate laws, integral components of reaction kinetics, provide the mathematical relationship that links the rate of reaction to the concentration of reactants. By inspecting these relationships, one can deduce the order of a reaction, which has profound implications on both understanding and manipulating chemical processes.