Chemical kinetics is the study of reaction rates, describing how quickly the concentration of reactants changes into products over time. The reaction rate is determined by a rate law, which incorporates a proportionality factor known as the rate constant, symbolized by \(k\). This rate constant is a specific value for a given reaction at a particular temperature and quantifies the reaction’s speed. For a first-order reaction, the rate is linearly dependent on the concentration of only one reactant.
The Integrated Rate Law
To relate a reactant’s concentration to the time elapsed during a first-order reaction, chemists use the integrated rate law. This equation is derived from the differential rate law using calculus, providing a direct connection between concentration and time. For a first-order reaction of the form \(A \rightarrow products\), the law is written as \(\ln[A]_t = -kt + \ln[A]_0\).
In this equation, \([A]_t\) is the concentration of reactant \(A\) remaining at time \(t\), and \([A]_0\) is the initial concentration. The rate constant, \(k\), is characteristic of the reaction under those conditions. The structure of this equation is useful because it resembles the algebraic form of a straight line, \(y = mx + b\).
The units for the rate constant \(k\) in a first-order reaction are always expressed as reciprocal time, such as \(s^{-1}\) or \(min^{-1}\). This unit structure ensures consistency, as the natural logarithm (\(\ln\)) of a concentration is unitless. The integrated rate law is the foundation for determining the rate constant experimentally.
Finding the Rate Constant Using Graphical Analysis
The most robust method for calculating the rate constant involves collecting experimental data and applying a graphical technique based on the integrated rate law. This approach begins by measuring the reactant concentration at various time intervals throughout the reaction. For a first-order reaction, the data collected will initially show a curved plot when concentration is plotted against time.
To transform this curve into a straight line, every measured concentration value must be converted to its natural logarithm (\(\ln[A]\)). This transformed data is then plotted with the natural logarithm of the concentration (\(\ln[A]\)) on the y-axis and time (\(t\)) on the x-axis.
If the reaction is indeed first-order, this plot yields a straight line, confirming the reaction’s order. The slope of this linear plot provides the rate constant directly, as the integrated rate law is structured so that the slope (\(m\)) equals the negative of the rate constant (Slope \(= -k\)).
To calculate the slope, select any two distinct points \((t_1, \ln[A]_1)\) and \((t_2, \ln[A]_2)\) on the line. The slope is calculated using the formula \(m = \frac{\ln[A]_2 – \ln[A]_1}{t_2 – t_1}\). The rate constant \(k\) is then found by taking the negative of the slope value. This graphical analysis is preferred because it uses all collected measurements, yielding the most accurate value for \(k\).
Finding the Rate Constant Using Half-Life Data
An alternative method for finding the rate constant involves using the reaction’s half-life, symbolized as \(t_{1/2}\). The half-life is the time required for the reactant concentration to decrease to half of its initial value. For first-order reactions, the half-life is constant and does not depend on the initial concentration.
The relationship between the rate constant and the half-life is mathematically simple. This relationship is expressed by the equation \(t_{1/2} = \frac{0.693}{k}\), where \(0.693\) is the approximate value of \(\ln 2\). By rearranging this formula, the rate constant can be calculated directly if the half-life is known: \(k = \frac{0.693}{t_{1/2}}\).
To use this method, experimentally determine the time it takes for the concentration to halve, which can be read directly from a concentration-versus-time graph. Dividing \(0.693\) by the measured half-life provides the rate constant \(k\). This technique is a quick estimation, but it relies on accurately determining a single time point, contrasting with the graphical method that incorporates multiple data points for higher precision.