The rate constant, symbolized as \(k\), is a proportionality factor that links the rate of a chemical reaction to the concentrations of the reactants. It measures how quickly a reaction proceeds, with a larger value indicating a faster reaction. Since \(k\) is temperature-dependent, its value changes if the reaction temperature is altered. While \(k\) can be calculated using instantaneous rate data, a more reliable method involves analyzing how reactant concentration changes over an extended period. This article focuses on the graphical method, which relies on plotting experimental data to find a straight line from which \(k\) can be directly determined.
The Integrated Rate Law Framework
Measuring the reaction rate directly at different concentrations is mathematically complex, so chemists use the integrated rate law to simplify analysis. The integrated rate law is a mathematical rearrangement of the reaction’s rate law that relates a reactant’s concentration directly to time. This transformation converts the typically curved plot of concentration versus time into a straight line, modeled after the algebraic equation \(y = mx + b\).
By testing three specific mathematical transformations of the concentration data against time, a researcher determines which plot yields a linear relationship. The linear plot identifies the reaction’s order, and its slope provides the numerical value for the rate constant \(k\).
Zero-Order Reactions
For a zero-order reaction, the rate does not depend on the concentration of the reactant, meaning the concentration decreases at a constant pace over time. The integrated rate law describing this behavior is \([A]_t = -kt + [A]_0\), where \([A]_t\) is the concentration at time \(t\), and \([A]_0\) is the initial concentration.
To determine \(k\), one must plot the concentration of the reactant, \([A]\), on the vertical axis against time \((t)\) on the horizontal axis. If the reaction is zero-order, this plot will produce a straight line with a negative slope. The slope of this linear plot is equal to the negative of the rate constant, \(m = -k\).
To calculate \(k\) from the graph, one first determines the slope \((m)\) using two points on the line: \(m = \frac{\Delta [A]}{\Delta t}\). Since the slope equals the negative of the rate constant (\(m = -k\)), the rate constant \(k\) is the absolute value of the slope. For example, if the calculated slope is \(-0.0050 \text{ M/s}\), the rate constant \(k\) is \(0.0050 \text{ M/s}\).
First-Order Reactions
A first-order reaction is one where the reaction rate is directly proportional to the concentration of a single reactant. Since the concentration decreases exponentially over time, the integrated rate law uses the natural logarithm of the concentration to achieve linearization.
The first-order integrated rate law is expressed as \(\ln[A]_t = -kt + \ln[A]_0\). In this equation, the natural logarithm of the concentration at time \(t\), \(\ln[A]_t\), plays the role of the \(y\)-variable. The natural logarithm of the initial concentration, \(\ln[A]_0\), serves as the \(y\)-intercept \((b)\).
The correct plot for a first-order reaction is the natural logarithm of the reactant concentration (\(\ln[A]\)) on the vertical axis versus time \((t)\) on the horizontal axis. A straight line in this plot confirms the reaction is first-order. The slope of this line is equal to the negative of the rate constant, \(m = -k\).
To find the rate constant \(k\), the slope \((m)\) of the \(\ln[A]\) versus \(t\) line is calculated using \(m = \frac{\Delta (\ln[A])}{\Delta t}\). If the calculated slope is, for example, \(-0.025 \text{ s}^{-1}\), the rate constant \(k\) is \(0.025 \text{ s}^{-1}\). The units for a first-order rate constant are typically inverse time, such as \(\text{s}^{-1}\).
Second-Order Reactions
Second-order reactions are those in which the rate is proportional to the square of one reactant’s concentration. Similar to the first-order case, a simple plot of concentration versus time yields a curve. The integrated rate law for a single-reactant second-order reaction is used to transform this curve into a straight line.
The integrated rate law for a single-reactant second-order reaction is \(\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}\). Here, the reciprocal of the concentration at time \(t\), \(\frac{1}{[A]_t}\), is the \(y\)-variable. The reciprocal of the initial concentration, \(\frac{1}{[A]_0}\), functions as the \(y\)-intercept \((b)\).
The graphical method requires plotting the inverse of the reactant concentration, \(\frac{1}{[A]}\), on the vertical axis against time \((t)\) on the horizontal axis. The resulting linear plot confirms the reaction is second-order. Unlike zero- and first-order reactions, the slope of this line is directly equal to the positive rate constant, \(m = +k\).
To calculate \(k\), the slope \((m)\) is found using \(m = \frac{\Delta (1/[A])}{\Delta t}\). If the calculated slope is \(0.15 \text{ M}^{-1}\text{s}^{-1}\), the rate constant \(k\) is simply \(0.15 \text{ M}^{-1}\text{s}^{-1}\). The units for a second-order rate constant are typically \(\text{M}^{-1}\text{s}^{-1}\).