Determining the order of a chemical reaction is a fundamental step in understanding how a reaction proceeds. The reaction order describes the mathematical relationship between reactant concentration and the speed, or rate, at which the reaction occurs. This relationship clarifies which reactant concentrations most directly influence the reaction’s speed. Observing how reactant concentration changes over time allows scientists to calculate the specific order of the reaction.
The Foundation: Integrated Rate Laws
To determine a reaction’s order graphically, chemists use the integrated rate law, moving beyond the differential rate law. The differential form relates the instantaneous reaction rate to concentration but is complex because the rate constantly changes as concentrations decrease. The integrated rate law shows how reactant concentration changes directly as a function of time.
Integration transforms the non-linear relationship between concentration and time into a linear equation, resembling the mathematical form \(y = mx + b\). A straight line on a graph provides immediate evidence of a specific mathematical relationship. The reaction order is confirmed when testing different mathematical manipulations of the concentration data against time yields a straight line. The data must be collected from a single experiment where the reactant concentration is monitored continuously.
Identifying Zero-Order Reactions Graphically
For a zero-order reaction, the rate does not depend on the reactant concentration. This means the reaction speed remains constant until the reactant is completely consumed. The integrated rate law for a single reactant, \(A\), is expressed as \([A]_t = -kt + [A]_0\).
This equation is already in the linear form \(y = mx + b\). To test for zero-order kinetics, plot the reactant concentration, \([A]\), on the y-axis against time, \(t\), on the x-axis. A perfectly straight line with a negative slope confirms zero-order kinetics. This plot visually represents that the reactant concentration decreases at a fixed rate over time.
Identifying First-Order Reactions Graphically
First-order reactions are common, including radioactive decay and many unimolecular decomposition reactions. In these reactions, the rate is directly proportional to the reactant concentration. Since concentration continuously decreases, the reaction rate continuously slows down, resulting in a curved line when plotting concentration versus time.
To achieve linearity for a first-order reaction, the concentration data must be transformed using the natural logarithm. The integrated rate law is \(\ln[A]_t = -kt + \ln[A]_0\). This equation fits the linear model when the natural logarithm of the concentration, \(\ln[A]\), is plotted against time, \(t\).
If the plot of \(\ln[A]\) versus time produces a straight line, the reaction is confirmed to be first order. The natural logarithm linearizes the exponential decay characteristic of first-order processes. The resulting straight line will have a negative slope, reflecting the decay of the reactant concentration over time.
Identifying Second-Order Reactions Graphically
For a second-order reaction, the rate is proportional to the square of the reactant concentration. This relationship means the rate decreases much more rapidly as the concentration drops compared to a first-order reaction. A simple plot of concentration versus time yields a curve that does not allow for easy rate determination.
To linearize the data for a second-order reaction, a different mathematical transformation must be applied. The integrated rate law is \(1/[A]_t = kt + 1/[A]_0\). To confirm second-order kinetics, the inverse of the reactant concentration, \(1/[A]\), is plotted against time, \(t\).
The key visual indicator for a second-order reaction is a straight line on this inverse concentration plot. Unlike the zero- and first-order plots, the linear graph for a second-order reaction will have a positive slope. This positive slope confirms that the reaction rate is dependent on the concentration squared.
Calculating the Rate Constant
Once a straight line is achieved on one of the three graphical plots, the reaction order is definitively determined. The final step is to calculate the rate constant, \(k\), which quantifies the reaction’s speed under the given conditions. This value is derived directly from the slope of the linear graph.
For zero-order and first-order reactions, the slope of the linear plot equals the negative of the rate constant. Therefore, \(k\) is found by calculating the slope and taking its absolute value. In contrast, for the second-order reaction, the slope of the linear plot equals the positive rate constant itself. Calculating the slope provides the specific, experimentally determined value of \(k\) needed to define the reaction’s rate law.