The speed at which reactants are consumed and products are formed is mathematically described by the rate law, an equation that links the reaction rate to the concentrations of the reactants. For a general reaction, the rate law is expressed as \(Rate = k[A]^x[B]^y\), where \(k\) is the rate constant, and \([A]\) and \([B]\) are the molar concentrations of the reactants. The exponents \(x\) and \(y\) are the reaction orders with respect to each reactant, indicating how sensitive the rate is to changes in concentration. Determining these exponents is a purely experimental process, using two primary methods to find the individual reaction orders and the overall rate law.
The Method of Initial Rates
The method of initial rates determines the exponents in the rate law by focusing on the reaction’s speed at the very beginning, before concentrations change significantly. This approach requires performing the reaction multiple times under identical conditions, varying only the starting concentration of one reactant at a time to isolate its effect on the rate.
To find the order with respect to reactant A, two experiments are compared where the initial concentration of A is changed, but all other reactant concentrations are held constant. The ratio of the initial rates from these trials is compared to the ratio of the corresponding initial concentrations of A. For example, if doubling the concentration of A doubles the initial rate, the reaction is first-order (\(x=1\)). If doubling the concentration of A quadruples the rate, the reaction is second-order (\(x=2\)).
This systematic comparison is repeated for every reactant to determine all individual orders, which are then summed to yield the overall reaction order.
The Method of Integrated Rate Laws
The method of integrated rate laws analyzes concentration versus time data from a single experiment to determine the reaction order. This method uses mathematical expressions that relate reactant concentration directly to time. The integrated rate laws for zero-order, first-order, and second-order reactions can be rearranged to resemble the equation for a straight line.
The technique relies on plotting the concentration data in three different ways to observe which plot yields a linear relationship. For a zero-order reaction, a plot of the reactant concentration \([A]\) directly against time will be linear. If the reaction is first-order, plotting the natural logarithm of the concentration, \(\ln[A]\), against time will produce a straight line. A second-order reaction is identified when a plot of the inverse of the reactant concentration, \(1/[A]\), versus time results in a straight line. The plot that is linear reveals the true order of the reaction because only the correct integrated rate law can linearize the experimental data.
Why Rate Order Differs from Reaction Stoichiometry
The exponents in the rate law cannot be determined by simply looking at the stoichiometric coefficients in the balanced chemical equation. The rate order is an experimentally determined value that reflects the reaction’s molecular mechanism, while stoichiometry only represents the overall mass balance. The balanced equation shows the net result of the reaction, but it does not reveal the step-by-step pathway by which reactants are converted into products.
Most chemical transformations occur through a sequence of elementary steps. The overall rate of the reaction is governed by the slowest step in this sequence, which is called the rate-determining step. The reaction order for an overall process is therefore dictated by the molecularity of the rate-determining step. Reactants that are not involved in this slow step will not appear in the experimentally determined rate law, even if they are present in the balanced equation, resulting in a zero-order dependence for that species. Only in the rare case where the overall reaction occurs in a single step—an elementary reaction—will the reaction orders match the stoichiometric coefficients.
Practical Significance of Reaction Order
Determining the reaction order provides insights fundamental to chemical research and industrial applications. The exponents in the rate law offer clues about the step-by-step mechanism of a reaction. By knowing which reactants, and how many of each, are involved in the rate-determining step, chemists can propose and confirm the likely sequence of molecular events.
Once the reaction order is established, the complete rate law allows for the accurate prediction of the reaction rate under any given set of reactant concentrations. This predictive capability is vital in fields like pharmaceutical development or materials science, where precise control over reaction speed is necessary for product consistency and quality. Furthermore, the integrated rate laws, derived from the reaction order, allow chemists to calculate the concentration of any reactant remaining at any point in the future, which is essential for optimizing industrial processes and maximizing yield.