In chemical kinetics, the rate constant (\(k\)) is a fundamental value that determines how quickly a chemical change occurs. This constant is a proportionality factor that connects the reaction rate directly to the concentrations of the reactants. Extracting this value from experimental data tables requires a systematic approach to first establish the reaction’s order. This guide provides practical steps for determining the rate constant from the two primary forms of kinetics data tables.
Understanding the Rate Law and the Rate Constant
The relationship between reaction rate and reactant concentration is formalized in the rate law: \(\text{Rate} = k[A]^m[B]^n\). Here, \(\text{Rate}\) is the speed of the reaction, typically measured in molarity per second (\(\text{M}/\text{s}\)), and \([A]\) and \([B]\) represent the molar concentrations of the reactants. The rate constant, \(k\), is the proportionality factor for a given reaction.
The exponents \(m\) and \(n\) are the reaction orders with respect to reactants \(A\) and \(B\). These orders must be determined experimentally, not from the balanced chemical equation. The overall reaction order is the sum of these exponents (\(m+n\)). While \(k\) is independent of reactant concentrations, its value is highly dependent on temperature.
Determining Reaction Order from Initial Rates Tables
The “method of initial rates” uses a table listing the initial concentrations of reactants and the corresponding initial reaction rates from a series of different experiments. The first step is to determine the reaction order by comparing two experiments where the concentration of only one reactant is changed. This comparison isolates the effect of that single reactant on the reaction rate.
To find the exponent \(m\) for reactant \(A\), select two experiments where the concentration of reactant \(B\) remains constant. Taking the ratio of the rates for these two experiments causes the rate constant \(k\) and the constant concentration of \(B\) to cancel out. The resulting equation compares the ratio of the rates to the ratio of the concentrations of \(A\) raised to the power \(m\).
For example, if doubling the concentration of \(A\) causes the rate to quadruple, the reaction is second order with respect to \(A\). If doubling the concentration of \(A\) only doubles the rate, the reaction is first order with respect to \(A\). Repeating this process for all reactants determines the full rate law.
Determining Reaction Order from Concentration vs. Time Tables
Another type of experimental data provides concentration measurements taken at various time intervals over the course of a single reaction. This data requires the use of integrated rate laws, which relate concentration directly to time. A graphical approach is the clearest method to find the reaction order in this case.
The technique involves plotting the experimental data in three different ways, each corresponding to a common reaction order. A plot of the concentration of the reactant, \([A]\), versus time yields a straight line if the reaction is zero order. For a first-order reaction, the straight line appears when the natural logarithm of the concentration, \(\ln[A]\), is plotted against time.
If a plot of the reciprocal of the concentration, \(1/[A]\), versus time results in a straight line, the reaction is second order. The linear plot reveals the correct reaction order, and its slope is directly related to the rate constant, \(k\). For zero and first-order reactions, the slope equals \(-k\) (or \(-k\) multiplied by a stoichiometric factor). For a second-order reaction, the slope is equal to positive \(k\).
Calculating the Rate Constant Value and Deriving Its Units
Once the full rate law is established by determining the reaction orders (\(m\) and \(n\)), the next step is to calculate the numerical value of the rate constant, \(k\). This is achieved by substituting the concentration and rate data from any single experiment back into the solved rate law equation. Using the rearranged formula, \(k = \text{Rate} / ([A]^m[B]^n)\), allows for the calculation of \(k\).
The rate constant’s units are not fixed but depend entirely on the overall reaction order. Since the reaction rate is always expressed in \(\text{M}/\text{s}\), the units of \(k\) must mathematically balance the units of the concentration terms in the rate law to result in \(\text{M}/\text{s}\). Calculating the units correctly serves as a final check to ensure the reaction order was determined accurately.
The units for common reaction orders are:
- First-order reaction (overall order of 1): \(\text{s}^{-1}\) (inverse time).
- Second-order reaction (overall order of 2): \(\text{M}^{-1}\text{s}^{-1}\).
- Third-order reaction: \(\text{M}^{-2}\text{s}^{-1}\).
- Zero-order reaction (overall order of 0): \(\text{M}\text{s}^{-1}\).