Chemical kinetics investigates the rate at which a chemical reaction occurs. Understanding the reaction rate dictates how quickly reactants are consumed and products are formed. The rate law is a mathematical expression that precisely describes the relationship between the reaction rate and the concentrations of the reactants.
The rate law allows chemists to predict how changes in reactant amounts will affect the overall speed of the chemical process. The rate law cannot be determined simply by looking at the balanced chemical equation; it must be found through careful experimentation. Determining the rate law is the first step in understanding a reaction’s mechanism, which is the detailed sequence of steps by which the reaction takes place.
Defining the Rate Law Components
The general form of the rate law for a reaction between reactants A and B is written as \(\text{Rate} = k[\text{A}]^m[\text{B}]^n\). The brackets \([\text{A}]\) and \([\text{B}]\) represent the molar concentrations of the reactants. To fully define the rate law, two components must be determined experimentally: the reaction orders (\(m\) and \(n\)) and the rate constant (\(k\)).
The reaction orders, represented by the exponents \(m\) and \(n\), indicate how sensitive the reaction rate is to changes in the concentration of each reactant. For instance, if a reaction is first-order (\(m=1\)) with respect to A, doubling the concentration of A will double the reaction rate. If it is second-order (\(m=2\)), doubling the concentration will quadruple the rate.
Reaction orders are typically small positive integers like 0, 1, or 2, but they can be fractional or negative. The rate constant, \(k\), is the proportionality factor that links the reaction rate to the reactant concentrations. It is an intrinsic measure of the reaction’s speed at a specific temperature, and a larger value of \(k\) signifies a faster reaction.
Determining Reaction Order Using the Method of Initial Rates
The Method of Initial Rates is a standard experimental technique used to determine the reaction orders for each reactant. This method involves performing a series of experiments where the initial reaction rate is measured at different starting concentrations. The key is to systematically vary the concentration of only one reactant at a time while keeping the concentrations of all others constant.
By isolating the effect of a single reactant, the corresponding reaction order can be mathematically determined. For a reaction with two reactants, A and B, one compares two experiments where the concentration of A is varied but the concentration of B is held constant. The ratio of the two initial rates is compared to the ratio of the two initial concentrations of A, and the difference reveals the exponent \(m\).
For example, if doubling the concentration of A causes the rate to increase by a factor of four, the reaction is second-order (\(m=2\)) with respect to A. This process is repeated for reactant B by comparing two experiments where the concentration of B is changed while A is held constant, thereby finding the exponent \(n\). Once all individual reaction orders are known, the rate law expression is complete except for the value of \(k\).
The rate constant, \(k\), is calculated by substituting the reaction orders, the concentration data, and the measured initial rate from any single experiment back into the full rate law expression. Since the initial rate and concentrations are known, the equation can be rearranged to solve for \(k\). The units of \(k\) will vary depending on the overall reaction order, which is the sum of all individual orders (\(m+n\)).
Determining Reaction Order Through Graphical Analysis (Integrated Rate Laws)
An alternative approach is to monitor the concentration of a single reactant over time in a single experiment, which uses the concept of integrated rate laws. Integrated rate laws are mathematical rearrangements of the differential rate law that relate concentration directly to time. This method is especially useful for reactions where the concentration of only one reactant significantly affects the rate.
For the three most common reaction orders (zero, first, and second), a specific plot of concentration data versus time will yield a straight line. The order of the reaction is determined by identifying which of these three characteristic plots is linear.
Zero-Order
If a plot of the reactant concentration, \([\text{A}]\), versus time (\(t\)) produces a straight line, the reaction is zero-order.
First-Order
If the natural logarithm of the concentration, \(\ln[\text{A}]\), versus time is linear, the reaction is first-order.
Second-Order
If the inverse of the concentration, \(1/[\text{A}]\), versus time yields a straight line, the reaction is second-order.
The rate constant, \(k\), is directly related to the slope of the linear plot. For zero-order and first-order reactions, the slope is equal to \(-k\), while for a second-order reaction, the slope is equal to \(+k\). This graphical analysis allows for the simultaneous determination of both the reaction order and the numerical value of the rate constant from a single set of continuous measurements.
Summarizing the Overall Process
Finding the complete rate law for a chemical reaction requires a two-part experimental process: first determining the reaction orders, and then calculating the rate constant. Whether the Method of Initial Rates is used to find the individual orders \(m\) and \(n\), or Integrated Rate Laws are used for graphical confirmation, the final goal is the same. The resulting rate law, \(\text{Rate} = k[\text{A}]^m[\text{B}]^n\), provides a concise summary of how reactant concentrations influence the reaction speed.
This mathematical model allows scientists to predict the rate of the reaction under any given set of reactant concentrations. Moreover, the form of the rate law provides strong evidence about the reaction mechanism by indicating which species are involved in the rate-determining step. It is important to remember that the rate constant, \(k\), is not a true constant but is highly sensitive to temperature.
The value of \(k\) typically increases significantly as the temperature of the reaction is raised, often doubling or tripling for every \(10^\circ \text{C}\) increase. This temperature dependence is described by the Arrhenius equation, which links the rate constant to the activation energy of the reaction. The complete rate law is a powerful predictive tool, but it is only accurate if the reaction is performed at the specific temperature for which \(k\) was determined.