Chemical kinetics is the study of how quickly chemical processes occur, focusing on the rate at which reactants are consumed and products are formed. The mathematical expression that relates the reaction rate to the concentration of its reactants is known as the Rate Law. This relationship cannot be predicted simply by looking at the balanced chemical equation. The rate law must be determined through experimental measurements, as the reaction’s speed depends on the sequence of molecular steps, called the reaction mechanism, which the overall equation does not reveal.
Understanding the Components of a Rate Law
The Rate Law is written in the general form: Rate = \(k[A]^x[B]^y\). The Rate is typically measured as the change in concentration per unit of time, such as molarity per second (M/s). The term \([A]\) represents the concentration of reactant A, and the exponent \(x\) is the reaction order with respect to A. The reaction order describes how sensitive the reaction rate is to a change in the concentration of that specific reactant.
If a reaction is first order in \(A\) (\(x=1\)), doubling the concentration of \(A\) will double the reaction rate. If it is second order (\(y=2\)), doubling the concentration will quadruple the rate. The overall reaction order is the sum of all the individual orders (\(x + y\)). The constant \(k\) is the rate constant, a proportionality factor characteristic of the specific reaction and highly dependent on temperature.
The stoichiometric coefficients from the balanced chemical equation rarely match the experimentally determined reaction orders (\(x\) and \(y\)). This difference arises because the balanced equation only shows the starting and ending materials, not the series of elementary steps that constitute the reaction mechanism. The rate law reflects the concentrations involved in the slowest step of that mechanism, which acts as the bottleneck for the entire process. Therefore, the exponents in the Rate Law must be found by measuring the change in rate as reactant concentrations are systematically varied.
Determining Reaction Orders Using the Method of Initial Rates
The Method of Initial Rates is the primary experimental technique used to determine the reaction orders (\(x\) and \(y\)). This involves measuring the reaction rate immediately after the reactants are mixed. The process requires running the reaction multiple times, with each trial starting with different initial concentrations. The initial rate is measured for each trial before concentrations change significantly, which simplifies the mathematical analysis.
To solve for the exponent \(x\), a chemist must compare two experimental trials where the initial concentration of A is changed, but the concentration of all other reactants (like B) is held constant. By taking the ratio of the rate law for Trial 2 to the rate law for Trial 1, the rate constant \(k\) and the constant concentration of \(B\) cancel out algebraically. This leaves the simplified relationship: \(\frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[A]_2}{[A]_1}\right)^x\).
If the concentration of A is doubled from Trial 1 to Trial 2, and the initial rate is observed to quadruple, the resulting equation is \(4 = 2^x\). This means the exponent \(x\) must be 2, indicating the reaction is second order with respect to A. If doubling the concentration of A only doubled the rate, the equation would be \(2 = 2^x\), making \(x=1\), or first order.
Alternatively, if changing the concentration of A has no effect on the rate, the ratio of the rates would be 1, leading to \(1 = 2^x\), meaning \(x=0\). A zero-order reaction means the rate is independent of that reactant’s concentration. Once the order for A is determined, the process is repeated by comparing trials where the concentration of A is held constant while the concentration of B is varied. This comparison allows the chemist to isolate and solve for the exponent \(y\) using the same ratio method.
Completing the Equation: Calculating the Rate Constant (\(k\))
After determining the reaction orders (\(x\) and \(y\)), the final step in establishing the complete Rate Law is to calculate the numerical value of the rate constant, \(k\). Since the orders are known, calculating \(k\) requires plugging in the measured values from any single experimental trial into the general Rate Law expression. A chemist can select the initial rate and corresponding initial concentrations of A and B from a trial and rearrange the Rate Law to solve for \(k\): \(k = \frac{\text{Rate}}{[A]^x[B]^y}\).
The value of \(k\) is typically calculated for every experimental trial, and the average of these values is reported as the final rate constant for the reaction under the specific experimental conditions, such as temperature. Because the rate constant is a true constant at a fixed temperature, the calculated \(k\) should be approximately the same regardless of which trial’s data is used.
Determining the correct units for \(k\) is important, as they are not fixed and depend entirely on the overall reaction order. The units for \(k\) must ensure that when multiplied by the concentration terms (in molarity, \(M\)) raised to their respective powers, the overall result yields the correct units for the Rate (\(M/s\)). For example, the units of \(k\) for a second-order overall reaction are \(M^{-1}s^{-1}\), while for a third-order reaction, they become \(M^{-2}s^{-1}\).