Rate equation

In chemistry, a rate equation describes how fast a chemical reaction occurs. This mathematical expression is also known as a rate law or an empirical differential rate equation. It expresses the reaction rate in terms of the concentrations of chemical species. It also includes constant parameters, such as the rate coefficient and partial orders of reaction. Understanding these rates is vital for predicting how substances interact over time. Scientists use these equations to understand the underlying processes of chemical changes.

The most common form for a rate equation is a power law. This law relates the reaction rate to the molar concentrations of the reactants. In this formula, the concentration of each reactant is raised to a specific exponent. These exponents are called the partial orders of reaction. The sum of all these exponents is known as the overall reaction order. The overall order quantifies how much the reaction rate depends on reactant concentrations. A higher order suggests the rate is very sensitive to concentration changes. The rate constant, or $k$, is a coefficient that stays constant for a specific reaction. However, its value can change based on temperature, light, or surface area.

Reaction orders can take several different forms. A zero-order reaction means the rate is independent of the reactant's concentration. This often happens when a bottleneck limits the reaction, such as when an enzyme is saturated. A first-order reaction depends on the concentration of only one reactant. The rate of a first-order reaction is proportional to that single concentration. A second-order reaction occurs when the overall order is two. This might mean the rate depends on one reactant squared. It could also mean the rate depends on the product of two different concentrations.

Scientists cannot simply look at a chemical equation to find the rate law. The numbers in a chemical recipe are called stoichiometric coefficients. While elementary reactions follow these coefficients, complex reactions often do not. An elementary reaction is a single-step process with one transition state. In these cases, the reaction order matches the molecularity of the reaction. However, complex reactions involve multiple steps. Because the mechanism is unknown, the rate law must be determined through experimentation. Once a rate law is found, it helps scientists deduce the actual reaction mechanism.

There are several ways to find a rate equation in a laboratory. The method of initial rates involves measuring the rate at the very start of a reaction. Scientists run multiple experiments with different starting concentrations to find the exponents. Another approach is the integral method. This method compares measured concentrations over a longer period of time. It is often used to verify the results from the initial rates method. Scientists also use the method of flooding, or isolation. In this method, one reactant is placed in a large excess. This keeps its concentration essentially constant, allowing the order of other reactants to be measured.

Specific examples illustrate how these different orders work in nature. For a zero-order reaction, the decomposition of phosphine on a hot tungsten surface is an example. Many enzyme-catalyzed reactions are also zero order when the enzyme is saturated. A classic first-order reaction is the decomposition of hydrogen peroxide into water and oxygen. Another example is the decomposition of dinitrogen pentoxide ($N_2O_5$). In organic chemistry, $S_N1$ nucleophilic substitution reactions are first-order. For second-order reactions, the alkaline hydrolysis of ethyl acetate is a common example. This specific reaction is first-order for each of its two reactants.

Reaction kinetics can also change in very specific environments. In highly dilute solutions, such as those below the micromolar level, molecular collisions change. In these cases, collisions are primarily governed by diffusion. Molecules must travel longer distances before they encounter one another. This causes the apparent reaction order to deviate from what is expected. This behavior is consistent with fractal reaction kinetics, which result in fractional reaction orders. By studying these deviations, scientists can better understand how molecules move and interact in complex systems.