Rate equation

In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as
where and are the molar concentrations of the species and usually in moles per liter. The exponents and are the partial orders of reaction for and, respectively, and the overall reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. The order of reaction is a number which quantifies the degree to which the rate of a chemical reaction depends on concentrations of the reactants. In other words, the order of reaction is the exponent to which the concentration of a particular reactant is raised. The constant is the reaction rate constant or rate coefficient and at very few places velocity constant or specific rate of reaction. Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent, or light irradiation. If the reaction goes to completion, the rate equation for the reaction rate applies throughout the course of the reaction.
Elementary reactions and reaction steps have reaction orders equal to the stoichiometric coefficients for each reactant. The overall reaction order, i.e. the sum of stoichiometric coefficients of reactants, is always equal to the molecularity of the elementary reaction. However, complex reactions may or may not have reaction orders equal to their stoichiometric coefficients. This implies that the order and the rate equation of a given reaction cannot be reliably deduced from the stoichiometry and must be determined experimentally, since an unknown reaction mechanism could be either elementary or complex. When the experimental rate equation has been determined, it is often of use for deduction of the reaction mechanism.
In highly dilute solutions, such as at concentrations below the micromolar level, molecular collisions are primarily governed by diffusion. Under these conditions, the apparent reaction order deviates from the stoichiometric expectation because reactant molecules require additional time to traverse longer distances before encountering one another. This behavior can be described by Fick's laws of diffusion and is consistent with fractal reaction kinetics, which yield fractional reaction orders.
The rate equation of a reaction with an assumed multi-step mechanism can often be derived theoretically using quasi-steady state assumptions from the underlying elementary reactions, and compared with the experimental rate equation as a test of the assumed mechanism. The equation may involve a fractional order, and may depend on the concentration of an intermediate species.
A reaction can also have an undefined reaction order with respect to a reactant if the rate is not simply proportional to some power of the concentration of that reactant; for example, one cannot talk about reaction order in the rate equation for a bimolecular reaction between adsorbed molecules:

Definition

Consider a typical chemical reaction in which two reactants A and B combine to form a product C:
This can also be written
The prefactors −1, −2 and 3 are known as stoichiometric coefficients. One molecule of A combines with two of B to form 3 of C, so if we use the symbol for the molar concentration of chemical X,
If the reaction takes place in a closed system at constant temperature and volume, without a build-up of reaction intermediates, the reaction rate is defined as
where is the stoichiometric coefficient for chemical X_i, with a negative sign for a reactant.
The initial reaction rate has some functional dependence on the concentrations of the reactants,
and this dependence is known as the rate equation or rate law. This law generally cannot be deduced from the chemical equation and must be determined by experiment.

Power laws

A common form for the rate equation is a power law:
The constant is called the rate constant. The exponents, which can be fractional, are called partial orders of reaction and their sum is the overall order of reaction.
In a dilute solution, an elementary reaction is empirically found to obey the law of mass action. This predicts that the rate depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients.
The differential rate equation for an elementary reaction using mathematical product notation is:
Where:

is the rate of change of reactant concentration with respect to time.
k is the rate constant of the reaction.
represents the concentrations of the reactants, raised to the powers of their stoichiometric coefficients and multiplied together.
Determination of reaction order

Method of initial rates

The natural logarithm of the power-law rate equation is
This can be used to estimate the order of reaction of each reactant. For example, the initial rate can be measured in a series of experiments at different initial concentrations of reactant with all other concentrations kept constant, so that
The slope of a graph of as a function of then corresponds to the order with respect to reactant.
However, this method is not always reliable because

measurement of the initial rate requires accurate determination of small changes in concentration in short times and is sensitive to errors, and
the rate equation will not be completely determined if the rate also depends on substances not present at the beginning of the reaction, such as intermediates or products.
Integral method

The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time with the integrated form of the rate equation; this assumes that the reaction goes to completion.
For example, the integrated rate law for a first-order reaction is
where is the concentration at time and is the initial concentration at zero time. The first-order rate law is confirmed if is in fact a linear function of time. In this case the rate constant is equal to the slope with sign reversed.

Method of flooding

The partial order with respect to a given reactant can be evaluated by the method of flooding of Ostwald. In this method, the concentration of one reactant is measured with all other reactants in large excess so that their concentration remains essentially constant. For a reaction with rate law the partial order with respect to is determined using a large excess of. In this case
with
and may be determined by the integral method. The order with respect to under the same conditions is determined by a series of similar experiments with a range of initial concentration so that the variation of can be measured.

Zero order

For zero-order reactions, the reaction rate is independent of the concentration of a reactant, so that changing its concentration has no effect on the rate of the reaction. Thus, the concentration changes linearly with time. The rate law for zero order reaction is
The unit of k is mol dm⁻³ s⁻¹. This may occur when there is a bottleneck which limits the number of reactant molecules that can react at the same time, for example if the reaction requires contact with an enzyme or a catalytic surface.
Many enzyme-catalyzed reactions are zero order, provided that the reactant concentration is much greater than the enzyme concentration which controls the rate, so that the enzyme is saturated. For example, the biological oxidation of ethanol to acetaldehyde by the enzyme liver alcohol dehydrogenase is zero order in ethanol.
Similarly, reactions with heterogeneous catalysis can be zero order if the catalytic surface is saturated. For example, the decomposition of phosphine on a hot tungsten surface at high pressure is zero order in phosphine, which decomposes at a constant rate.
In homogeneous catalysis zero order behavior can come about from reversible inhibition. For example, ring-opening metathesis polymerization using third-generation Grubbs catalyst exhibits zero order behavior in catalyst due to the reversible inhibition that occurs between pyridine and the ruthenium center.

First order

A first order reaction depends on the concentration of only one reactant. Other reactants can be present, but their concentration has no effect on the rate. The rate law for a first order reaction is
The unit of k is s⁻¹. Although not affecting the above math, the majority of first order reactions proceed via intermolecular collisions. Such collisions, which contribute the energy to the reactant, are necessarily second order. However according to the Lindemann mechanism the reaction consists of two steps: the bimolecular collision which is second order and the reaction of the energized molecule which is unimolecular and first order. The rate of the overall reaction depends on the slowest step, so the overall reaction will be first order when the reaction of the energized reactant is slower than the collision step.
The half-life is independent of the starting concentration and is given by. The mean lifetime is τ = 1/k.
Examples of such reactions are:

2N2O5 -> 4NO2 + O2
^2+ + H2O -> ^3+ + Cl-
H2O2 -> H2O + 1/2O2

In organic chemistry, the class of S_N1 reactions consists of first-order reactions. For example, in the reaction of aryldiazonium ions with nucleophiles in aqueous solution,, the rate equation is where Ar indicates an aryl group.