Rate equation
The rate law or rate equation for a chemical reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters. For many reactions the initial rate is given by a power law such as
where and express the concentration of the species A and B, usually in moles per liter. The exponents x and y are the partial orders of reaction for A and B 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 constant k is the reaction rate constant or rate coefficient of the 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.
The rate equation of a reaction with an assumed multistep mechanism can often be derived theoretically using quasisteady 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 number of moles of chemical X,
If the reaction takes place in a closed system at constant temperature and volume, without a buildup 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 _{0} = 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.
Determination of reaction order
Method of initial rates
The natural logarithm of the powerlaw rate equation isThis 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 A with all other concentrations , ,... kept constant, so that
The slope of a graph of as a function of then corresponds to the order x with respect to reactant A.
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
For example, the integrated rate law for a firstorder reaction is
where is the concentration at time t and _{0} is the initial concentration at zero time. The firstorder 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 a·A + b·B → c·C with rate law:, the partial order x with respect to A is determined using a large excess of B. In this casewith,
and x may be determined by the integral method. The order y with respect to B under the same conditions is determined by a series of similar experiments with a range of initial concentration _{0} so that the variation of k' can be measured.
Zero order
For zeroorder reactions, the reaction rate is independent of the concentration of a reactant, so that changing its concentration has no effect on the speed of the reaction. Thus, the concentration changes linearly with time. 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 enzymecatalyzed 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, ringopening metathesis polymerization using thirdgeneration Grubbs catalyst exhibits zero order behavior in catalyst due to the reversible inhibition that is occur between the 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 each will be zero order. The rate law for such a reaction isThe halflife is independent of the starting concentration and is given by.
Examples of such reactions are:

H2O2 > H2O + 1/2O2 
SO2Cl2 > SO2 + Cl2 
2N2O5 > 4NO2 + O2
Second order
A reaction is said to be second order when the overall order is two. The rate of a secondorder reaction may be proportional to one concentration squared, or to the product of two concentrations. As an example of the first type, the reaction NO_{2} + CO → NO + CO_{2} is secondorder in the reactant NO_{2} and zero order in the reactant CO. The observed rate is given by, and is independent of the concentration of CO.For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
The time dependence for a rate proportional to two unequal concentrations is
if the concentrations are equal, they satisfy the previous equation.
The second type includes nucleophillic additionelimination reactions, such as the alkaline hydrolysis of ethyl acetate:
This reaction is firstorder in each reactant and secondorder overall: _{0} = k
If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes v = k. The rate is firstorder in one reactant, and also firstorder in imidazole which as a catalyst does not appear in the overall chemical equation.
Another wellknown class of secondorder reactions are the S_{N}2 reactions, such as the reaction of nbutyl bromide with sodium iodide in acetone:
This same compound can be made to undergo a bimolecular elimination reaction, another common type of secondorder reaction, if the sodium iodide and acetone are replaced with sodium tertbutoxide as the salt and tertbutanol as the solvent:
Pseudofirst order
If the concentration of a reactant remains constant, its concentration can be included in the rate constant, obtaining a pseudo–firstorder rate equation. For a typical secondorder reaction with rate equation v = k, if the concentration of reactant B is constant then _{0} = k = k', where the pseudo–firstorder rate constantk' = k. The secondorder rate equation has been reduced to a pseudo–firstorder rate equation, which makes the treatment to obtain an integrated rate equation much easier.
One way to obtain a pseudofirst order reaction is to use a large excess of one reactant so that, as the reaction progresses, only a small fraction of the reactant in excess is consumed, and its concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudofirst order kinetics where the concentration of water is present in large excess:
The hydrolysis of sucrose in acid solution is often cited as a firstorder reaction with rate r = k. The true rate equation is thirdorder, r = k; however, the concentrations of both the catalyst H^{+} and the solvent H_{2}O are normally constant, so that the reaction is pseudo–firstorder.
Summary for reaction orders 0, 1, 2, and ''n''
Elementary reaction steps with order 3 are rare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any order.Zero order  First order  Second order  nth order  
Rate Law  
Integrated Rate Law  
Units of Rate Constant  
Linear Plot to determine k  vs.  
Halflife 
Where M stands for concentration in molarity, t for time, and k for the reaction rate constant. The halflife of a first order reaction is often expressed as t_{1/2} = 0.693/k.
Fractional order
In fractional order reactions, the order is a noninteger, which often indicates a chemical chain reaction or other complex reaction mechanism. For example, the pyrolysis of acetaldehyde into methane and carbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde: r = k^{3/2}. The decomposition of phosgene to carbon monoxide and chlorine has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine: v = k ^{1/2}.The order of a chain reaction can be rationalized using the steady state approximation for the concentration of reactive intermediates such as free radicals. For the pyrolysis of acetaldehyde, the RiceHerzfeld mechanism is
;Initiation :CH_{3}CHO → •CH_{3} + •CHO
;Propagation :•CH_{3} + CH_{3}CHO → CH_{3}CO• + CH_{4}
;Termination :2 •CH_{3} → C_{2}H_{6}
where • denotes a free radical. To simplify the theory, the reactions of the •CHO to form a second •CH_{3} are ignored.
In the steady state, the rates of formation and destruction of methyl radicals are equal, so that
so that the concentration of methyl radical satisfies
The reaction rate equals the rate of the propagation steps which form the main reaction products CH_{4} and CO:
in agreement with the experimental order of 3/2.
Complex laws
Mixed order
More complex rate laws have been described as being mixed order if they approximate to the laws for more than one order at different concentrations of the chemical species involved. For example, a rate law of the form represents concurrent first order and second order reactions reactions, and can be described as mixed first and second order. For sufficiently large values of such a reaction will approximate second order kinetics, but for smaller the kinetics will approximate first order. As the reaction progresses, the reaction can change from second order to first order as reactant is consumed.Another type of mixedorder rate law has a denominator of two or more terms, often because the identity of the ratedetermining step depends on the values of the concentrations. An example is the oxidation of an alcohol to a ketone by hexacyanoferrate ion with ruthenate ion as catalyst. For this reaction, the rate of disappearance of hexacyanoferrate is
This is zeroorder with respect to hexacyanoferrate at the onset of the reaction, but changes to firstorder when its concentration decreases and the regeneration of catalyst becomes ratedetermining.
Notable mechanisms with mixedorder rate laws with twoterm denominators include:
 MichaelisMenten kinetics for enzymecatalysis: firstorder in substrate at low substrate concentrations, zero order in substrate at higher substrate concentrations; and
 the Lindemann mechanism for unimolecular reactions: secondorder at low pressures, firstorder at high pressures.
Negative order
When a partial order is negative, the overall order is usually considered as undefined. In the above example for instance, the reaction is not described as first order even though the sum of the partial orders is, because the rate equation is more complex than that of a simple firstorder reaction.
Opposed reactions
A pair of forward and reverse reactions may occur simultaneously with comparable speeds. For example, A and B react into products P and Q and vice versa :The reaction rate expression for the above reactions can be written as:
where: k_{1} is the rate coefficient for the reaction that consumes A and B; k_{−1} is the rate coefficient for the backwards reaction, which consumes P and Q and produces A and B.
The constants k_{1} and k_{−1} are related to the equilibrium coefficient for the reaction by the following relationship :
Simple example
In a simple equilibrium between two species:where the reaction starts with an initial concentration of reactant A,
Then the constant K at equilibrium is expressed as:
Where and are the concentrations of A and P at equilibrium, respectively.
The concentration of A at time t,, is related to the concentration of P at time t,, by the equilibrium reaction equation:
The term
This applies even when time t is at infinity; i.e., equilibrium has been reached:
then it follows, by the definition of K, that
and, therefore,
These equations allow us to uncouple the system of differential equations, and allow us to solve for the concentration of A alone.
The reaction equation was given previously as:
For
The derivative is negative because this is the rate of the reaction going from A to P, and therefore the concentration of A is decreasing. To simplify notation, let x be, the concentration of A at time t. Let be the concentration of A at equilibrium. Then:
Since:
The reaction rate becomes:
which results in:
A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope k_{1} + k_{−1}. By measurement of _{e} and _{e} the values of K and the two reaction rate constants will be known.
Generalization of simple example
If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions:When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy.
Consecutive reactions
If the rate constants for the following reaction are and ;With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation. The differential equations can be solved analytically and the integrated rate equations are
The steady state approximation leads to very similar results in an easier way.
Parallel or competitive reactions
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.Two first order reactions
The integrated rate equations are then ; and
One important relationship in this case is
One first order and one second order reaction
This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example, A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct:The integrated rate equation for the main product is, which is equivalent to. Concentration of B is related to that of C through
The integrated equations were analytically obtained but during the process it was assumed that therefeore, previous equation for can only be used for low concentrations of compared to _{0}
Stoichiometric reaction networks
The most general description of a chemical reaction network considers a number of distinct chemical species reacting via reactions.The chemical equation of the th reaction can then be written in the generic form
which is often written in the equivalent form
Here
The rate of such reaction can be inferred by the law of mass action
which denotes the flux of molecules per unit time and unit volume. Here
 zero order reactions
 first order reactions
 second order reactions
Each of which are discussed in detail below. One can define the stoichiometric matrix
denoting the net extent of molecules of in reaction. The reaction rate equations can then be written in the general form
This is the product of the stoichiometric matrix and the vector of reaction rate functions.
Particular simple solutions exist in equilibrium,, for systems composed of merely reversible reactions. In this case the rate of the forward and backward reactions are equal, a principle called detailed balance. Detailed balance is a property of the stoichiometric matrix alone and does not depend on the particular form of the rate functions. All other cases where detailed balance is violated are commonly studied by flux balance analysis which has been developed to understand metabolic pathways.
General dynamics of unimolecular conversion
For a general unimolecular reaction involving interconversion of different species, whose concentrations at time are denoted by through, an analytic form for the timeevolution of the species can be found. Let the rate constant of conversion from species to species be denoted as, and construct a rateconstant matrix whose entries are the.Also, let be the vector of concentrations as a function of time.
Let be the vector of ones.
Let be the × identity matrix.
Let be the function that takes a vector and constructs a diagonal matrix whose ondiagonal entries are those of the vector.
Let be the inverse Laplace transform from to.
Then the timeevolved state is given by
thus providing the relation between the initial conditions of the system and its state at time.