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In chemistry, the rate law or rate equation for a
reaction Reaction may refer to a process or to a response to an action, event, or exposure: Physics and chemistry *Chemical reaction *Nuclear reaction * Reaction (physics), as defined by Newton's third law *Chain reaction (disambiguation). Biology and m ...
is an equation that links the initial or forward reaction rate with the concentrations or pressures of the
reactant In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
s and constant parameters (normally rate coefficients and partial reaction orders). For many reactions, the initial rate is given by a power law such as :v_0\; =\; k mathrmx mathrmy where and express the concentration of the species and usually in
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
per liter (
molarity Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a solution, in terms of amount of substance per unit volume of solu ...
, ). The exponents and are the partial ''orders of reaction'' for and 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 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 Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a f ...
, or light
irradiation Irradiation is the process by which an object is exposed to radiation. The exposure can originate from various sources, including natural sources. Most frequently the term refers to ionizing radiation, and to a level of radiation that will serve ...
. If the reaction goes to completion, the rate equation for the reaction rate v\; =\; k cex cey applies throughout the course of the reaction. Elementary (single-step) reactions and
reaction step A reaction step of a chemical reaction is defined as: ''"An elementary reaction, constituting one of the stages of a stepwise reaction in which a reaction intermediate (or, for the first step, the reactants) is converted into the next reaction inter ...
s have reaction orders equal to the
stoichiometric coefficients Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions. Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equals ...
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 (multi-step) 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 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: :v_0=k \frac.


Definition

Consider a typical
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
in which two
reactant In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
s A and B combine to form a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
C: :\ce. This can also be written :0 = -\mathrm - 2\mathrm + 3\mathrm. The prefactors −1, −2 and 3 (with negative signs for reactants because they are consumed) are known as
stoichiometric coefficients Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions. Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equals ...
. 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 Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
of chemical X, :-\frac = - \frac \frac = \frac \frac. If the reaction takes place in a
closed system A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed. In ...
at constant temperature and volume, without a build-up of reaction intermediates, the ''reaction rate'' v is defined as :v = \frac \frac, where is the stoichiometric coefficient for chemical X''i'', with a negative sign for a reactant. The initial reaction rate v_0 = v(t = 0) has some functional dependence on the concentrations of the reactants, :v_0 = f\left( mathrm mathrm \ldots\right), 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: :v_0 = k ce Ax ce By \cdots 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 An elementary reaction is a chemical reaction in which one or more chemical species react directly to form products in a single reaction step and with a single transition state. In practice, a reaction is assumed to be elementary if no reaction ...
(one having a single step with a single transition state) is empirically found to obey the
law of mass action In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...
. 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 power-law rate equation is :\ln v_0 = \ln k + x\ln ce A+ y\ln ce B+ \cdots 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 :\ln v_0 = x\ln ce A+ \textrm. The
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of a graph of as a function of \ln ce A/math> 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 (compared to the reaction half-life) 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 (several half-lives) 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 :\ln = -kt + \ln, where is the concentration at time and is the initial concentration at zero time. The first-order rate law is confirmed if \ln 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 (or of isolation) 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 v_0 = k \cdot []^x \cdot []^y, the partial order with respect to is determined using a large excess of . In this case v_0 = k' \cdot []^x with k' = k \cdot []^y, and may be determined by the integral method. The order with respect to under the same conditions (with in excess) 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. 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 Enzymes () are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products ...
or a
catalytic Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
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 Ethanol (abbr. EtOH; also called ethyl alcohol, grain alcohol, drinking alcohol, or simply alcohol) is an organic compound. It is an alcohol with the chemical formula . Its formula can be also written as or (an ethyl group linked to a ...
to acetaldehyde by the enzyme liver alcohol dehydrogenase (LADH) is zero order in ethanol. Similarly reactions with
heterogeneous catalysis In chemistry, heterogeneous catalysis is catalysis where the phase of catalysts differs from that of the reactants or products. The process contrasts with homogeneous catalysis where the reactants, products and catalyst exist in the same phase. ...
can be zero order if the catalytic surface is saturated. For example, the decomposition of phosphine () on a hot
tungsten Tungsten, or wolfram, is a chemical element with the symbol W and atomic number 74. Tungsten is a rare metal found naturally on Earth almost exclusively as compounds with other elements. It was identified as a new element in 1781 and first isol ...
surface at high pressure is zero order in phosphine, which decomposes at a constant rate. In
homogeneous catalysis In chemistry, homogeneous catalysis is catalysis by a soluble catalyst in a solution. Homogeneous catalysis refers to reactions where the catalyst is in the same phase as the reactants, principally in solution. In contrast, heterogeneous catalysi ...
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 (a ''unimolecular reaction''). Other reactants can be present, but their concentration has no effect on the rate. The rate law for a first order reaction is : -\frac = k ce A 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. The rate of these collisions is, however, masked by the fact that the rate determining step remains the unimolecular breakdown of the energized reactant. The half-life is independent of the starting concentration and is given by t_ = \frac. Examples of such reactions are: * 2N2O5 -> 4NO2 + O2 * oCl(NH3)52+ + H2O -> o(H2O)(NH3)53+ + Cl- * H2O2 -> H2O + 1/2O2 In organic chemistry, the class of SN1 (nucleophilic substitution unimolecular) reactions consists of first-order reactions. For example, in the reaction of aryldiazonium ions with
nucleophiles In chemistry, a nucleophile is a chemical species that forms bonds by donating an electron pair. All molecules and ions with a free pair of electrons or at least one pi bond can act as nucleophiles. Because nucleophiles donate electrons, they are ...
in aqueous solution, , the rate equation is v_0 = k ce where Ar indicates an aryl group.


Second order

A reaction is said to be second order when the overall order is two. The rate of a second-order reaction may be proportional to one concentration squared, v_0 = k ce A2, or (more commonly) to the product of two concentrations, v_0 = k ce A\ce B]. As an example of the first type, the reaction is second-order in the reactant and zero order in the reactant CO. The observed rate is given by v_0 = k ce2, 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 : \frac = \frac + kt. The time dependence for a rate proportional to two unequal concentrations is : \frac = \frac e^; if the concentrations are equal, they satisfy the previous equation. The second type includes nucleophilic addition-elimination reactions, such as the alkaline hydrolysis of
ethyl acetate Ethyl acetate ( systematically ethyl ethanoate, commonly abbreviated EtOAc, ETAC or EA) is the organic compound with the formula , simplified to . This colorless liquid has a characteristic sweet smell (similar to pear drops) and is used in glues ...
: :CH3COOC2H5 + OH- -> CH3COO- + C2H5OH This reaction is first-order in each reactant and second-order overall: :v_0 = k ce\ce] If the same hydrolysis reaction is catalysis, catalyzed by imidazole, the rate equation becomes :v_0 = k
text Text may refer to: Written word * Text (literary theory), any object that can be read, including: **Religious text, a writing that a religious tradition considers to be sacred **Text, a verse or passage from scripture used in expository preachin ...
\ce]. The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole, which as a catalyst does not appear in the overall chemical equation. Another well-known class of second-order reactions are the SN2, SN2 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in
acetone Acetone (2-propanone or dimethyl ketone), is an organic compound with the formula . It is the simplest and smallest ketone (). It is a colorless, highly volatile and flammable liquid with a characteristic pungent odour. Acetone is miscib ...
: :CH3CH2CH2CH2Br + NaI -> CH3CH2CH2CH2I + NaBr(v) This same compound can be made to undergo a bimolecular (E2)
elimination reaction An elimination reaction is a type of organic reaction in which two substituents are removed from a molecule in either a one- or two-step mechanism. The one-step mechanism is known as the E2 reaction, and the two-step mechanism is known as the E1 r ...
, another common type of second-order reaction, if the sodium iodide and acetone are replaced with
sodium tert-butoxide Sodium ''tert''-butoxide is the chemical compound with the formula (CH3)3CONa. It is a strong base and a non-nucleophilic base. It is flammable and moisture sensitive. It is sometimes written in chemical literature as sodium ''t''-butoxide. It ...
as the salt and
tert-butanol ''tert''-Butyl alcohol is the simplest tertiary alcohol, with a formula of (CH3)3COH (sometimes represented as ''t''-BuOH). Its isomers are 1-butanol, isobutanol, and butan-2-ol. ''tert''-Butyl alcohol is a colorless solid, which melts near r ...
as the solvent: :CH3CH2CH2CH2Br + NaO\mathit-Bu -> CH3CH2CH=CH2 + NaBr + HO\mathit-Bu


Pseudo-first order

If the concentration of a reactant remains constant (because it is a
catalyst Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
, or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, leading to a ''pseudo–first-order'' (or occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate equation v_0 = k ce A\ce B], if the concentration of reactant B is constant then v_0 = k ce\ce] = k' ce where the pseudo–first-order rate constant k' = k ce B The second-order rate equation has been reduced to a pseudo–first-order rate equation, which makes the treatment to obtain an integrated rate equation much easier. One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, �� so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudo- first order kinetics, where the concentration of water is constant because it is present in large excess: : CH3COOCH3 + H2O -> CH3COOH + CH3OH The hydrolysis of sucrose () in acid solution is often cited as a first-order reaction with rate v_0 = k ce The true rate equation is third-order, v_0 = k ce\ce] ce however, the concentrations of both the catalyst and the solvent are normally constant, so that the reaction is pseudo–first-order.


Summary for reaction orders 0, 1, 2, and ''n''

Elementary reaction steps with order 3 (called ''ternary reactions'') are rare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order. Here stands for concentration in
molarity Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a solution, in terms of amount of substance per unit volume of solu ...
(mol · L−1), for time, and for the reaction rate constant. The half-life of a first order reaction is often expressed as ''t''1/2 = 0.693/''k'' (as ln(2)≈0.693).


Fractional order

In fractional order reactions, the order is a non-integer, which often indicates a chemical chain reaction or other complex reaction mechanism. For example, the
pyrolysis The pyrolysis (or devolatilization) process is the thermal decomposition of materials at elevated temperatures, often in an inert atmosphere. It involves a change of chemical composition. The word is coined from the Greek-derived elements ''py ...
of acetaldehyde () into
methane Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Ea ...
and
carbon monoxide Carbon monoxide (chemical formula CO) is a colorless, poisonous, odorless, tasteless, flammable gas that is slightly less dense than air. Carbon monoxide consists of one carbon atom and one oxygen atom connected by a triple bond. It is the simple ...
proceeds with an order of 1.5 with respect to acetaldehyde: v_0 = k ce. The decomposition of phosgene () to carbon monoxide and
chlorine Chlorine is a chemical element with the symbol Cl and atomic number 17. The second-lightest of the halogens, it appears between fluorine and bromine in the periodic table and its properties are mostly intermediate between them. Chlorine i ...
has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine: v_0 = k \ce^. The order of a chain reaction can be rationalized using the
steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
approximation for the concentration of reactive intermediates such as
free radical A daughter category of ''Ageing'', this category deals only with the biological aspects of ageing. Ageing Ailments of unknown cause Biogerontology Biological processes Causes of death Cellular processes Gerontology Life extension Metabo ...
s. For the pyrolysis of acetaldehyde, the Rice- Herzfeld mechanism is ;Initiation :CH3CHO -> .CH3 + .CHO ;Propagation :.CH3 + CH3CHO -> CH3CO. + CH4 :CH3CO. -> .CH3 + CO ;Termination :2 .CH3 -> C2H6 where • denotes a free radical. To simplify the theory, the reactions of the to form a second are ignored. In the steady state, the rates of formation and destruction of methyl radicals are equal, so that :\frac = k_i cek_t ce2 = 0 , so that the concentration of methyl radical satisfies :
CH3 CH3 may refer to: *Channel 3 (band) * Channel 3 (Thai television network) * Methenium (methyl cation) *Methyl group In organic chemistry, a methyl group is an alkyl derived from methane, containing one carbon atom bonded to three hydrogen atoms ...
\quad\propto \quad H3CHO.
The reaction rate equals the rate of the propagation steps which form the main reaction products and CO: :v_0 = \frac, _0 = k_p\ce \quad\propto \quad\ce 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 v_0 = k_1 k_2 2 represents concurrent first order and second order reactions (or more often concurrent pseudo-first order and second order) 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 (or pseudo-first order). As the reaction progresses, the reaction can change from second order to first order as reactant is consumed. Another type of mixed-order rate law has a denominator of two or more terms, often because the identity of the
rate-determining step In chemical kinetics, the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step (RDS or RD-step or r/d step) or rate-limiting step. For a given reaction mechanism, the prediction of the ...
depends on the values of the concentrations. An example is the oxidation of an alcohol to a ketone by hexacyanoferrate (III) ion e(CN)63−with ruthenate (VI) ion (RuO42−) as
catalyst Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
. For this reaction, the rate of disappearance of hexacyanoferrate (III) is v_0 = \frac This is zero-order with respect to hexacyanoferrate (III) at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated), but changes to first-order when its concentration decreases and the regeneration of catalyst becomes rate-determining. Notable mechanisms with mixed-order rate laws with two-term denominators include: * Michaelis-Menten kinetics for enzyme-catalysis: first-order in substrate (second-order overall) at low substrate concentrations, zero order in substrate (first-order overall) at higher substrate concentrations; and * the
Lindemann mechanism In chemical kinetics, the Lindemann mechanism (also called the Lindemann–Christiansen mechanism or the Lindemann–Hinshelwood mechanism) is a schematic reaction mechanism for Molecularity, unimolecular reactions. Frederick Lindemann and J.A. Chri ...
for unimolecular reactions: second-order at low pressures, first-order at high pressures.


Negative order

A reaction rate can have a negative partial order with respect to a substance. For example, the conversion of
ozone Ozone (), or trioxygen, is an inorganic molecule with the chemical formula . It is a pale blue gas with a distinctively pungent smell. It is an allotrope of oxygen that is much less stable than the diatomic allotrope , breaking down in the lo ...
(O3) to
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as ...
follows the rate equation v_0 =k\ce\ce in an excess of oxygen. This corresponds to second order in ozone and order (−1) with respect to oxygen. 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 2 + (-1) = 1, because the rate equation is more complex than that of a simple first-order 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 (''a, b, p'', and ''q'' are the
stoichiometric coefficient A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities on the right-hand side with a plus sign between ...
s): : + <=> + The reaction rate expression for the above reactions (assuming each one is elementary) can be written as: : v = k_1 ce Aa ce Bb - k_ ce Pp ce Qq 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 (K) by the following relationship (set ''v''=0 in balance): : k_1 ce Aa ce Bb = k_ ce Pp ce Qq : K = \frac = \frac


Simple example

In a simple equilibrium between two species: : A <=> P where the reaction starts with an initial concentration of reactant A, , and an initial concentration of 0 for product P at time ''t''=0. Then the equilibrium constant ''K'' is expressed as: : K \ \stackrel\ \frac = \frac where ce Ae and ce Pe are the concentrations of A and P at equilibrium, respectively. The concentration of A at time ''t'', ce At, is related to the concentration of P at time ''t'', ce Pt, by the equilibrium reaction equation: : \mathit = - \mathit The term is not present because, in this simple example, the initial concentration of P is 0. This applies even when time ''t'' is at infinity; i.e., equilibrium has been reached: : \mathit = - \mathit then it follows, by the definition of ''K'', that : ce Pe = \frac\ce and, therefore, : \ ce Ae = \ce - ce Pe = \frac\ce These equations allow us to uncouple the
system of differential equations In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial dif ...
, and allow us to solve for the concentration of A alone. The reaction equation was given previously as: : v = k_1 ce Aa ce Bb - k_ ce Pp ce Qq For A <=> P this is simply : -\frac = k_1 ce At - k_ ce Pt 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 ce At, the concentration of A at time ''t''. Let x_e be the concentration of A at equilibrium. Then: : \begin -\frac &= - \\ -\frac &= - \\ &= - \\ &= - \end Since: : k_1 + k_ = k_ \frac the reaction rate becomes: : \frac = \frac (x_e - x) which results in: : \ln \left(\frac\right) = (k_1 + k_)t . 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 ''k1'' + ''k−1''. By measurement of sub>''e'' and sub>''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: :\left \ce A \right\ce\frac\left( k_+k_e^ \right)+\ce\frac\left( 1-e^ \right) :\left \ce P \right\ce\frac\left( 1-e^ \right)+\ce\frac\left( k_+k_e^ \right) 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 Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique to observe local magnetic fields around atomic nuclei. The sample is placed in a magnetic fie ...
.


Consecutive reactions

If the rate constants for the following reaction are k_1 and k_2; A -> B -> C , then the rate equation is: :For reactant A: \frac = -k_1 ce A :For reactant B: \frac = k_1 ce A- k_2 ce B/math> :For product C: \frac = k_2 ce B/math> 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 In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined ...
. The differential equations can be solved analytically and the integrated rate equations are : ce A\ce e^ :\left \ce B \right \begin \ce\frac\left( e^-e^ \right)+\cee^& k_\ne k_ \\ \cek_te^+\cee^ & \text \\ \end :\left \ce C \right \begin \ce\left( 1+\frac \right)+\ce\left( 1-e^ \right)+\ce & k_\ne k_ \\ \ce\left( 1-e^-k_te^ \right)+\ce\left( 1-e^ \right)+\ce & \text \\ \end The
steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
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

A -> B and A -> C , with constants k_1 and k_2 and rate equations -\frac=(k_1+k_2) ce A/math>; \frac=k_1 ce A/math> and \frac=k_2 ce A/math> The integrated rate equations are then ce A= \ce e^; ce B= \frac\ce \left(1 - e^\right) and ce C= \frac\ce \left(1 - e^\right). One important relationship in this case is \frac=\frac


One first order and one second order reaction

This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated as pseudo order one) 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: A + H2O -> B and A + R -> C . The rate equations are: \frac=k_1\ce=k_1' ce A/math> and \frac=k_2\ce, where k_1' is the pseudo first order constant. The integrated rate equation for the main product is \ce \left 1-e^ \right , which is equivalent to \ln \frac=\frac\left(1 - e^\right). Concentration of B is related to that of C through ce B-\frac \ln \left ( 1 - \frac \right ) The integrated equations were analytically obtained but during the process it was assumed that \ce - \ce\approx \ce. Therefore, previous equation for can only be used for low concentrations of compared to sub>0


Stoichiometric reaction networks

The most general description of a chemical reaction network considers a number N of distinct chemical species reacting via R reactions. The chemical equation of the j-th reaction can then be written in the generic form : r_ \ce X_1 + r_ \ce X_2 + \cdots + r_ \ce X_ \ce \ p_ \ce X_ + \ p_ \ce X_ + \cdots + p_ \ce X_, which is often written in the equivalent form : \sum_^ r_ \ce X_i \ce \sum_^\ p_ \ce X_. Here * j is the reaction index running from 1 to R, * \ce X_i denotes the i-th chemical species, * k_j is the rate constant of the j-th reaction and * r_ and p_ are the stoichiometric coefficients of reactants and products, respectively. The rate of such a reaction can be inferred by the
law of mass action In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...
: f_j( vec= k_j \prod_^N ce X_z which denotes the flux of molecules per unit time and unit volume. Here vec( 1 2 \ldots , _\mathit is the vector of concentrations. This definition includes the
elementary reaction An elementary reaction is a chemical reaction in which one or more chemical species react directly to form products in a single reaction step and with a single transition state. In practice, a reaction is assumed to be elementary if no reaction ...
s: ; zero order reactions :for which r_=0 for all z, ; first order reactions :for which r_=1 for a single z, ; second order reactions :for which r_=1 for exactly two z; that is, a bimolecular reaction, or r_ = 2 for a single z; that is, a dimerization reaction. Each of these is discussed in detail below. One can define the stoichiometric matrix : N_=p_-r_, denoting the net extent of molecules of i in reaction j. The reaction rate equations can then be written in the general form : \frac =\sum_^ r_ f_j( vec. This is the product of the stoichiometric matrix and the vector of reaction rate functions. Particular simple solutions exist in equilibrium, \frac=0, 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 The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...
. Detailed balance is a property of the stoichiometric matrix N_ alone and does not depend on the particular form of the rate functions f_j. All other cases where detailed balance is violated are commonly studied by
flux balance analysis Flux balance analysis (FBA) is a mathematical method for simulating metabolism in genome-scale reconstructions of metabolic networks. In comparison to traditional methods of modeling, FBA is less intensive in terms of the input data required for c ...
, which has been developed to understand
metabolic pathway In biochemistry, a metabolic pathway is a linked series of chemical reactions occurring within a cell. The reactants, products, and intermediates of an enzymatic reaction are known as metabolites, which are modified by a sequence of chemical reac ...
s.


General dynamics of unimolecular conversion

For a general unimolecular reaction involving interconversion of N different species, whose concentrations at time t are denoted by X_1(t) through X_N(t), an analytic form for the time-evolution of the species can be found. Let the rate constant of conversion from species X_i to species X_j be denoted as k_, and construct a rate-constant matrix K whose entries are the k_. Also, let X(t) = (X_1(t), X_2(t), \ldots,X_N(t))^T be the vector of concentrations as a function of time. Let J=(1,1,1,\ldots,1)^T be the vector of ones. Let I be the N \times N identity matrix. Let \operatorname be the function that takes a vector and constructs a diagonal matrix whose on-diagonal entries are those of the vector. Let \mathcal^ be the inverse Laplace transform from s to t. Then the time-evolved state X(t) is given by : X(t) = \mathcal^ sI + \operatorname(KJ)-K^T)^X(0) thus providing the relation between the initial conditions of the system and its state at time t.


See also

*
Michaelis–Menten kinetics In biochemistry, Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rat ...
* Molecularity *
Petersen matrix The Petersen matrix is a comprehensive description of systems of biochemical reactions used to model reactors for pollution control (engineered decomposition) as well as in environmental systems. It has got as many columns as the number of relevan ...
* Reaction–diffusion system *
Reactions on surfaces Reactions on surfaces are reactions in which at least one of the steps of the reaction mechanism is the adsorption of one or more reactants. The mechanisms for these reactions, and the rate equations are of extreme importance for heterogeneous c ...
: rate equations for reactions where at least one of the reactants adsorbs onto a surface *
Reaction progress kinetic analysis In chemistry, reaction progress kinetic analysis (RPKA) is a subset of a broad range of kinetic techniques utilized to determine the rate laws of chemical reactions and to aid in elucidation of reaction mechanisms. While the concepts guiding reacti ...
* Reaction rate * Reaction rate constant *
Steady state approximation In chemistry, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, ...
* Gillespie algorithm * Balance equation * Belousov-Zhabotinsky reaction * Lotka–Volterra equations


References


Books cited

* * * * *


External links


Chemical kinetics, reaction rate, and order
(needs flash player)

(lecture with audio).

{{DEFAULTSORT:Rate Equation Chemical kinetics Chemical reaction engineering cy:Cyfradd adwaith#Hafaliadau cyfradd