Inchemistry, therate equation (also known as therate law orempirical differential rate equation) is anempiricaldifferentialmathematical expression for thereaction rate of a given reaction in terms of concentrations of chemical species and constant parameters (normally rate coefficients and partial orders of reaction) only.[1] For many reactions, the initial rate is given by a power law such as
where and are themolar concentrations of the species and usually inmoles perliter (molarity,). The exponents and are the partialorders of reaction for and, respectively, and theoverall reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. Theorder of reaction is a number which quantifies the degree to which the rate of a chemical reaction depends on concentrations of the reactants.[2] In other words, the order of reaction is the exponent to which the concentration of a particular reactant is raised.[2] The constant is thereaction rate constant orrate coefficient and at very few placesvelocity constant orspecific rate of reaction. Its value may depend on conditions such as temperature, ionic strength, surface area of anadsorbent, or lightirradiation. If the reaction goes to completion, the rate equation for thereaction rate applies throughout the course of the reaction.
Elementary (single-step) reactions andreaction steps have reaction orders equal to thestoichiometric coefficients for each reactant. The overall reaction order, i.e. the sum of stoichiometric coefficients of reactants, is always equal to themolecularity 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 unknownreaction mechanism could be either elementary or complex. When the experimental rate equation has been determined, it is often of use for deduction of thereaction mechanism.
In highly dilute solutions, such as at concentrations below the micromolar level, molecular collisions are primarily governed bydiffusion. 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 byFick'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 usingquasi-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 afractional order, and may depend on the concentration of anintermediate species.
A reaction can also have anundefined 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 betweenadsorbed molecules:
Consider a typicalchemical reaction in which tworeactants A and B combine to form aproduct C:
This can also be written
The prefactors −1, −2 and 3 (with negative signs for reactants because they are consumed) are known asstoichiometric coefficients. One molecule of A combines with two of B to form 3 of C, so if we use the symbol [X] for themolar concentration of chemical X,[3]
If the reaction takes place in aclosed system at constant temperature and volume, without a build-up ofreaction intermediates, thereaction rate is defined as
whereνi is the stoichiometric coefficient for chemical Xi, with a negative sign for a reactant.[4]
The initial reaction rate has some functional dependence on the concentrations of the reactants,
and this dependence is known as therate equation orrate law.[5] This law generally cannot be deduced from the chemical equation and must be determined by experiment.[6]
A common form for the rate equation is a power law:[6]
The constant is called therate constant. The exponents, which can be fractional,[6] are calledpartial orders of reaction and their sum is the overall order of reaction.[7]
In a dilute solution, anelementary reaction (one having a single step with a singletransition state) is empirically found to obey thelaw of mass action. This predicts that the rate depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients.[8]
Thedifferential rate equation for an elementary reaction usingmathematical product notation is:
Where:
Thenatural 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
Theslope of a graph of as a function of then corresponds to the order with respect to reactant.[9][10]
However, this method is not always reliable because
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
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 therate constant is equal to the slope with sign reversed.[11][12]
The partial order with respect to a given reactant can be evaluated by the method of flooding (or of isolation) ofOstwald. 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 reactiona·A +b·B →c·C 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 (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.[13]
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 ofk ismol dm−3 s−1.[14] 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 anenzyme or acatalytic surface.[15]
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 issaturated. For example, the biological oxidation ofethanol toacetaldehyde by the enzymeliver alcohol dehydrogenase (LADH) is zero order in ethanol.[16]
Similarly, reactions withheterogeneous catalysis can be zero order if the catalytic surface is saturated. For example, the decomposition ofphosphine (PH3) on a hottungsten surface at high pressure is zero order in phosphine, which decomposes at a constant rate.[15]
Inhomogeneous catalysis zero order behavior can come about from reversible inhibition. For example,ring-opening metathesis polymerization usingthird-generation Grubbs catalyst exhibits zero order behavior in catalyst due to the reversibleinhibition that occurs betweenpyridine and the ruthenium center.[17]
Afirst order reaction depends on the concentration of only one reactant (aunimolecular reaction). 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 ofk iss−1.[14] 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 theLindemann 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.
Thehalf-life is independent of the starting concentration and is given by. Themean lifetime isτ = 1/k.[18]
Examples of such reactions are:
In organic chemistry, the class ofSN1 (nucleophilic substitution unimolecular) reactions consists of first-order reactions. For example, in the reaction ofaryldiazonium ions withnucleophiles in aqueous solution,ArN+2 + X− → ArX + N2, the rate equation is where Ar indicates anaryl group.[22]
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, or (more commonly) to the product of two concentrations, As an example of the first type, the reactionNO2 + CO → NO + CO2 is second-order in the reactantNO2 and zero order in the reactant CO. The observed rate is given by and is independent of the concentration of CO.[23]
For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
The unit ofk ismol−1 dm3 s−1.[14]
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 nucleophilicaddition-elimination reactions, such as thealkalinehydrolysis ofethyl acetate:[22]
This reaction is first-order in each reactant and second-order overall:
If the same hydrolysis reaction iscatalyzed byimidazole, the rate equation becomes[22]
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 theSN2 (bimolecular nucleophilic substitution) reactions, such as the reaction ofn-butyl bromide withsodium iodide inacetone:
This same compound can be made to undergo a bimolecular (E2)elimination reaction, another common type of second-order reaction, if the sodium iodide and acetone are replaced withsodium tert-butoxide as the salt andtert-butanol as the solvent:
If the concentration of a reactant remains constant (because it is acatalyst, or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, leading to apseudo–first-order (or occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate equation if the concentration of reactant B is constant then where the pseudo–first-order rate constant 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, [B]≫[A]) 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:
The hydrolysis of sucrose (C12H22O11) in acid solution is often cited as a first-order reaction with rate The true rate equation is third-order, however, the concentrations of both the catalystH+ and the solventH2O are normally constant, so that the reaction is pseudo–first-order.[24]
Elementary reaction steps with order 3 (calledternary reactions) arerare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order.
| Parameter | Zero order | First order | Second order | nth order (g = 1−n) |
|---|---|---|---|---|
| Rate Law | [25] | |||
| Integrated Rate Law | [25] | [Except first order] | ||
| Units of Rate Constant (k) | ||||
| Linear Plot to determinek | [A] vs.t | vs.t | vs.t | vs.t [Except first order] |
| Half-life | [25] | [Limit is necessary for first order] |
Here stands for concentration inmolarity (mol · L−1), for time, and for the reaction rate constant. The half-life of a first-order reaction is often expressed ast1/2 = 0.693/k (as ln(2)≈0.693).
In fractional order reactions, the order is a non-integer, which often indicates achemical chain reaction or other complexreaction mechanism. For example, thepyrolysis ofacetaldehyde (CH3CHO) intomethane andcarbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde:[26] The decomposition ofphosgene (COCl2) to carbon monoxide andchlorine has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine:[27]
The order of a chain reaction can be rationalized using thesteady state approximation for the concentration of reactive intermediates such asfree radicals. For the pyrolysis of acetaldehyde, the Rice-Herzfeld mechanism is
where • denotes a free radical.[26][28] To simplify the theory, the reactions of the*CHO to form a second*CH3 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 productsCH4 and CO:
in agreement with the experimental order of 3/2.[26][28]
In highly diluted solutions, such as at concentrations below the micromolar level, molecular collisions are primarily governed bydiffusion. 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 byFick's laws of diffusion and is consistent with fractal reaction kinetics, which yield fractional reaction orders.
More complex rate laws have been described as beingmixed 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 (or more often concurrent pseudo-first order and second order) reactions, and can be described as mixed first and second order.[29] For sufficiently large values of [A] such a reaction will approximate second order kinetics, but for smaller [A] 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 therate-determining step depends on the values of the concentrations. An example is the oxidation of analcohol to aketone byhexacyanoferrate (III) ion [Fe(CN)63−] withruthenate (VI) ion (RuO42−) ascatalyst.[30] For this reaction, the rate of disappearance of hexacyanoferrate (III) is
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:
A reaction rate can have a negative partial order with respect to a substance. For example, the conversion ofozone (O3) tooxygen follows the rate equation in an excess of oxygen. This corresponds to second order in ozone and order (−1) with respect to oxygen.[31]
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 first-order reaction.
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, andq are thestoichiometric coefficients):
The reaction rate expression for the above reactions (assuming each one is elementary) can be written as:
where:k1 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 constantsk1 andk−1 are related to the equilibrium coefficient for the reaction (K) by the following relationship (setv=0 in balance):
In a simple equilibrium between two species:
where the reaction starts with an initial concentration of reactant A,, and an initial concentration of 0 for product P at timet=0.
Then theequilibrium constantK is expressed as:
where and are the concentrations of A and P at equilibrium, respectively.
The concentration of A at timet,, is related to the concentration of P at timet,, by the equilibrium reaction equation:
The term is not present because, in this simple example, the initial concentration of P is 0.
This applies even when timet is at infinity; i.e., equilibrium has been reached:
then it follows, by the definition ofK, that
and, therefore,
These equations allow us to uncouple thesystem of differential equations, and allow us to solve for the concentration of A alone.
The reaction equation was given previously as:
For this is simply
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, letx be, the concentration of A at timet. Let be the concentration of A at equilibrium. Then:
Since:
thereaction rate becomes:
which results in:
A plot of the negativenatural logarithm of the concentration of A in time minus the concentration at equilibrium versus timet gives a straight line with slopek1 +k−1. By measurement of [A]e and [P]e the values ofK and the tworeaction rate constants will be known.[32]
If the concentration at the timet = 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 inconformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis inNMR spectroscopy.
If the rate constants for the following reaction are and;, then the rate equation is:
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 amaster equation. The differential equations can be solved analytically and the integrated rate equations are
Thesteady state approximation leads to very similar results in an easier way.
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.
and, with constants and and rate equations; and
The integrated rate equations are then; and.
One important relationship in this case is
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: and. The rate equations are: and, where is the pseudo first order constant.[33]
The integrated rate equation for the main product [C] 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. Therefore, previous equation for [C] can only be used for low concentrations of [C] compared to [A]0
The most general description of achemical reaction network considers a number of distinct chemical species reacting via reactions.[34][35] 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 a reaction can be inferred by thelaw of mass action
which denotes the flux of molecules per unit time and unit volume. Here is the vector of concentrations. This definition includes theelementary reactions:
Each of these is discussed in detail below. One can define thestoichiometric 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 calleddetailed 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 byflux balance analysis, which has been developed to understandmetabolic pathways.[36][37]
For a general unimolecular reaction involving interconversion of different species, whose concentrations at time are denoted by through, an analytic form for the time-evolution of the species can be found. Let the rate constant of conversion from species to species be denoted as, and construct a rate-constant 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 on-diagonal entries are those of the vector.
Let be the inverse Laplace transform from to.
Then the time-evolved state is given by
thus providing the relation between the initial conditions of the system and its state at time.