Rate equation
In chemistry, the rate equation (also known as the rate law or empirical differential 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 (normally rate coefficients and partial orders of reaction) only.[1] For many reactions, the initial rate is given by a power law such as
![{\displaystyle v_{0}\;=\;k[\mathrm {A} ]^{x}[\mathrm {B} ]^{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0762360082ad1693fa36fa98c82deab6a72004a)
where and are the molar concentrations of the species and usually in moles per liter (molarity, ). 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.[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 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.
![{\displaystyle [\mathrm {A} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba12364ea89576c100fa535be4f06beaebc1c4b)
![{\displaystyle [\mathrm {B} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/474e97de049c58c8b606696e58cb04a9f806da29)
![{\displaystyle v\;=\;k[{\mathrm {A} }]^{x}[{\mathrm {B} }]^{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abd7a471d2c6b2c7cf00bf532f2ae5f8987f8657)
Elementary (single-step) 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 (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.
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
[edit]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 (with negative signs for reactants because they are consumed) 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 [X] for the molar concentration of chemical X,[3]
![{\displaystyle -{\frac {d[\mathrm {A} ]}{dt}}=-{\frac {1}{2}}{\frac {d[\mathrm {B} ]}{dt}}={\frac {1}{3}}{\frac {d[\mathrm {C} ]}{dt}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7ab4335384a8b42d4eb22635f09b6ce2fc1e78)
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
![{\displaystyle v={\frac {1}{\nu _{i}}}{\frac {d[\mathrm {X} _{i}]}{dt}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea228d3374efd357bf05b5de4163c8b1a47970e8)
where νi 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,
![{\displaystyle v_{0}=f\left([\mathrm {A} ],[\mathrm {B} ],\ldots \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5da1c62967b03e4ecb41f15ba0f766acf6935bf1)
and this dependence is known as the rate equation or rate law.[5] This law generally cannot be deduced from the chemical equation and must be determined by experiment.[6]
Power laws
[edit]A common form for the rate equation is a power law:[6]
![{\displaystyle v_{0}=k[{\ce {A}}]^{x}[{\ce {B}}]^{y}\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f5036fbee7ae0524bc05dfbde11a5edecc752d)
The constant is called the rate constant. The exponents, which can be fractional,[6] are called partial orders of reaction and their sum is the overall order of reaction.[7]
In a dilute solution, an elementary reaction (one having a single step with a single transition state) 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.[8]
The differential rate equation for an elementary reaction using mathematical product notation is:
![{\displaystyle -{d \over dt}[{\text{Reactants}}]=k\prod _{i}[{\text{Reactants}}_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe460cc7cfad705ef4e7661f987011bd5bda1a3)
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.
![{\textstyle -{d \over dt}[{\text{Reactants}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64dee387ec28997405c2f2c06937753b0909669)
![{\textstyle \prod _{i}[{\text{Reactants}}_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/395c02bdb1531daf49438bc3772dbd6c70d26d53)
Determination of reaction order
[edit]Method of initial rates
[edit]The natural logarithm of the power-law rate equation is
![{\displaystyle \ln v_{0}=\ln k+x\ln[{\ce {A}}]+y\ln[{\ce {B}}]+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/936423ce7dae2600b48a3e39b6af680b03b33b0c)
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
![{\displaystyle [{\rm {B],[{\rm {C],\dots }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13aa682efd2d727d056664ab8e2a5a5cd06618e9)
![{\displaystyle \ln v_{0}=x\ln[{\ce {A}}]+{\textrm {constant}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/362ce9e21412c332bd36c0b1c32c61c4a6f132d9)
The slope of a graph of as a function of then corresponds to the order with respect to reactant .[9][10]
![{\displaystyle \ln[{\ce {A}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a4300d2f50163eebda325d43deeeaaa4531a15)
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
[edit]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
![{\displaystyle \ln {[{\ce {A}}]}=-kt+\ln {[{\ce {A}}]_{0}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2d67f733c840fb62603b4a72a564dbce3a0c15)
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.[11][12]
![{\displaystyle [{\rm {A]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fdf5b161b5bc659e3286aaf42973c24fef2b86d)
![{\displaystyle [{\rm {A]_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b60aba64e49e430e6847c972809217980228f06)
![{\displaystyle \ln {[{\ce {A}}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7dcef6e387bdeae2628a6d0404a483f7b0c9033)
Method of flooding
[edit]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 a·A + b·B → c·C with rate law the partial order with respect to is determined using a large excess of . In this case
![{\displaystyle v_{0}=k\cdot [{\rm {A}}]^{x}\cdot [{\rm {B}}]^{y},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a04eaa260be40ad0d7a26d8e1549ae6f0b8b88)
with
![{\displaystyle v_{0}=k'\cdot [{\rm {A}}]^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e40575b055c31938d04c0d8415c5c1ac837d343f)
![{\displaystyle k'=k\cdot [{\rm {B}}]^{y},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c467869f67ce13f4db3e5e9f0146bb1cb96f73b9)
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]
![{\displaystyle [{\rm {B]_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387e005ca5eb51ab686690ed6a3ef3eef228f4b3)
Zero order
[edit]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
![{\displaystyle -{d[A] \over dt}=k[A]^{0}=k,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/471ea54f9fae44679d1294b067634c969d3b86a9)
The unit of k is mol 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 an enzyme or a catalytic 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 is saturated. For example, the biological oxidation of ethanol to acetaldehyde by the enzyme liver alcohol dehydrogenase (LADH) is zero order in ethanol.[16]
Similarly, reactions with heterogeneous catalysis can be zero order if the catalytic surface is saturated. For example, the decomposition of phosphine (PH3) on a hot tungsten surface at high pressure is zero order in phosphine, which decomposes at a constant rate.[15]
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.[17]
First order
[edit]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
![{\displaystyle -{\frac {d[{\ce {A}}]}{dt}}=k[{\ce {A}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7785d62f81535b155883b6662c78c18637db73)
The unit of k is s−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 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.[18]
Examples of such reactions are:
![{\displaystyle {2\,\mathrm {N} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {\vphantom {A}}_{\smash[{t}]{5}}{}\mathrel {\longrightarrow } {}4\,\mathrm {NO} {\vphantom {A}}_{\smash[{t}]{2}}{}+{}\mathrm {O} {\vphantom {A}}_{\smash[{t}]{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9314adfb218b8a0071c6e65cf5ae0aa2a147d414)
![{\displaystyle {[\mathrm {CoCl} (\mathrm {NH} {\vphantom {A}}_{\smash[{t}]{3}}){\vphantom {A}}_{\smash[{t}]{5}}]{\vphantom {A}}^{2+}{}+{}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {}\mathrel {\longrightarrow } {}[\mathrm {Co} (\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} )(\mathrm {NH} {\vphantom {A}}_{\smash[{t}]{3}}){\vphantom {A}}_{\smash[{t}]{5}}]{\vphantom {A}}^{3+}{}+{}\mathrm {Cl} {\vphantom {A}}^{-}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2493a5d0c36d527c7ddcd5c8867aae0ff5dc01)
![{\displaystyle {\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {\vphantom {A}}_{\smash[{t}]{2}}{}\mathrel {\longrightarrow } {}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {}+{}{\mathchoice {\textstyle {\frac {1}{2}}}{\frac {1}{2}}{\frac {1}{2}}{\frac {1}{2}}}\,\mathrm {O} {\vphantom {A}}_{\smash[{t}]{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71bdd6b540794e1839bf5647c6db4097f84e1bde)
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 aqueous solution, ArN+2 + X− → ArX + N2, the rate equation is where Ar indicates an aryl group.[22]
![{\displaystyle v_{0}=k[{\mathrm {ArN} {\vphantom {A}}_{\smash[{t}]{2}}{\vphantom {A}}^{+}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31345a634bc30803dc4282945995f6380b5eb57)
Second order
[edit]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 reaction NO2 + CO → NO + CO2 is second-order in the reactant NO2 and zero order in the reactant CO. The observed rate is given by and is independent of the concentration of CO.[23]
![{\displaystyle v_{0}=k[{\ce {A}}]^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ec098a03148c709861ce7aaa3991fb858ee912)
![{\displaystyle v_{0}=k[{\ce {A}}][{\ce {B}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1d646a84133df46ef9e6bc938ac57cd7bd80e1)
![{\displaystyle v_{0}=k[{\mathrm {NO} {\vphantom {A}}_{\smash[{t}]{2}}}]^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/207c00607e954cc615e7910ba761726575cd6289)
For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
![{\displaystyle {\frac {1}{[\mathrm {A} ]}}={\frac {1}{[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}}+kt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc35f39f5c1da3a2981359e7627f5bf3275b467)
The unit of k is mol−1 dm3 s−1.[14]
The time dependence for a rate proportional to two unequal concentrations is
![{\displaystyle {\frac {[\mathrm {A} ]}{[\mathrm {B} ]}}={\frac {[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}{[\mathrm {B} ]{\vphantom {A}}_{\smash[{t}]{0}}}}e^{\left({[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}-{[\mathrm {B} ]{\vphantom {A}}_{\smash[{t}]{0}}}\right)kt};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3929803fc5b572a71c5c1a1f10ebe15b16cf064)
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:[22]
![{\displaystyle {\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COOC} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}{}+{}\mathrm {OH} {\vphantom {A}}^{-}{}\mathrel {\longrightarrow } {}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COO} {\vphantom {A}}^{-}{}+{}\mathrm {C} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}\mathrm {OH} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d3291356c97284a43040f6b6b3196f09ffd15a)
This reaction is first-order in each reactant and second-order overall:
![{\displaystyle v_{0}=k[{\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COOC} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}}][{\mathrm {OH} {\vphantom {A}}^{-}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f3ae610e7d5da1ada0191e0a70a7eb370acde2)
If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes[22]
![{\displaystyle v_{0}=k[{\text{imidazole}}][{\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COOC} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{5}}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6dc0db7bb4d21f15740c7fbb5920e8fda3889b)
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 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in acetone:
![{\displaystyle {\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {Br} {}+{}\mathrm {NaI} {}\mathrel {\longrightarrow } {}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {I} {}+{}\mathrm {NaBr} {\mskip {2mu}}(\mathrm {v} )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd8ca88d3886903ff48014314e14cd5a364b295)
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 with sodium tert-butoxide as the salt and tert-butanol as the solvent:
Pseudo-first order
[edit]If the concentration of a reactant remains constant (because it is a catalyst, 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 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.
![{\displaystyle v_{0}=k[{\ce {A}}][{\ce {B}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2fe1617a5e3a93b8bbf5b0756ffefcd9dc28f4)
![{\displaystyle v_{0}=k[{\mathrm {A} }][{\mathrm {B} }]=k'[{\mathrm {A} }],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c0be6e042cc175260fbabff18a70782998d58f)
![{\displaystyle k'=k[{\ce {B}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca0bd2f22e5152c93668a42aa2482f5ba874044)
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:
![{\displaystyle {\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COOCH} {\vphantom {A}}_{\smash[{t}]{3}}{}+{}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} {}\mathrel {\longrightarrow } {}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {COOH} {}+{}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {OH} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab5593899b2695bb76324eaf9592815876d70d1)
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 catalyst H+ and the solvent H2O are normally constant, so that the reaction is pseudo–first-order.[24]
![{\displaystyle v_{0}=k[{\mathrm {C} {\vphantom {A}}_{\smash[{t}]{12}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{22}}\mathrm {O} {\vphantom {A}}_{\smash[{t}]{11}}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88db83cb7651a3287a903ac87536255a8f00ec7d)
![{\displaystyle v_{0}=k[{\mathrm {C} {\vphantom {A}}_{\smash[{t}]{12}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{22}}\mathrm {O} {\vphantom {A}}_{\smash[{t}]{11}}}][{\mathrm {H} {\vphantom {A}}^{+}}][{\mathrm {H} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {O} }];}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4e8c000c14b6227afc7e7c975391d3cde7e584)
Summary for reaction orders 0, 1, 2, and n
[edit]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.
| 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 determine k | [A] vs. t | vs. t | vs. t | vs. t
[Except first order] |
| Half-life | [25] |
[Limit is necessary for first order] |
![{\displaystyle -{d[{\ce {A}}]}/{dt}=k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0bbeea17cc80e39fb00450c1ea86631cbe0cfd)
![{\displaystyle -{d[{\ce {A}}]}/{dt}=k[{\ce {A}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f96ec3225358636e6401453fece14afa5283b7e)
![{\displaystyle -{d[{\ce {A}}]}/{dt}=k[{\ce {A}}]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4c186bb0e3fc1fce4a7d9c8021bc8e49948507)
![{\displaystyle -{d[{\ce {A}}]}/{dt}=k[{\ce {A}}]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21385ff7a769c369af7dcf9f990ae8130eea98cb)
![{\displaystyle {[\mathrm {A} ]{}={}[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}-kt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee628dae90349b5938ea4b7b0638159bb3df8d8)
![{\displaystyle {[\mathrm {A} ]{}={}[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}e^{-kt}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b216310f1c50dec1c5c60b887635a9e2bb87b21e)
![{\displaystyle {\frac {1}{[\mathrm {A} ]}}={\frac {1}{[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}}+kt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/577fdc157e8a32cd138f2f7a020a328e6561908e)
![{\displaystyle [{\ce {A}}]^{g}={{\ce {[A]0}}^{g}}-gkt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e50d01d5c9fe65a8021aa812fbcb5e0508bc97c)
![{\displaystyle {\ln([\mathrm {A} ])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8378871d4c6cf8ce28a1a4c9aff69e59ba35fc8)
![{\displaystyle {\frac {1}{[\mathrm {A} ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/104bf3c6ecace101c498f948eaf16bf55240619f)
![{\displaystyle {\rm {[A]}}^{g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/593f78dff943f1d79c44a2d0c476a13c9a369822)
![{\displaystyle t_{\frac {1}{2}}={\frac {[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}{2k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/039f23e90f62bb6d112b1523e7dff9d7969863a0)
![{\displaystyle t_{\frac {1}{2}}={\frac {1}{k{[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b25f9b5e7675b3e3f9027241a74ec2d751810c43)
![{\displaystyle t_{\frac {1}{2}}={\frac {{[\mathrm {A} ]{\vphantom {A}}_{\smash[{t}]{0}}}^{g}(1-2^{-g})}{gk}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84b39ae4e9f2af3ec33f9b6dd699fab9d88dc346)
Here stands for concentration in molarity (mol · L−1), for time, and for the reaction rate constant. The half-life of a first-order reaction is often expressed as t1/2 = 0.693/k (as ln(2)≈0.693).
Fractional order
[edit]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 of acetaldehyde (CH3CHO) into methane and carbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde: [26] The decomposition of phosgene (COCl2) to carbon monoxide and chlorine has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine: [27]
![{\displaystyle v_{0}=k[{\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} }]^{3/2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e06de5f10f593f434ec230d9866aae1c07d2db)
![{\displaystyle v_{0}=k{[\mathrm {COCl} {\vphantom {A}}_{\smash[{t}]{2}}]~[\mathrm {Cl} {\vphantom {A}}_{\smash[{t}]{2}}]}^{1/2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89dccea99e65793f9bae7402feef1481c7dd6b2c)
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 Rice-Herzfeld mechanism is
- Initiation
- Propagation
- Termination
![{\displaystyle {\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} {}\mathrel {\longrightarrow } {}\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}{}+{}\,{\cdot }\,\mathrm {CHO} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ca6ef80321dcb3d0610084a759d3e98d3bce33)
![{\displaystyle {\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}{}+{}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} {}\mathrel {\longrightarrow } {}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CO} \,{\cdot }\,{}+{}\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ab12587267d9410e785aebc0c3eb35b2d67da9)
![{\displaystyle {\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CO} \,{\cdot }\,{}\mathrel {\longrightarrow } {}\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}{}+{}\mathrm {CO} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f0b9f427aed08473bb829f708458a2d75141cd)
![{\displaystyle {2\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}{}\mathrel {\longrightarrow } {}\mathrm {C} {\vphantom {A}}_{\smash[{t}]{2}}\mathrm {H} {\vphantom {A}}_{\smash[{t}]{6}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a259fae7bcb271c2a7fceddef5abc82d8a4d82)
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
![{\displaystyle {\frac {d[{\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}}]}{dt}}=k_{i}[{\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} }]-k_{t}[{\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}}]^{2}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8da5841bfe73d0dee54fea810ec768a76fc4b283)
so that the concentration of methyl radical satisfies
![{\displaystyle {[\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}]~\quad \propto \quad [\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} ]{\vphantom {A}}^{1/2}\,{\cdot }\,}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73f063268e034796003bb7ef150ba3c18b8403b7)
The reaction rate equals the rate of the propagation steps which form the main reaction products CH4 and CO:
![{\displaystyle v_{0}={\frac {d[{\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{4}}}]}{dt}}|_{0}=k_{p}{[\,{\cdot }\,\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}][\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} ]}\quad \propto \quad {[\mathrm {CH} {\vphantom {A}}_{\smash[{t}]{3}}\mathrm {CHO} ]{\vphantom {A}}^{3/2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15732a37687f161125dd08e529b16b2fc33a2460)
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 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.
Complex laws
[edit]Mixed order
[edit]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 (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.
![{\displaystyle v_{0}=k_{1}[A]+k_{2}[A]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72738096215f782ef9094a2979f6eb81f90932d1)
Another type of mixed-order rate law has a denominator of two or more terms, often because the identity of the rate-determining step depends on the values of the concentrations. An example is the oxidation of an alcohol to a ketone by hexacyanoferrate (III) ion [Fe(CN)63−] with ruthenate (VI) ion (RuO42−) as catalyst.[30] For this reaction, the rate of disappearance of hexacyanoferrate (III) is