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[[Image:Rust03102006.JPG|thumb|225px|Iron [[rusting]] has a ''low'' reaction rate. This process is slow.]]
[[File:Large bonfire.jpg|thumb|225px|Wood [[combustion]] has a ''high'' reaction rate. This process is fast.]]

The '''reaction rate''' or '''rate of reaction''' is the speed at which reactants are converted into products. For example, the [[Rusting#Chemical reactions|oxidative rusting]] of [[iron]] under [[Earth's atmosphere]] is a slow reaction that can take many years, but the combustion of [[cellulose]] in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. 

[[Chemical kinetics]] is the part of [[physical chemistry]] that studies reaction rates. The concepts of chemical kinetics are applied in many disciplines, such as [[chemical engineering]]<ref>{{Cite journal|last=Silva|first=Camylla K. S.|last2=Baston|first2=Eduardo P.|last3=Melgar|first3=Lisbeth Z.|last4=Bellido|first4=Jorge D. A.|date=2019-10-01|title=Ni/Al2O3-La2O3 catalysts synthesized by a one-step polymerization method applied to the dry reforming of methane: effect of precursor structures of nickel, perovskite and spinel|url=https://doi.org/10.1007/s11144-019-01644-3|journal=Reaction Kinetics, Mechanisms and Catalysis|language=en|volume=128|issue=1|pages=251–269|doi=10.1007/s11144-019-01644-3|issn=1878-5204}}</ref><ref>{{Cite book|url=https://pure.mpg.de/rest/items/item_1199619_5/component/file_1199618/content|title=Kinetic studies of propane oxidation on Mo and V based mixed oxide catalysts|last=|first=|publisher=|year=2011|isbn=|location=|pages=}}</ref><ref>{{Cite journal|last=|first=|date=2014|title=The reaction network in propane oxidation over phase-pure MoVTeNb M1 oxide catalysts|url=https://pure.mpg.de/rest/items/item_1896844_6/component/file_1896843/content|journal=J. Catal.|volume=311|pages=369-385|via=}}</ref><ref>{{Cite journal|last=Elizalde|first=Ignacio|last2=Mederos|first2=Fabián S.|last3=del Carmen Monterrubio|first3=Ma.|last4=Casillas|first4=Ninfa|last5=Díaz|first5=Hugo|last6=Trejo|first6=Fernando|date=2019-02-01|title=Mathematical modeling and simulation of an industrial adiabatic trickle-bed reactor for upgrading heavy crude oil by hydrotreatment process|url=https://doi.org/10.1007/s11144-018-1489-7|journal=Reaction Kinetics, Mechanisms and Catalysis|language=en|volume=126|issue=1|pages=31–48|doi=10.1007/s11144-018-1489-7|issn=1878-5204}}</ref>, [[enzymology]] and [[environmental engineering]]<ref>{{Cite journal|last=Liu|first=Jiaqi|last2=Shen|first2=Meiqing|last3=Li|first3=Chenxu|last4=Wang|first4=Jianqiang|last5=Wang|first5=Jun|date=2019-10-01|title=Enhanced hydrothermal stability of a manganese metavanadate catalyst based on WO3–TiO2 for the selective catalytic reduction of NOx with NH3|url=https://doi.org/10.1007/s11144-019-01624-7|journal=Reaction Kinetics, Mechanisms and Catalysis|language=en|volume=128|issue=1|pages=175–191|doi=10.1007/s11144-019-01624-7|issn=1878-5204}}</ref><ref>{{Cite journal|last=Li|first=Xiaoliang|last2=Feng|first2=Jiangjiang|last3=Xu|first3=Zhigang|last4=Wang|first4=Junqiang|last5=Wang|first5=Yujie|last6=Zhao|first6=Wei|date=2019-10-01|title=Cerium modification for improving the performance of Cu-SSZ-13 in selective catalytic reduction of NO by NH3|url=https://doi.org/10.1007/s11144-019-01621-w|journal=Reaction Kinetics, Mechanisms and Catalysis|language=en|volume=128|issue=1|pages=163–174|doi=10.1007/s11144-019-01621-w|issn=1878-5204}}</ref><ref>{{Cite journal|last=Vedyagin|first=Aleksey A.|last2=Stoyanovskii|first2=Vladimir O.|last3=Kenzhin|first3=Roman M.|last4=Slavinskaya|first4=Elena M.|last5=Plyusnin|first5=Pavel E.|last6=Shubin|first6=Yury V.|date=2019-06-01|title=Purification of gasoline exhaust gases using bimetallic Pd–Rh/δ-Al2O3 catalysts|url=https://doi.org/10.1007/s11144-019-01573-1|journal=Reaction Kinetics, Mechanisms and Catalysis|language=en|volume=127|issue=1|pages=137–148|doi=10.1007/s11144-019-01573-1|issn=1878-5204}}</ref>.

==Formal definition==
Consider a typical [[chemical reaction]]:

:<chem>{\mathit{a}A} + {\mathit{b}B} -> {\mathit{p}P} + {\mathit{q}Q}</chem>

The lowercase letters (''a'', ''b'', ''p'', and ''q'') represent [[stoichiometric coefficients]], while the capital letters represent the [[reactant]]s (A and B) and the [[product (chemistry)|products]] (P and Q).

According to [[IUPAC]]'s [[Gold Book]] definition<ref name=IUPACrate>{{GoldBookRef|file=R05156|title=Rate of reaction}}</ref>
the reaction rate ''v'' for a chemical reaction occurring in a [[closed system]] under [[Isochoric process|isochoric conditions]], without a build-up of [[reaction intermediate]]s, is defined as:

:<math>v = - \frac{1}{a} \frac{d[\mathrm{A}]}{dt} = - \frac{1}{b} \frac{d[\mathrm{B}]}{dt} = \frac{1}{p} \frac{d[\mathrm{P}]}{dt} = \frac{1}{q} \frac{d[\mathrm{Q}]}{dt}</math>

where [X] denotes the [[concentration]] of the substance X (= A, B, P or Q). Reaction rate thus defined has the units of mol/L/s.

The rate of a reaction is always positive. A [[Plus and minus signs|negative sign]] is present to indicate that the reactant concentration is decreasing. The IUPAC<ref name="IUPACrate"/> recommends that the unit of time should always be the second. The rate of reaction differs from the rate of increase of concentration of a product P by a constant factor (the reciprocal of its [[stoichiometric number]]) and for a reactant A by minus the reciprocal of the stoichiometric number. The stoichiometric numbers are included so that the defined rate is independent of which reactant or product species is chosen for measurement.<ref name=LM>{{cite book|authorlink1=Keith J. Laidler|last1=Laidler |first1=K. J.|last2=Meiser|first2=J.H. |title=Physical Chemistry |publisher=Benjamin/Cummings |date=1982 |isbn=0-8053-5682-7}}</ref>{{rp|349}} For example, if ''a = 1'' and ''b = 3'' then B is consumed three times more rapidly than A, but v ''= -d[''A'']/dt = -(1/3)d[''B'']/dt'' is uniquely defined. An additional advantage of this definition is that for an [[elementary reaction | elementary]] and [[reversible reaction | irreversible]] reaction, ''v'' is equal to the product of probability of overcoming the [[transition state]] [[activation energy]] and the number of times per second the transition state is approached by reactant molecules. When so defined, for an elementary and irreversible reaction, ''v'' is the rate of successful chemical reaction events leading to the product.  

The above definition is only valid for a ''single reaction'', in a ''[[closed system]]'' of ''constant volume'', an assumption which should be stated explicitly in the definition. If water is added to a pot containing salty water, the concentration of salt decreases, although there is no chemical reaction.

For an open system, the full [[mass balance]] must be taken into account: {{nowrap|1=in − out + generation − consumption = accumulation}}

:<math>F_{\mathrm{A}0} - F_\mathrm{A} + \int_{0}^{V} v\, dV = \frac{dN_\mathrm{A}}{dt}</math>,
where ''F''<sub>A0</sub> is the inflow rate of A in molecules per second, ''F''<sub>A</sub> the outflow, and ''v'' is the instantaneous reaction rate of A (in [[number concentration]] rather than molar) in a given differential volume, integrated over the entire system volume ''V'' at a given moment. When applied to the closed system at constant volume considered previously, this equation reduces to:
:<math>v= \frac{d[A]}{dt}</math>,
where the concentration [A] is related to the number of molecules ''N''<sub>A</sub> by [A]&nbsp;= {{sfrac|''N''<sub>A</sub>|''N''<sub>0</sub>''V''}}. Here ''N''<sub>0</sub> is the [[Avogadro constant]].

For a single reaction in a closed system of varying volume the so-called ''rate of conversion'' can be used, in order to avoid handling concentrations. It is defined as the derivative of the [[extent of reaction]] with respect to time.

:<math>v =\frac{d\xi}{dt} = \frac{1}{\nu_i} \frac{dn_i}{dt} = \frac{1}{\nu_i} \frac{d(C_i V)}{dt} = \frac{1}{\nu_i} \left(V\frac{dC_i}{dt} + C_i \frac{dV}{dt} \right)  </math>

Here ''ν<sub>i</sub>'' is the stoichiometric coefficient for substance ''i'', equal to ''a'', ''b'', ''p'', and ''q'' in the typical reaction above. Also ''V'' is the volume of reaction and ''C<sub>i</sub>'' is the concentration of substance ''i''.

When side products or reaction intermediates are formed, the IUPAC<ref name="IUPACrate"/> recommends the use of the terms '''the rate of increase of concentration''' and '''rate of the decrease of concentration''' for products and reactants, properly.

Reaction rates may also be defined on a basis that is not the volume of the reactor.  When a [[catalyst]] is used the reaction rate may be stated on a catalyst weight (mol&nbsp;g<sup>−1</sup>&nbsp;s<sup>−1</sup>) or surface area (mol&nbsp;m<sup>−2</sup>&nbsp;s<sup>−1</sup>) basis.  If the basis is a specific catalyst site that may be rigorously counted by a specified method, the rate is given in units of s<sup>−1</sup> and is called a turnover frequency.

==Influencing factors==
Factors that influence the reaction rate are the nature of the reaction, [[concentration]], [[pressure]], [[Order (chemistry)|reaction order]], [[temperature]], [[solvent]], [[electromagnetic radiation]], [[catalyst]], [[isotopes]], surface area, [[Mixing (process engineering)|stirring]], and [[diffusion limit]]. Some reactions are naturally faster than others. The number of reacting species, their [[Phase (matter)|physical state]] (the particles that form solids move much more slowly than those of gases or those in [[solution]]), the complexity of the reaction and other factors can greatly influence the rate of a reaction.

Reaction rate increases with concentration, as described by the [[rate law]] and explained by [[collision theory]]. As reactant concentration increases, the [[frequency]] of [[collision]] increases. The rate of gaseous reactions increases with pressure, which is, in fact, equivalent to an increase in concentration of the gas. The reaction rate increases in the direction where there are fewer moles of gas and decreases in the reverse direction. For condensed-phase reactions, the pressure dependence is weak.

The order of the reaction controls how the reactant concentration (or pressure) affects reaction rate.

Usually conducting a reaction at a higher temperature delivers more energy into the system and increases the reaction rate by causing more collisions between particles, as explained by collision theory. However, the main reason that temperature increases the rate of reaction is that more of the colliding particles will have the necessary [[activation energy]] resulting in more successful collisions (when bonds are formed between reactants). The influence of temperature is described by the [[Arrhenius equation]]. For example, coal burns in a fireplace in the presence of oxygen, but it does not when it is stored at [[room temperature]]. The reaction is spontaneous at low and high temperatures but at room temperature its rate is so slow that it is negligible. The increase in temperature, as created by a match, allows the reaction to start and then it heats itself, because it is [[exothermic]]. That is valid for many other fuels, such as [[methane]], [[butane]], and [[hydrogen]].

Reaction rates can be independent of temperature (''non-Arrhenius'') or decrease with increasing temperature (''anti-Arrhenius''). Reactions without an activation barrier (e.g., some [[radical (chemistry)|radical]] reactions), tend to have anti Arrhenius temperature dependence: the rate constant decreases with increasing temperature.

Many reactions take place in solution and the properties of the solvent affect the reaction rate. The [[ionic strength]] also has an effect on reaction rate.

Electromagnetic radiation is a form of energy. As such, it may speed up the rate or even make a reaction spontaneous as it provides the particles of the reactants with more energy.{{cn|date=January 2019}} This energy is in one way or another stored in the reacting particles (it may break bonds, promote molecules to electronically or vibrationally excited states...) creating intermediate species that react easily. As the intensity of light increases, the particles absorb more energy and hence the rate of reaction increases. For example, when [[methane]] reacts with [[chlorine]] in the dark, the reaction rate is slow. It can be sped up when the [[mixture]] is put under diffused light. In bright sunlight, the reaction is explosive.

The presence of a catalyst increases the reaction rate (in both the forward and reverse reactions) by providing an alternative pathway with a lower [[activation energy]]. For example, [[platinum]] catalyzes the combustion of hydrogen with oxygen at room temperature.

The [[kinetic isotope effect]] consists in a different reaction rate for the same molecule if it has different isotopes, usually [[hydrogen]] isotopes, because of the relative mass difference between hydrogen and [[deuterium]].
In [[reactions on surfaces]], which take place for example during [[heterogeneous catalysis]], the rate of reaction increases as the surface area does. That is because more particles of the solid are exposed and can be hit by reactant molecules.

Stirring can have a strong effect on the rate of reaction for [[Homogeneous and heterogeneous reactions|heterogeneous reactions]].

Some reactions are limited by diffusion. All the factors that affect a reaction rate, except for concentration and reaction order, are taken into account in the reaction [[rate coefficient]] (the coefficient in the rate equation of the reaction).

==Rate equation==
{{main|Rate equation}}
For a [[chemical reaction]] ''a''&nbsp;A + ''b''&nbsp;B → ''p''&nbsp;P + ''q''&nbsp;Q, the rate equation or rate law is a [[mathematical expression]] used in chemical kinetics to link the rate of a reaction to the [[concentration]] of each reactant. It is often of the type:
:<math>\,v = k(T)[\mathrm{A}]^{n}[\mathrm{B}]^{m}</math>
For gas phase reaction the rate is often alternatively expressed by [[partial pressure]]s.

In these equations ''k''(''T'') is the ''reaction rate coefficient'' or ''rate constant'', although it is not really a constant, because it includes all the parameters that affect reaction rate, except for concentration, which is explicitly taken into account.  Of all the parameters influencing reaction rates, temperature is normally the most important one and is accounted for by the [[Arrhenius equation]].

The exponents ''n'' and ''m'' are called reaction [[order (chemistry)|orders]] and depend on the [[reaction mechanism]]. For elementary (single-step) reactions the order with respect to each reactant is equal to its stoichiometric coefficient. For complex (multistep) reactions, however, this is often not true and the rate equation is determined by the detailed mechanism, as illustrated below for the reaction of H<sub>2</sub> and NO.

For elementary reactions or reaction steps, the order and stoichiometric coefficient are both equal to the [[molecularity]] or number of molecules participating. For a unimolecular reaction or step the rate is proportional to the concentration of molecules of reactant, so that the rate law is first order. For a bimolecular reaction or step, the [[collision theory|number of collisions]] is proportional to the product of the two reactant concentrations, or second order. A termolecular step is predicted to be third order, but also very slow as simultaneous collisions of three molecules are rare.

By using the [[mass balance]] for the system in which the reaction occurs, an expression for the rate of change in concentration can be derived. For a closed system with constant volume, such an expression can look like

:<math>\frac{d[\mathrm{P}]}{dt} = k(T)[\mathrm{A}]^{n}[\mathrm{B}]^{m}</math>

===Example of a complex reaction: hydrogen and nitric oxide===
For the reaction

:<chem>2H2(g) + 2NO(g) -> N2(g) + 2H2O(g)</chem>

the observed rate equation (or rate expression) is:
<math chem> v = k [\ce{H2}] [\ce{NO}]^2 \,</math><!--Please do not change this to second order in H2. The experimental result is first order in H2. Read the section to find out why.-->

As for many reactions, the experimental rate equation does not simply reflect the stoichiometric coefficients in the overall reaction: It is [[order of reaction|third order]] overall: first order in H<sub>2</sub> and second order in NO,  even though the stoichiometric coefficients of both reactants are equal to 2.<ref>{{cite book|authorlink=Keith J. Laidler|last=Laidler |first=K. J. |title=Chemical Kinetics |edition=3rd |publisher=Harper & Row |date=1987 |page=277 |isbn=0060438622}}</ref>

In chemical kinetics, the overall reaction rate is often explained using a mechanism consisting of a number of elementary steps. Not all of these steps affect the rate of reaction; normally the slowest elementary step controls the reaction rate. For this example, a possible mechanism is:

# <chem>\quad 2NO(g) <=> N2O2(g) \qquad(fast\ equilibrium)</chem> 
# <chem>\quad {N2O2} + H2 -> {N2O} + H2O \qquad(slow)</chem>
# <chem>\quad {N2O} + H2 ->{N2} + H2O\qquad(fast) </chem>

Reactions 1 and 3 are very rapid compared to the second, so the slow reaction 2 is the rate determining step. This is a [[bimolecular]] elementary reaction whose rate is given by the second order equation:
:<math chem> v = k_2 [\ce{H2}] [\ce{N2O2}] \,</math>,
where ''k''<sub>2</sub> is the rate constant for the second step.

However N<sub>2</sub>O<sub>2</sub> is an unstable intermediate whose concentration is determined by the fact that the first step is in [[chemical equilibrium|equilibrium]], so that {{math|1=[N<sub>2</sub>O<sub>2</sub>]&nbsp;= ''K''<sub>1</sub>[NO]<sup>2</sup>}}, where ''K''<sub>1</sub> is the [[equilibrium constant]] of the first step. Substitution of this equation in the previous equation leads to a rate equation expressed in terms of the original reactants
:<math chem> v = k_2 K_1 [\ce{H2}] [\ce{NO}]^2 \,</math>

This agrees with the form of the observed rate equation if it is assumed that {{math|1=''k''&nbsp;= ''k''<sub>2</sub>''K''<sub>1</sub>}}. In practice the rate equation is used to suggest possible mechanisms which predict a rate equation in agreement with experiment.

The second molecule of H<sub>2</sub> does not appear in the rate equation because it reacts in the third step, which is a rapid step ''after'' the rate-determining step, so that it does not affect the overall reaction rate.

==Temperature dependence==
{{main|Arrhenius equation}}
Each reaction rate coefficient k has a temperature dependency, which is usually given by the [[Arrhenius equation]]:

:<math> k = A e^{ - \frac{E_\mathrm{a}}{RT} }</math>

''E''<sub>a</sub> is the [[activation energy]] and ''R'' is the [[gas constant]].  Since at [[temperature]] ''T'' the molecules have energies given by a [[Boltzmann distribution]], [[Arrhenius equation#Kinetic theory's interpretation of Arrhenius equation|one can expect]] the number of collisions with energy greater than ''E<sub>a</sub>'' to be proportional to e<sup>{{frac|−''E''<sub>a</sub>|''RT''}}</sup>. ''A'' is the pre-exponential factor or [[frequency factor (chemistry)|frequency factor]].

The values for ''A'' and ''E''<sub>a</sub> are dependent on the reaction. There are also more complex equations possible, which describe temperature dependence of other rate constants that do not follow this pattern.

A chemical reaction takes place only when the reacting particles collide. However, not all collisions are effective in causing the reaction. Products are formed only when the colliding particles possess a certain minimum energy called threshold energy. As a [[rule of thumb]], reaction rates for many reactions double for every 10 degrees [[Celsius]] increase in temperature,<ref name="Connors">{{cite book|first=Kenneth |last=Connors|title= Chemical Kinetics:The Study of Reaction Rates in Solution|date= 1990|publisher= VCH Publishers |page=14 |ISBN=978-0-471-72020-1}}</ref> For a given reaction, the ratio of its rate constant at a higher temperature to its rate constant at a lower temperature is known as its temperature coefficient (''Q'').  [[Q10 (temperature coefficient)|''Q''<sub>10</sub>]] is commonly used as the ratio of rate constants that are 10&nbsp;°C apart.

==Pressure dependence==
The pressure dependence of the rate constant for [[Condensed matter physics|condensed]]-phase reactions (i.e., when reactants and products are solids or liquid) is usually sufficiently weak in the range of pressures normally encountered in industry that it is neglected in practice.

The pressure dependence of the rate constant is associated with the activation volume. For the reaction proceeding through an activation-state complex:

:A + B {{eqm}} |A⋯B|<sup>‡</sup> → P

the activation volume, Δ''V''<sup>‡</sup>, is:

:<math>\Delta V^{\ddagger} =  \bar{V}_{\ddagger} - \bar{V}_\mathrm{A} - \bar{V}_\mathrm{B} </math>

where ''V̄'' denotes the [[partial molar volume]] of a species and ‡ indicates the activation-state complex.

For the above reaction, one can expect the change of the [[reaction rate constant]] (based either on [[mole fraction]] or on [[molar concentration]]) with pressure at constant temperature to be:<ref name=LM/>{{rp|390}}

:<math> \left(\frac{\partial \ln k_x}{\partial P} \right)_T = -\frac{\Delta V^{\ddagger}} {RT}</math>

In practice, the matter can be complicated because the partial molar volumes and the activation volume can themselves be a function of pressure.

Reactions can increase or decrease their rates with pressure, depending on the value of Δ''V''<sup>‡</sup>. As an example of the possible magnitude of the pressure effect, some organic reactions were shown to double the reaction rate when the pressure was increased from atmospheric (0.1&nbsp;MPa) to 50&nbsp;MPa (which gives Δ''V''<sup>‡</sup>&nbsp;=&nbsp;−0.025&nbsp;L/mol).<ref>{{cite book|last=Isaacs|first= Neil S.|title=Physical Organic Chemistry |edition=2nd|chapter= Section 2.8.3 |publisher=Addison Wesley Longman |location=Harlow |date=1995|ISBN=9780582218635}}</ref>

==See also==
* [[Rate of solution]]
* [[Dilution (equation)]]
* [[Diffusion-controlled reaction]]
* [[Steady state approximation]]
* Collision theory and [[transition state]] are chemical theories that attempt to predict and explain reaction rates.
* [[Isothermal microcalorimetry]]

==Notes==
{{reflist}}

== External links ==
* [http://www.citycollegiate.com/chemical_kineticsXIa.htm Chemical kinetics, reaction rate, and order] (needs flash player)
* [https://web.archive.org/web/20120420055947/http://itl.chem.ufl.edu/4411/2041/lec_k.html Reaction kinetics, examples of important rate laws] (lecture with audio).
* [http://www.chemguide.co.uk/physical/basicratesmenu.html#top Rates of reaction]
* [https://web.archive.org/web/20081201014313/http://monte.chem.ttu.edu/group/tutorial/br_sn2.html Overview of Bimolecular Reactions (Reactions involving two reactants)]
* [http://www.nrcresearchpress.com/doi/pdf/10.1139/v66-330 pressure dependence [[Can. J. Chem.]]]

{{DEFAULTSORT:Reaction Rate}}
[[Category:Chemical kinetics]]
[[Category:Chemical reaction engineering]]
[[Category:Temporal rates]]