Editing Reaction rate (section)

==Rate equation==
{{Main|Rate equation}}
For a chemical reaction {{math|''a''A + ''b''B → ''p''P + ''q''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. For a closed system at constant volume, this is often of the form 
<math display=block>v = k [\mathrm{A}]^{n}[\mathrm{B}]^{m} - k_r[\mathrm{P}]^{i}[\mathrm{Q}]^{j}.</math>

For reactions that go to completion (which implies very small {{math|''k''<sub>r</sub>}}), or if only the initial rate is analyzed (with initial vanishing product concentrations), this simplifies to the commonly quoted form

<math display=block>v = k(T)[\mathrm{A}]^{n}[\mathrm{B}]^{m}.</math>

For gas phase reaction the rate equation is often alternatively expressed in terms of [[partial pressure]]s.

In these equations {{math|''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 time and concentration. Of all the parameters influencing reaction rates, temperature is normally the most important one and is accounted for by the [[Arrhenius equation]].

The exponents {{mvar|n}} and {{mvar|m}} are called reaction [[order (chemistry)|orders]] and depend on the reaction mechanism. For an elementary (single-step) reaction, 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 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 display=block>\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

<math chem display=block>\ce{2H2_{(g)}} + \ce{2NO_{(g)} -> N2_{(g)}} + \ce{2H2O_{(g)}},</math>

the observed rate equation (or rate expression) is
<math chem display=block> 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|author-link=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

<math chem display=block> \begin{array}{rll}
1) & \quad \ce{2NO_{(g)} <=> N2O2_{(g)}} & (\text{fast equilibrium}) \\
2) & \quad \ce{N2O2 + H2 -> N2O + H2O} & (\text{slow}) \\
3) & \quad \ce{N2O + H2 -> N2 + H2O} & (\text{fast}).
\end{array}</math>

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 display=block> 
v = k_2 [\ce{H2}] [\ce{N2O2}] ,
</math>
where {{math|''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 chem>\ce{[N2O2] = \mathit{K}_1[NO]^2},</math> where {{math|''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 display=block> 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'' = ''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.