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## Rate equation

Main article: Rate equation

For a chemical reaction 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 theconcentration of each reactant. It is of the kind:

{\displaystyle \,r=k(T)[\mathrm {A} ]^{n}[\mathrm {B} ]^{m}}

For gas phase reaction the rate is often alternatively expressed by partial pressures.

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 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 H2 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 number of collisionsis 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

{\displaystyle {\frac {d[\mathrm {P} ]}{dt}}=k(T)[\mathrm {A} ]^{n}[\mathrm {B} ]^{m}}

### Example of a complex reaction: Reaction of hydrogen and nitric oxide

For the reaction

2 H2(g) + 2 NO(g) → N2(g) + 2 H2O(g) the observed rate equation (or rate expression) is: {\displaystyle r=k[\mathrm {H_{2}} ][\mathrm {NO} ]^{2}\,}

As for many reactions, the experimental rate equation does not simply reflect the stoichiometric coefficients in the overall reaction: It is third order overall: first order in H2 and second order in NO, even though the stoichiometric coefficients of both reactants are equal to 2.[2]

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:

1. 2 NO(g) ⇌ N2O2(g) (fast equilibrium)

2. N2O2 + H2 → N2O + H2O (slow)

3. N2O + H2 → N2 + H2O (fast)

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:

{\displaystyle r=k_{2}[\mathrm {H_{2}} ][\mathrm {N_{2}O_{2}} ]\,}where k2 is the rate constant for the second step.

However N2O2 is an unstable intermediate whose concentration is determined by the fact that the first step is in equilibrium, so that [N2O2] = K1[NO]2, where K1 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

{\displaystyle r=k_{2}K_{1}[\mathrm {H_{2}} ][\mathrm {NO} ]^{2}\,}This agrees with the form of the observed rate equation if it is assumed that k = k2K1. In practice the rate equation is used to suggest possible mechanisms which predict a rate equation in agreement with experiment.

The second molecule of H2 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 article: Arrhenius equation

Each reaction rate coefficient k has a temperature dependency, which is usually given by the Arrhenius equation:

{\displaystyle k=Ae^{-{\frac {E_{\mathrm {a} }}{RT}}}}Ea is the activation energy and R is the gas constant. Since at temperature T the molecules have energies given by a Boltzmann distributionone can expect the number of collisions with energy greater than Ea to be proportional to eEaRTA is the pre-exponential factor or frequency factor.

The values for A and Ea 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,[3] 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 is commonly used as the ratio of rate constants that are 10 °C apart.

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