# Formal definition of reaction rate

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## Formal definition of reaction rate

Consider a typical chemical reaction:

a A + b B → p P + q Q

The lowercase letters (abp, and q) represent stoichiometric coefficients, while the capital letters represent the reactants (A and B) and the products (P and Q).

According to IUPAC's Gold Book definition[1] the reaction rate r for a chemical reaction occurring in a closed system under isochoric conditions, without a build-up of reaction intermediates, is defined as:

{\displaystyle r=-{\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}}}where [X] denotes the concentration of the substance X. (Note: The rate of a reaction is always positive. A negative sign is present to indicate the reactant concentration is decreasing.) The IUPAC[1] recommends that the unit of time should always be the second. In such a case 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. Reaction rate usually has the units of mol L−1 s−1. It is important to bear in mind that the previous definition is only valid for a single reaction, in a closed system of constant volume. This usually implicit assumption must be stated explicitly, otherwise the definition is incorrect: If water is added to a pot containing salty water, the concentration of salt decreases, although there is no chemical reaction.

For any open system, the full mass balance must be taken into account: in − out + generation − consumption = accumulation

{\displaystyle F_{\mathrm {A} 0}-F_{\mathrm {A} }+\int _{0}^{V}v\,dV={\frac {dN_{\mathrm {A} }}{dt}}}where FA0 is the inflow rate of A in molecules per second, FA 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:

{\displaystyle r={\frac {d[A]}{dt}}}where the concentration [A] is related to the number of molecules NA by [A] = NA/N0V. Here N0 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.{\displaystyle r={\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)}

Here νi is the stoichiometric coefficient for substance i, equal to abp, and q in the typical reaction above. Also V is the volume of reaction and Ci is the concentration of substance i.

When side products or reaction intermediates are formed, the IUPAC[1] recommends the use of the terms rate of appearance and rate of disappearance 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 g−1 s−1) or surface area (mol m−2 s−1) 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−1 and is called a turnover frequency.

## - Temperature: 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 necessaryactivation energyresulting in more successful collisions (when bonds are formed between reactants). The influence of temperature is described by theArrhenius 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 methanebutane, 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., someradical reactions), tend to have anti Arrhenius temperature dependence: the rate constant decreases with increasing temperature.

- Solvent: 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 and intensity of light: 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. 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 very slow. It can be sped up when the mixture is put under diffused light. In bright sunlight, the reaction is explosive.

- A catalyst: 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.

- Isotopes: 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.

- Surface Area: 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: Stirring can have a strong effect on the rate of reaction for heterogeneous reactions.

- Diffusion limit: 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).

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