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Rate equation (rate law) in chemical kinetics

A concise account of the rate equation (rate law): its form, interpretation, relation to mechanism, common integrated laws, and practical uses in chemistry and engineering.

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The rate equation, often called the rate law, is the empirical relationship that links the speed of a chemical reaction to the concentrations of reactants and a proportionality constant. It is a central concept in chemical kinetics because it summarizes how changing concentrations, temperature, pressure or catalysts affects how rapidly products form. The law must be established experimentally for most reactions, although simple mechanisms can sometimes predict its form. Rate equation and reaction rate are closely related terms used across physical chemistry, biochemistry and industrial processes.

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Basic form and terms

For a reaction written generically as aA + bB → products, the usual algebraic form is r = k [A]^x [B]^y, where r is the instantaneous rate, [A] and [B] are species concentrations, k is the rate constant, and x and y are the reaction orders with respect to A and B. The overall order is the sum x + y. The rate constant k carries units that depend on this overall order and is sensitive to conditions such as temperature and catalysts. The exponents x and y are determined by the reaction mechanism rather than the stoichiometric coefficients in general; see concentration and rate-determining step concepts.

Orders, mechanisms and experimental determination

Reaction orders (zero, first, second, fractional, etc.) are found by measuring how the rate changes when concentrations are varied. A first-order process satisfies r = k[A], producing an exponential decay of [A] with time; integration gives ln[A] = −kt + ln[A]0 so a plot of ln[A] versus t is linear with slope −k. More complex mechanisms and intermediate steps can produce different apparent orders; for example, if one reactant remains in large excess its concentration can be treated as constant and the observed rate may appear first order (a pseudo-first-order condition). The concepts of reaction mechanism and transition state often guide interpretation.

Mathematical character and condition dependence

Rate equations are differential expressions: they relate the time derivative of concentration to a function of concentrations, and are integrated to predict concentration vs. time profiles. Because k commonly depends on temperature (Arrhenius behavior) and can be influenced by pressure, solvent, ionic strength or catalysts, kinetic parameters are typically reported with the conditions used to measure them. For more on these mathematical aspects see entries on temperature effects, pressure dependence, the underlying differential equation form, and techniques for integration.

Applications and examples

  • Laboratory kinetics: determining mechanisms by comparing observed orders with proposed elementary steps.
  • Enzymology: enzyme-catalyzed reactions use modified rate laws (e.g., Michaelis–Menten) to describe substrate dependence and saturation.
  • Industrial reactors: design and scale-up rely on integrated rate laws to size reactors and predict yields over time.
  • Environmental chemistry and atmospheric models: rate expressions predict pollutant formation and decay under varying conditions.

Practical analysis often combines experimental data, mechanistic hypotheses and kinetic modeling. Simple integrated laws (zero, first, second order) serve as diagnostics, while numerical integration of coupled rate equations is used for complex networks.

Notable distinctions and cautions

Important distinctions include the difference between molecularity (an attribute of an elementary step) and reaction order (an empirical exponent), and the fact that stoichiometric coefficients do not always equal orders. Reaction orders can be non-integer or change with conditions when mechanisms shift. Kinetic data should always be reported alongside experimental conditions and the method used to determine orders and rate constants, because those parameters are not intrinsic constants in the same sense as molecular masses. Careful interpretation links rate laws to plausible microscopic steps and to thermodynamic constraints without over-interpreting limited data.

For further reading, introductory texts and specialized reviews discuss experimental techniques, integrated forms for common orders, and how to extract mechanistic information from kinetic measurements. Additional resources and detailed examples are available through linked references above.

Derivation

The rate equations can be derived for all species involved by setting up the continuity equation with source and sink terms (or a balance equation) for the particle concentrations:

{\displaystyle {\frac {\partial c_{i}}{\partial t}}+\operatorname {div} {\vec {j}}_{i}=f_{i}(\{a_{i}\})},

where {\displaystyle f_{i}(\{a_{i}\})}is the source term which {\displaystyle a_{i}(\{c_{i}\})}depends on the activities These activities are in general again non-trivially dependent on all concentrations.

Since an equilibrium reaction always has an outward reaction and a reverse reaction, the outward reaction rate {\displaystyle k_{\text{+,r}}}and the reverse reaction rate {\displaystyle k_{\text{-,r}}}exist. The source term is given by a sum over all reactions:

{\displaystyle f_{i}(\{a_{i}\})=\sum _{r}|\nu _{i,r}|\left[-k_{\text{-,r}}\prod _{{\text{Reaktanden j}},r}a_{r,j}^{|\nu _{r,j}|}+k_{\text{+,r}}\prod _{{\text{Produkte j}},r}a_{r,j}^{|\nu _{r,j}|}\right].}

Note that the partial reaction order (the exponent with which the concentrations enter) only corresponds to the amount of the stoichiometric coefficients if activities are used. If concentrations are also used in the source term instead of activities and particle interactions are present, the amounts of the stoichiometric coefficients are to be replaced with the partial reaction orders. The partial reaction order can assume any values (e.g. 0) and is determined experimentally.

Various cases

  • At equilibrium, there are no particle flows ( {\displaystyle {\vec {j}}_{i}={\vec {0}}\,\forall i}) and the particle concentrations no longer change in time. Therefore, at equilibrium:

{\displaystyle 0=f_{i}(\{a_{i}\}).}

Assuming that each reaction (as a pair of outward and backward reactions) is balanced in detail, the law of mass action is obtained for each reaction by transformation:

{\displaystyle K_{r}={\frac {k_{\text{-,r}}}{k_{\text{+,r}}}}={\frac {\prod _{{\text{Produkte j}},r}a_{r,j}^{|\nu _{r,j}|}}{\prod _{{\text{Reaktanden j}},r}a_{r,j}^{|\nu _{r,j}|}}}}

  • If the system is in non-equilibrium but homogeneous, no particle currents occur, but the concentrations change in time until equilibrium is reached:

{\displaystyle {\frac {\partial c_{i}}{\partial t}}=f_{i}(\{a_{i}\})}

  • In the case where one considers an inhomogeneous system in non-equilibrium, the particle flow {\displaystyle {\vec {j}}_{i}\neq 0}and can be given by Fick's first law

{\displaystyle {\vec {j}}_{i}=-{\vec {\nabla }}(D_{i}c_{i}({\vec {r}}))+{\vec {j}}_{i}^{\text{excess}}}

(whereby the non-ideal excess term only occurs for non-ideal systems). One then obtains a reaction diffusion equation.

  • In the case that there is additional flow in the system, convection in the particle flow must be taken into account and the convection-diffusion equation is obtained.

Examples

Hydrogen oxidation

Hydrogen oxidation is used to illustrate this:

{\displaystyle \displaystyle \mathrm {2H_{2}+O_{2}\longrightarrow 2H_{2}O} }

(Rate coefficient: \displaystyle k_{1})

a part dissociates

{\displaystyle \displaystyle \mathrm {2H_{2}O\longrightarrow H_{3}O^{+}+OH^{-}} }

(Rate coefficient: \displaystyle k_{2})

The rate equations (Eq.1) for the five species are:

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{2}]} =-2k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} }

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[O_{2}]} =-k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} }

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{2}O]} =+2k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} -2k_{2}\mathrm {[H_{2}O]^{2}} }

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{3}O^{+}]} =+k_{2}\mathrm {[H_{2}O]^{2}} }

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[OH^{-}]} =+k_{2}\mathrm {[H_{2}O]^{2}} }

The concentrations of the species:

{\displaystyle c_{1}\equiv \mathrm {[H_{2}]} ,\;\;c_{2}\equiv \mathrm {[O_{2}]} ,\;\;c_{3}\equiv \mathrm {[H_{2}O]} ,\;\;c_{4}\equiv \mathrm {[H_{3}O^{+}]} ,\;\;c_{5}\equiv \mathrm {[OH^{-}]} }

Belousov-Zhabotinsky reaction

→ Main article: Belousov-Zhabotinsky reaction

Oscillating reactions are described by rate equations. For special models on this, see Oregonator and Brusselsator. The numerical solution of such systems of differential equations then yields oscillating chemical concentrations.

Lotka-Volterra equations

→ Main article: Lotka-Volterra equations

The interaction of predator and prey populations is described by the Lotka-Volterra equations.

Numerical solution, using scipy.integrate.odeint. The reaction rates {\displaystyle k_{1},k_{2}}and the initial concentrations are arbitrarily chosen in this example. c_{0}is a reference concentration.

Numerical solution methods

Since the rate equations are a system of stiff differential equations, one is forced to choose a method with as large a stability region as possible so that the integration steps do not become too small. A-stable methods are the most favourable.

For the rate equations, 'stiff' means that the time constants of the different species differ very much: Relative to others, some concentrations change very slowly. Two examples of absolutely stiff-stable integration methods are the Implicit Trapezoidal Method and the Implicit Euler Method, as well as some BDF (backward differentiation formula) methods.

Building block conservation

The principle of building block conservation provides a way to check the goodness of the numerical solutions, because it applies at all times:

{\displaystyle \sum _{i=1}^{N_{\mathrm {Sp} }}c_{i}\beta _{ik}=\gamma _{k}={\text{const.}},\quad \forall k=1,\dots ,N_{\mathrm {B} }}

where

N_{{\mathrm {B}}}\,   Minimum number of building blocks,

N_{{\mathrm {Sp}}}\,  Number of species involved in the reactions.

Derivation

A species i, here written as A_{i}the building blocks B_{k}as follows:

A_{i}=\sum _{{k=1}}^{{N_{{\mathrm {B}}}}}\beta _{{ik}}B_{k}\quad \Rightarrow \quad \sum _{{i=1}}^{{N_{{\mathrm {Sp}}}}}\nu _{{ij}}\beta _{{ik}}=\underline {\underline {0}}.

into the rate equation (Eq.1) and summed over all species, yields the above building block conservation because of \sum _{{i=1}}^{{N_{{\mathrm {Sp}}}}}{\dot c}_{i}\beta _{{ik}}=0.

Example for the matrix βik

{\displaystyle {\begin{pmatrix}\mathrm {H_{2}} \\\mathrm {O_{2}} \\\mathrm {H_{2}O} \\\mathrm {H_{3}O^{+}} \\\mathrm {OH^{-}} \end{pmatrix}}=\underbrace {\begin{pmatrix}2&0&0\\0&2&2\\2&1&1\\3&1&0\\1&1&2\end{pmatrix}} _{\beta _{ik}}\cdot {\begin{pmatrix}\mathrm {H} \\\mathrm {O^{+}} \\e^{-}\end{pmatrix}}}

See also

  • Master equation

Questions and answers

Q: What is the rate equation?

A: The rate equation (or rate law) is an equation used to calculate the speed of a chemical reaction. It takes into account the concentrations of reactants and products, as well as other conditions such as temperature and pressure.

Q: How can the rate constant be calculated?

A: In special cases, it is possible to solve the differential equation and find k by integrating it. For example, in a first-order reaction, a plot of ln[A] against time t will give a straight line with a slope of -k.

Q: What does x and y represent in the general reaction formula?

A: x and y depend on which step is rate-determining. If the reaction mechanism is very simple, where A and B hit each other then go to products through one transition state, then x=a and y=b.

Q: Is there another way to calculate k if one reagent has a high concentration?

A: Yes, if one reagent has a high concentration that can be thought of as constant then it becomes what's known as pseudo-first order rate constant (k'). This can also be used to calculate k'.

Q: How does temperature affect the rate constant?

A: The rate constant changes with temperature, pressure and other conditions.

Q: What type of equation is the rate equation?

A: The rate equation is a differential equation.

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AlegsaOnline.com Rate equation (rate law) in chemical kinetics

URL: https://en.alegsaonline.com/art/81260

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