Rate equation
Encyclopedia
The rate law or rate equation for a chemical reaction
is an equation that links the reaction rate
with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders). To determine the rate equation for a particular system one combines the reaction rate with a mass balance
for the system. For a generic reaction with no intermediate steps in its reaction mechanism
(that is, an elementary reaction
), the rate is given by
where [A] and [B] express the concentration of the species A and B, respectively (usually in moles per liter (molarity, M)); x and y are not the respective stoichiometric coefficients of the balanced equation; they must be determined experimentally. k is the rate coefficient or rate constant of the reaction. The value of this coefficient k depends on conditions such as temperature, ionic strength, surface area of the adsorbent or light irradiation. For elementary reactions, the rate equation can be derived from first principles using collision theory
.
Again, x and y are NOT always derived from the balanced equation.
The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometric coefficients of the overall reaction; it must be determined experimentally. The equation may involve fractional exponential coefficients, or it may depend on the concentration of an intermediate species.
The rate equation is a differential equation
, and it can be integrated
to obtain an integrated rate equation that links concentrations of reactants or products with time.
If the concentration of one of the reactants remains constant (because it is a catalyst or it is in great excess with respect to the other reactants), its concentration can be grouped with the rate constant, obtaining a pseudo constant: If B is the reactant whose concentration is constant, then
. The second-order rate equation has been reduced to a pseudo-first-order rate equation. This makes the treatment to obtain an integrated rate equation much easier.
where r is the reaction rate and k is the reaction rate coefficient with units of concentration/time. If, and only if, this zeroth-order reaction 1) occurs in a closed system, 2) there is no net build-up of intermediates, and 3) there are no other reactions occurring, it can be shown by solving a mass balance
equation for the system that:
If this differential equation
is integrated
it gives an equation often called the integrated zero-order rate law.
where
represents the concentration of the chemical of interest at a particular time, and 
represents the initial concentration.
A reaction is zeroth order if concentration data are plotted versus time and the result is a straight line. The slope of this resulting line is the negative of the zero order rate constant k.
The half-life of a reaction describes the time needed for half of the reactant to be depleted (same as the half-life
involved in nuclear decay, which is a first-order reaction). For a zero-order reaction the half-life is given by
Example of a zeroth-order reaction
It should be noted that the order of a reaction cannot be deduced from the chemical equation of the reaction.
A first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero-order. The rate law for an elementary reaction that is first order with respect to a reactant A is
k is the first order rate constant, which has units of 1/s.
The integrated first-order rate law is
A plot of
vs. time t gives a straight line with a slope of 
.
The half-life of a first-order reaction is independent of the starting concentration and is given by
.
Examples of reactions that are first-order with respect to the reactant:
is usually written in the form of the exponential decay equation
A different (but equivalent) way of considering first order kinetics is as follows: The exponential decay equation can be rewritten as:
where
corresponds to a specific time period and 
is an integer corresponding to the number of time periods. At the end of each time period, the fraction of the reactant population remaining relative to the amount present at the start of the time period, 
, will be:
Such that after
time periods, the fraction of the original reactant population will be:
where:
corresponds to the fraction of the reactant population that will break down in each time period.
This equation indicates that the fraction of the total amount of reactant population that will break down in each time period is independent of the initial amount present. When the chosen time period corresponds to
, the fraction of the population that will break down in each time period will be exactly ½ the amount present at the start of the time period (i.e. the time period corresponds to the half-life of the first-order reaction).
The average rate of the reaction for the nth time period is given by:
Therefore, the amount remaining at the end of each time period will be related to the average rate of that time period and the reactant population at the start of the time period by:
Since the fraction of the reactant population that will break down in each time period can be expressed as:
The amount of reactant that will break down in each time period can be related to the average rate over that time period by:
Such that the amount that remains at the end of each time period will be related to the amount present at the start of the time period according to:
This equation is a recursion allowing for the calculation of the amount present after any number of time periods, without need of the rate constant, provided that the average rate for each time period is known.
For a second order reaction, its reaction rate is given by:
or 
or 
In several popular kinetics books, the definition of the rate law for second-order reactions is
. Conflating the 2 inside the constant for the first, derivative, form will only make it required in the second, integrated form (presented below). The option of keeping the 2 out of the constant in the derivative form is considered more correct, as it is almost always used in peer-reviewed literature, tables of rate constants, and simulation software.
The integrated second-order rate laws are respectively
or
[A]0 and [B]0 must be different to obtain that integrated equation.
The half-life equation for a second-order reaction dependent on one second-order reactant is
. For a second-order reaction half-lives progressively double.
Another way to present the above rate laws is to take the log of both sides:
Examples of a Second-order reaction:
If either [A] or [B] remains constant as the reaction proceeds, then the reaction can be considered pseudo-first-order because, in fact, it depends on the concentration of only one reactant. If, for example, [B] remains constant, then:
where
(k' or kobs with units s−1) and an expression is obtained identical to the first order expression above.
One way to obtain a pseudo-first-order reaction is to use a large excess of one of the reactants ([B]>>[A] would work for the previous example) so that, as the reaction progresses, only a small amount of the reactant is consumed, and its concentration can be considered to stay constant. By collecting
for many reactions with different (but excess) concentrations of [B], a plot of 
versus [B] gives 
(the regular second order rate constant) as the slope.
Example:
The hydrolysis of esters by dilute mineral acids follows pseudo-first-order kinetics where the concentration of water is present in large excess.
Where M stands for concentration in molarity (mol · L−1), t for time, and k for the reaction rate constant. The half-life of a first-order reaction is often expressed as t1/2 = 0.693/k (as ln2 = 0.693).
process. For example, A and B react into X and Y and vice versa (s, t, u, and v are the stoichiometric coefficients):
The reaction rate expression for the above reactions (assuming each one is elementary) can be expressed as:
where: k1 is the rate coefficient for the reaction that consumes A and B; k2 is the rate coefficient for the backwards reaction, which consumes X and Y and produces A and B.
The constants k1 and k2 are related to the equilibrium coefficient for the reaction (K) by the following relationship (set r=0 in balance):
Where the reactions starts with an initial concentration of A,
, with an initial concentration of 0 for B at time t=0.
Then the constant K at equilibrium is expressed as:
Where
and 
are the concentrations of A and B at equilibrium, respectively.
The concentration of A at time t,
, is related to the concentration of B at time t, 
, by the equilibrium reaction equation:
Note that the term
is not present because, in this simple example, the initial concentration of B is 0.
This applies even when time t is at infinity; i.e., equilibrium has been reached:
then it follows, by the definition of K, that
and, therefore,
These equations allow us to uncouple the system of differential equations, and allow us to solve for the concentration of A alone.
The reaction equation, given previously as:
The derivative is negative because this is the rate of the reaction going from A to B, and therefore the concentration of A is decreasing. To simplify annotation, let x be
, the concentration of A at time t. Let 
be the concentration of A at equilibrium. Then:
Since:
The reaction rate
becomes:
which results in:
A plot of the negative natural logarithm
of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope kf + kb. By measurement of Ae and Be the values of K and the two reaction rate constants will be known.
When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis
of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy
.
and 
; 
, then the rate equation is:
For reactant A:
For reactant B:
For product C:
With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation
. The differential equations can be solved analytically and the integrated rate equations are
The steady state
approximation leads to very similar results in an easier way.
and 
, with constants 
and 
and rate equations 
, 
and 
The integrated rate equations are then
; 
and
.
One important relationship in this case is
This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated as pseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct:
and 
. The rate equations are: 
and 
. Where 
is the pseudo first order constant.
The integrated rate equation for the main product [C] is
, which is equivalent to 
. Concentration of B is related to that of C through 
The integrated equations were analytically obtained but during the process it was assumed that
therefeore, previous equation for [C] can only be used for low concentrations of [C] compared to [A]0
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
is an equation that links the reaction rate
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders). To determine the rate equation for a particular system one combines the reaction rate with a mass balance
Mass balance
A mass balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique...
for the system. For a generic reaction with no intermediate steps in its reaction mechanism
Reaction mechanism
In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs.Although only the net chemical change is directly observable for most chemical reactions, experiments can often be designed that suggest the possible sequence of steps in...
(that is, an elementary reaction
Elementary reaction
An elementary reaction is a chemical reaction in which one or more of the chemical species react directly to form products in a single reaction step and with a single transition state....
), the rate is given by
where [A] and [B] express the concentration of the species A and B, respectively (usually in moles per liter (molarity, M)); x and y are not the respective stoichiometric coefficients of the balanced equation; they must be determined experimentally. k is the rate coefficient or rate constant of the reaction. The value of this coefficient k depends on conditions such as temperature, ionic strength, surface area of the adsorbent or light irradiation. For elementary reactions, the rate equation can be derived from first principles using collision theory
Collision theory
Collision theory is a theory proposed by Max Trautz and William Lewis in 1916 and 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. For a reaction to occur the reactant particles must collide. Only a certain fraction of the total...
.
Again, x and y are NOT always derived from the balanced equation.
The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometric coefficients of the overall reaction; it must be determined experimentally. The equation may involve fractional exponential coefficients, or it may depend on the concentration of an intermediate species.
The rate equation is a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
, and it can be integrated
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
to obtain an integrated rate equation that links concentrations of reactants or products with time.
If the concentration of one of the reactants remains constant (because it is a catalyst or it is in great excess with respect to the other reactants), its concentration can be grouped with the rate constant, obtaining a pseudo constant: If B is the reactant whose concentration is constant, then
. The second-order rate equation has been reduced to a pseudo-first-order rate equation. This makes the treatment to obtain an integrated rate equation much easier.Zeroth-order reactions
A Zeroth-order reaction has a rate that is independent of the concentration of the reactant(s). Increasing the concentration of the reacting species will not speed up the rate of the reaction. Zeroth-order reactions are typically found when a material that is required for the reaction to proceed, such as a surface or a catalyst, is saturated by the reactants. The rate law for a zeroth-order reaction iswhere r is the reaction rate and k is the reaction rate coefficient with units of concentration/time. If, and only if, this zeroth-order reaction 1) occurs in a closed system, 2) there is no net build-up of intermediates, and 3) there are no other reactions occurring, it can be shown by solving a mass balance
Mass balance
A mass balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique...
equation for the system that:
If this differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
is integrated
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
it gives an equation often called the integrated zero-order rate law.
where
represents the concentration of the chemical of interest at a particular time, and represents the initial concentration.A reaction is zeroth order if concentration data are plotted versus time and the result is a straight line. The slope of this resulting line is the negative of the zero order rate constant k.
The half-life of a reaction describes the time needed for half of the reactant to be depleted (same as the half-life
Half-life
Half-life, abbreviated t½, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms , but it may apply to any quantity which follows a set-rate decay.The original term, dating to...
involved in nuclear decay, which is a first-order reaction). For a zero-order reaction the half-life is given by
Example of a zeroth-order reaction
- Reversed Haber processHaber processThe Haber process, also called the Haber–Bosch process, is the nitrogen fixation reaction of nitrogen gas and hydrogen gas, over an enriched iron or ruthenium catalyst, which is used to industrially produce ammonia....
:
It should be noted that the order of a reaction cannot be deduced from the chemical equation of the reaction.
First-order reactions
- See also Order of reactionOrder of reactionIn chemical kinetics, the order of reaction with respect to certain reactant, is defined as the power to which its concentration term in the rate equation is raised .For example, given a chemical reaction 2A + B → C with a rate equation...
.
A first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero-order. The rate law for an elementary reaction that is first order with respect to a reactant A is
k is the first order rate constant, which has units of 1/s.
The integrated first-order rate law is
A plot of
vs. time t gives a straight line with a slope of .The half-life of a first-order reaction is independent of the starting concentration and is given by
.Examples of reactions that are first-order with respect to the reactant:
Further Properties of First-Order Reaction Kinetics
The integrated first-order rate lawis usually written in the form of the exponential decay equation
A different (but equivalent) way of considering first order kinetics is as follows: The exponential decay equation can be rewritten as:
where
corresponds to a specific time period and is an integer corresponding to the number of time periods. At the end of each time period, the fraction of the reactant population remaining relative to the amount present at the start of the time period, , will be:Such that after
time periods, the fraction of the original reactant population will be:where:
corresponds to the fraction of the reactant population that will break down in each time period.This equation indicates that the fraction of the total amount of reactant population that will break down in each time period is independent of the initial amount present. When the chosen time period corresponds to
, the fraction of the population that will break down in each time period will be exactly ½ the amount present at the start of the time period (i.e. the time period corresponds to the half-life of the first-order reaction).The average rate of the reaction for the nth time period is given by:
Therefore, the amount remaining at the end of each time period will be related to the average rate of that time period and the reactant population at the start of the time period by:
Since the fraction of the reactant population that will break down in each time period can be expressed as:
The amount of reactant that will break down in each time period can be related to the average rate over that time period by:
Such that the amount that remains at the end of each time period will be related to the amount present at the start of the time period according to:
This equation is a recursion allowing for the calculation of the amount present after any number of time periods, without need of the rate constant, provided that the average rate for each time period is known.
Second-order reactions
A second-order reaction depends on the concentrations of one second-order reactant, or two first-order reactants.For a second order reaction, its reaction rate is given by:
or or In several popular kinetics books, the definition of the rate law for second-order reactions is
. Conflating the 2 inside the constant for the first, derivative, form will only make it required in the second, integrated form (presented below). The option of keeping the 2 out of the constant in the derivative form is considered more correct, as it is almost always used in peer-reviewed literature, tables of rate constants, and simulation software.The integrated second-order rate laws are respectively
or
[A]0 and [B]0 must be different to obtain that integrated equation.
The half-life equation for a second-order reaction dependent on one second-order reactant is
. For a second-order reaction half-lives progressively double.Another way to present the above rate laws is to take the log of both sides:
Examples of a Second-order reaction:
Pseudo-first-order
Measuring a second-order reaction rate with reactants A and B can be problematic: The concentrations of the two reactants must be followed simultaneously, which is more difficult; or measure one of them and calculate the other as a difference, which is less precise. A common solution for that problem is the pseudo-first-order approximationIf either [A] or [B] remains constant as the reaction proceeds, then the reaction can be considered pseudo-first-order because, in fact, it depends on the concentration of only one reactant. If, for example, [B] remains constant, then:
where
(k' or kobs with units s−1) and an expression is obtained identical to the first order expression above.One way to obtain a pseudo-first-order reaction is to use a large excess of one of the reactants ([B]>>[A] would work for the previous example) so that, as the reaction progresses, only a small amount of the reactant is consumed, and its concentration can be considered to stay constant. By collecting
for many reactions with different (but excess) concentrations of [B], a plot of versus [B] gives (the regular second order rate constant) as the slope.Example:
The hydrolysis of esters by dilute mineral acids follows pseudo-first-order kinetics where the concentration of water is present in large excess.
- CH3COOCH3 + H2O → CH3COOH + CH3OH
Summary for reaction orders 0, 1, 2, and n
Elementary reaction steps with order 3 (called ternary reactions) are rare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order.| Zero-Order | First-Order | Second-Order | nth-Order | |
|---|---|---|---|---|
| Rate Law |  |
 |
 |
 |
| Integrated Rate Law |  |
 |
 |
 [Except first order] |
| Units of Rate Constant (k) |  |
 |
 |
 |
| Linear Plot to determine k |  |
 |
 |
 [Except first order] |
| Half-life |  |
 |
 |
 [Except first order] |
Where M stands for concentration in molarity (mol · L−1), t for time, and k for the reaction rate constant. The half-life of a first-order reaction is often expressed as t1/2 = 0.693/k (as ln2 = 0.693).
Equilibrium reactions or opposed reactions
A pair of forward and reverse reactions may define an equilibriumChemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...
process. For example, A and B react into X and Y and vice versa (s, t, u, and v are the stoichiometric coefficients):
The reaction rate expression for the above reactions (assuming each one is elementary) can be expressed as:
where: k1 is the rate coefficient for the reaction that consumes A and B; k2 is the rate coefficient for the backwards reaction, which consumes X and Y and produces A and B.
The constants k1 and k2 are related to the equilibrium coefficient for the reaction (K) by the following relationship (set r=0 in balance):
Simple Example
In a simple equilibrium between two species:Where the reactions starts with an initial concentration of A,
, with an initial concentration of 0 for B at time t=0.Then the constant K at equilibrium is expressed as:
Where
and are the concentrations of A and B at equilibrium, respectively.The concentration of A at time t,
, is related to the concentration of B at time t, , by the equilibrium reaction equation:Note that the term
is not present because, in this simple example, the initial concentration of B is 0.This applies even when time t is at infinity; i.e., equilibrium has been reached:
then it follows, by the definition of K, that
and, therefore,
These equations allow us to uncouple the system of differential equations, and allow us to solve for the concentration of A alone.
The reaction equation, given previously as:
The derivative is negative because this is the rate of the reaction going from A to B, and therefore the concentration of A is decreasing. To simplify annotation, let x be
, the concentration of A at time t. Let be the concentration of A at equilibrium. Then:Since:
The reaction rate
Reaction rate
The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
becomes:
which results in:
A plot of the negative natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope kf + kb. By measurement of Ae and Be the values of K and the two reaction rate constants will be known.
Generalization of Simple Example
If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions:When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis
Conformational isomerism
In chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted exclusively by rotations about formally single bonds...
of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy
NMR spectroscopy
Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy, is a research technique that exploits the magnetic properties of certain atomic nuclei to determine physical and chemical properties of atoms or the molecules in which they are contained...
.
Consecutive reactions
If the rate constants for the following reaction are and ; , then the rate equation is:For reactant A:
For reactant B:
For product C:
With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation
Master equation
In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
. The differential equations can be solved analytically and the integrated rate equations are
The steady state
Steady state (chemistry)
In chemistry, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, there must be a flow through the system...
approximation leads to very similar results in an easier way.
Parallel or competitive reactions
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.- Two first order reactions:
and , with constants and and rate equations , and The integrated rate equations are then
; and.One important relationship in this case is
- One first order and one second order reaction:
This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated as pseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct:
and . The rate equations are: and . Where is the pseudo first order constant.The integrated rate equation for the main product [C] is
, which is equivalent to . Concentration of B is related to that of C through The integrated equations were analytically obtained but during the process it was assumed that
therefeore, previous equation for [C] can only be used for low concentrations of [C] compared to [A]0See also
- Reaction rateReaction rateThe reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how fast or slow a reaction takes place...
- Reaction rate constant
- Reactions on surfacesReactions on surfacesBy reactions on surfaces it is understood reactions in which at least one of the steps of the reaction mechanism is the adsorption of one or more reactants...
: rate equations for reactions where at least one of the reactants adsorbsAdsorptionAdsorption is the adhesion of atoms, ions, biomolecules or molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. It differs from absorption, in which a fluid permeates or is dissolved by a liquid or solid...
onto a surface - Reaction-diffusion equation
- Steady state approximationSteady state (chemistry)In chemistry, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, there must be a flow through the system...