Lindemann mechanism

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Lindemann mechanism

In chemical kinetics

Chemical kinetics

Chemical kinetics, also known as reaction kinetics, is the study of reaction rate of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction mechanism and transition states, as well as the construction of ma...
, the Lindemann mechanism, sometimes called the Lindemann-Hinshelwood mechanism, is a schematic 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 observation for most chemical reactions, experiments can often be designed that suggest the possible sequence of steps in a reaction mechanism....
. Frederick Lindemann discovered the concept in 1921 and Cyril Hinshelwood developed it.

It breaks down a stepwise reaction

Stepwise reaction

A stepwise reaction is a chemical reaction with one or more reaction intermediates and involving at least two consecutive elementary reactions....
into two or more elementary steps, then it gives a rate constant for each elementary step. The rate law and rate equation

Rate equation

The rate law or rate equation for a chemical reaction is an equation which links the reaction rate with concentrations or pressures of reactants and constant parameters ....
for the entire reaction can be derived from this information.

Lindemann mechanisms have been used to model gas phase decomposition

Chemical decomposition

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Encyclopedia

In chemical kinetics

Chemical kinetics

Reaction mechanism

Stepwise reaction

Rate equation

Chemical decomposition

Chemical decomposition or analysis is the separation of a chemical compound into chemical element or smaller compounds. It is sometimes defined as the opposite of a chemical synthesis....
reactions. Although the net formula for a decomposition may appear to be first-order (unimolecular) in the reactant, a Lindemann mechanism may show that the reaction is actually second-order (bimolecular).

Activated reaction intermediates

A Lindemann mechanism typically includes an activated reaction intermediate

Reaction intermediate

A reaction intermediate or an intermediate is a molecular entity that is formed from the reactants and reacts further to give the directly observed products of a chemical reaction....
, labeled A* (where A can be any element or compound). The activated intermediate is produced from the reactants only after a sufficient activation energy

Activation energy

In chemistry, activation energy is a term introduced in 1889 by the Swedish scientist Svante Arrhenius, that is defined as the energy that must be overcome in order for a chemical reaction to occur....
is applied. It then decomposes or reacts with another reactant in order to form the products

Product (chemistry)

A product is a substance that forms as a result of a biological- or chemical reaction. While the end product of some chemical reactions may be the result of a relatively rapid reaction, nanoseconds to seconds, chemical equilibrium in complex systems may require years or even centuries to be established....
or a second intermediate.

The steady-state approximation

In some cases, one of the elementary steps is much slower than the other steps. This slow step is called the rate-determining step

Rate-determining step

The rate-determining step is a chemistry term for the slowest reaction step in a chemical reaction. The rate-determining step is often compared to the neck of a funnel; the rate at which water flows through the funnel is determined by the width of the neck, not by the speed at which water is poured in....
because it is the only step that affects the rate. In layman's terms, a rate-determining step could be compared to traveling through a traffic jam: the time it takes to complete a journey is most severely affected by the time spent waiting in the traffic jam, which is the slow step of the journey.

In the steady-state approximation, it is assumed that each of the elementary steps influences the rate, so there is no "fast" or "slow" step. Therefore, all of the steps must be accounted for in calculating the rate equation. It is also assumed that the concentration of intermediate A* remains constant over time because the concentration of A* builds up very quickly but decays very slowly over the course of a reaction, and the concentration of A* never becomes large. This assumption simplifies the calculation of the rate equation.

Examples

General schematic example

The schematic reaction A + M → P is assumed to consist of two elementary steps:

A + M → A* + M (forward reaction rate = k₁; reverse reaction rate = k_-1)
A* → P (forward reaction rate = k₂)

Assuming that the concentration

Concentration

In chemistry, concentration is the measure of how much of a given chemical substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently the concept is limited to homogeneous solutions, where it refers to the amount of solute in the solvent....
of intermediate A* is held constant according to the quasi steady-state approximation, what is the rate of formation of product P?

First, find the rates of production and consumption of intermediate A*. The rate of production of A* in the first elementary step is simply:

d[A*]/dt = k₁ [A] [M] (forward first step)

A* is consumed both in the reverse first step and in the forward second step. The respective rates of consumption of A* are:

-d[A*]/dt = k_-1 [A*] [M] (reverse first step)

-d[A*]/dt = k₂ [A*] (forward second step)

According to the steady-state approximation, the rate of production of A* equals the rate of consumption. Therefore:

k₁ [A] [M] = k_-1 [A*] [M] + k₂ [A*]

Solving for [A*], it is found that

[A*] = (k₁ [A] [M]) / (k_-1 [M] + k₂)

The overall reaction rate is

d[P]/dt = k₂ [A*]

Now, by substituting the calculated value for [A*], the overall reaction rate can be expressed in terms of the original reactants A and M as follows:

d[P]/dt = (k₁k₂ [A] [M]) / (k_-1 [M] + k₂)

Specific practical example

The decomposition of dinitrogen pentoxide

Dinitrogen pentoxide

Dinitrogen pentoxide is the chemical compound with the chemical formula N2O5. Also known as nitrogen pentoxide, N2O5 is one of the binary nitrogen oxygens, a family of compounds that only contain nitrogen and oxygen....
to nitrogen dioxide

Nitrogen dioxide

Nitrogen dioxide is the chemical compound with the chemical formula NitrogenOxygen2. One of several nitrogen oxides, NO2 is an intermediate in the industrial synthesis of nitric acid, millions of tons of which are produced each year....
and nitrogen trioxide

N₂O₅ → NO₂ + NO₃

is postulated to take place via two elementary steps, which are similar in form to the schematic example given above:

N₂O₅ + N₂O₅ → N₂O₅* + N₂O₅
N₂O₅* → NO₂ + NO₃

Using the quasi steady-state approximation, the rate equation is calculated to be

Rate = k₂ [N₂O₅]* = k₁k₂ [N₂O₅]² / (k_-1[N₂O₅] + k₂)

Experiment has shown that the rate is observed as first-order in the original concentration of N₂O₅ sometimes, and second order at other times.

If k₂ >> k_-1 (>> means "much larger than"), then the rate equation may be simplified by assuming that k_-1 ~= 0. Then the rate equation is

Rate = k₁[N₂O₅]²

which is second order. In contrast, if k₂ << k_-1 (<< means "much less than"), then the rate equation may be simplified by assuming k₂ ~= 0. Then the rate equation is

Rate = k₁k₂[N₂O₅] / k_-1

which is first order.