Lindemann mechanism

In chemical kinetics, the Lindemann mechanism, sometimes called the Lindemann-Hinshelwood mechanism, is a schematic reaction mechanism. Frederick Lindemann discovered the concept in 1921 and Cyril Hinshelwood developed it.^[1]

It breaks down a stepwise reaction into two or more elementary steps, then it gives a rate constant for each elementary step. The rate law and rate equation for the entire reaction can be derived from this information.

Lindemann mechanisms have been used to model gas phase decomposition 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).^[2] An example is given below.

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Activated reaction intermediates

A Lindemann mechanism typically includes an activated reaction intermediate, labeled A* (where A can be any element or compound). The activated intermediate is produced from the reactants only after a sufficient activation energy is applied. It then decomposes or reacts with another reactant in order to form the products 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 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:

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

Assuming that the concentration of intermediate A* is held constant according to the 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 calculate 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)^[3]

Specific practical example

The decomposition of dinitrogen pentoxide to nitrogen dioxide 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:

1. N₂O₅ + N₂O₅ → N₂O₅* + N₂O₅
2. N₂O₅* → NO₂ + NO₃

Using the 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.^[4]

References