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 1922 and Cyril Hinshelwood developed it.^{[1] [2]}

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.

The Lindemann mechanism is used to model gas phase decomposition or isomerization reactions. Although the net formula for a decomposition or isomerization appears to be unimolecular and suggests first-order kinetics in the reactant, the Lindemann mechanism shows that the unimolecular reaction step is preceded by a bimolecular activation step so that the kinetics may actually be second-order in certain cases.^[3]

Activated reaction intermediates

The overall equation for a unimolecular reaction may be written A → P, where A is the initial reactant molecule and P is one or more products (one for isomerization, more for decomposition).

A Lindemann mechanism typically includes an activated reaction intermediate, labeled A*. The activated intermediate is produced from the reactant only after a sufficient activation energy is acquired by collision with a second molecule M, which may or may not be similar to A. It then either deactivates from A* back to A by another collision, or reacts in a unimolecular step to produce the product(s) P.

The two-step mechanism is then

$A + M \rightleftharpoons A^* + M$

$A^* \rightarrow P$

Rate equation in steady-state approximation

The rate equation for the rate of formation of product P may be obtained by using the steady-state approximation, in which the concentration of intermediate A* is assumed constant because its rates of production and consumption are (almost) equal.^[4] This assumption simplifies the calculation of the rate equation.

For the schematic mechanism of two elementary steps above, rate constants are defined as k₁ for the forward reaction rate of the first step, k_-1 for the reverse reaction rate of the first step, and k₂ for the forward reaction rate of the second step. For each elementary step, the order of reaction is equal to the molecularity

The rate of production of the intermediate A* in the first elementary step is simply:

$\frac{d[A^*]}{dt} = k_1 [A] [M]$ (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:

$\frac{-d[A^*]}{dt} = k_{-1} [A^*] [M]$ (reverse first step)

$\frac{-d[A^*]}{dt} = k_2 [A^*]$ (forward second step)

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

$k_1 [A] [M] = k_{-1} [A^*] [M] + k_2 [A^*]$

Solving for $[A^*]$, it is found that

$[A^*] = \frac{k_1 [A] [M]}{k_{-1} [M] + k_2}$

The overall reaction rate is

$\frac{d[P]}{dt} = k_2 [A^*]$

Now, by substituting the calculated value for [A*], the overall reaction rate can be expressed in terms of the original reactants A and M:^[5][4]

$\frac{d[P]}{dt} = \frac{k_1 k_2 [A] [M]}{k_{-1} [M] + k_2}$

Reaction order and rate-determining step

The steady-state rate equation is of mixed order and predicts that a unimolecular reaction can be of either first or second order, depending on which of the two terms in the denominator is larger. At sufficiently low pressures, $k_{-1}[M] \ll k_2$ so that $d[P]/dt = k_1 [A][M]$, which is second order. That is, the rate-determining step is the first, bimolecular activation step.^[4][5]

At higher pressures, however, $k_{-1}[M] \gg k_2$ so that $\frac{d[P]}{dt} = \frac{k_1k_2}{k_{-1}}[A]$ which is first order, and the rate-determining step is the second step, i.e. the unimolecular reaction of the activated molecule.

The theory can be tested by defining an effective rate constant (or coefficient) $k_{uni}$ which would be constant if the reaction were first order at all pressures: $\frac{d[P]}{dt} = k_{uni}[A], \quad k_{uni} = \frac{1}{[A]} \frac{d[P]}{dt}$. The Lindemann mechanism predicts that k decreases with pressure, and that its reciprocal $1/k$ is a linear function of $1/p$. Experimentally for many reactions, $k$ does decrease at low pressure, but the graph of $1/k$ as a function of $1/p$ is quite curved. To account accurately for the pressure-dependence of rate constants for unimolecular reactions, more elaborate theories are required such as the RRKM theory.^[5][4]

Example: Decomposition of dinitrogen pentoxide

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

$Rate = k_2 [N_2 O_5]^* = \frac{k_1 k_2 [N_2 O_5]^2}{k_{-1} [N_2 O_5 ] + k_2}$

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 $k_2 \gg k_{-1}$ ($\gg$ means "much larger than"), then the rate equation may be simplified by assuming that $k_{-1} [N_2 O_5 ] + k_2 \simeq k_2$. Then the rate equation is

$Rate = k_1 [N_2 O_5 ]^2$

which is second order. In contrast, if $k_2 \ll k_{-1}$ ($\ll$ means "much less than"), then the rate equation may be simplified by assuming $k_{-1} [N_2 O_5 ] + k_2 \simeq k_{-1} [N_2 O_5 ]$. Then the rate equation is

$Rate = \frac{k_1 k_2 [N_2 O_5]}{k_{-1}}$

which is first order.^[6]

References

1. Di Giacomo, F. (2015). "A Short Account of RRKM Theory of Unimolecular Reactions and of Marcus Theory of Electron Transfer in a Historical Perspective". Journal of Chemical Education 92 (3): 476. doi:10.1021/ed5001312.
2. Lindemann, F. A.; Arrhenius, S.; Langmuir, I.; Dhar, N. R.; Perrin, J.; Mcc. Lewis, W. C. (1922). "Discussion on ?the radiation theory of chemical action?". Transactions of the Faraday Society 17: 598. doi:10.1039/TF9221700598.
3. "Gas phase decomposition by the Lindemann mechanism" by S. L. Cole and J. W. Wilder. SIAM Journal on Applied Mathematics, Vol. 51, No. 6 (Dec., 1991), pp. 1489-1497.
4. Atkins P. and de Paula J., Physical Chemistry (8th ed., W.H. Freeman 2006) p.820-1 ISBN 0-7167-8759-8
5. Steinfeld J.I., Francisco J.S. and Hase W.L. Chemical Kinetics and Dynamics (2nd ed., Prentice-Hall 1999), p.335 ISBN 0-13-737123-3
6. "Lindemann Mechanism" by W. R. Salzman at the University of Arizona, 2004. Access date 8 December 2007.