# 12.7: The Lindemann Mechanism

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The Lindemann mechanism (Lindemann, Arrhenius, Langmuir, Dhar, Perrin, & Lewis, 1922) is a useful one to demonstrate some of the techniques we use for relating chemical mechanisms to rate laws. In this mechanism, a reactant is collisionally activated to a highly energetic form that can then go on to react to form products.

$A + A \xrightleftharpoons [k_1]{k_{-1}} A^* \nonumber$

$A^* \xrightarrow{k_2} P \nonumber$

If the steady state approximation is applied to the intermediate $$A^*$$

$\dfrac{d[A^*]}{dt} = k_1[A]^2 - k_{-1}[A^*][A] - k_2[A^*] \approx 0 \nonumber$

an expression can be derived for $$[A^*]$$.

$A^*]= \dfrac{ k_1[A]^2 }{k_{-1}[A] + k_2} \nonumber$

Substituting this into an expression for the rate of the production of the product $$P$$

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

yields

$\dfrac{d[P]}{dt} = \dfrac{ k_2 k_1[A]^2 }{k_{-1}[A] + k_2} \nonumber$

In the limit that $$k_{-1}[A] \ll k_2$$, the rate law becomes first order in $$[A]$$ since $$k_{-1}[A] + k_2 \approx k_{-1}[A]$$.

$\dfrac{d[P]}{dt} = \dfrac{ k_2 k_1 }{k_{-1}} [A] \nonumber$

This will happen if the second step is very slow (and is the rate determining step), such that the reverse of the first step “wins” in the competition for [A*]. However, in the other limit, that $$k_2 \gg k_{-1}[A]$$, the reaction becomes second order in $$[A]$$ since $$k_{-1}[A] + k_2 \approx k_2$$.

$\dfrac{d[P]}{dt} = k_1[A]^2 \nonumber$

which is consistent with the forward reaction of the first step being the rate determining step, since $$A^*$$ is removed from the reaction (through the formation of products) very quickly as soon as it is formed.

## Third-body Collisions

Sometimes, the third-body collision is provided by an inert species $$M$$, perhaps by filling the reaction chamber with a heavy non-reactive species, such as Ar. In this case, the mechanism becomes

$A + M \xrightleftharpoons [k_1]{k_{-1}} A^* + M \nonumber$

$A^* \xrightarrow{k_2} P \nonumber$

And in the limit that $$[A^*]$$ can be treated using the steady state approximation, the rate of production of the product becomes

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

And if the concentration of the third body collider is constant, it is convenient to define an effective rate constant, $$k_{uni}$$.

$k_{uni} = \dfrac{ k_2 k_1[M] }{k_{-1}[M] + k_2 } \nonumber$

The utility is that important information about the individual step rate constants can be extracted by plotting $$1/k_{uni}$$ as a function of $$1/[M]$$.

$\dfrac{1}{k_{uni}} = \dfrac{k_{-1}}{ k_2 k_1 } + k_2 \left( \dfrac{1}{[M]} \right) \nonumber$

The plot should yield a straight line, the slope of which gives the value of $$k_2$$, and the intercept gives $$(k_{-1}/k_2k_1)$$.

This page titled 12.7: The Lindemann Mechanism is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Patrick Fleming.