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 \((k_{-1}/k_2k_1)\).