12.7: The Lindemann Mechanism
- Page ID
- 84601
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \((k_{-1}/k_2k_1)\).