Abstract
This chapter guides the reader through the phenomenology of the simplest kinetics of chemical systems to the modeling of the rate coefficients governing their time evolution. From the analysis of the weakness of the transition state (TS) model approach (that is, phenomenologically valid but useless for predicting), the rate of chemical processes is rationalized in terms of collisions of two structureless bodies using classical mechanics. In this way, it is possible to follow the space and time evolution of the colliding partners.
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Notes
- 1.
The TS is usually associated with a saddle. However, this association is an arbitrary assumption because (as it will be discussed in some detail in Chap. 4) the regions of the PES dividing trajectories back reflected from those crossing over are associated with periodic orbits dividing the (potential energy) surface (PODS). PODS can be more than one, do not necessarily sit on a saddle, and can be recrossed by the system.
- 2.
Ionic is the interaction between charged particles (ions) in which the number of positive components (e.g., protons) differs from that of negatively charged particles (e.g., electrons). Covalent is the interaction associated with evenly shared particles (e.g., two atoms equally sharing the electrons). Permanent is a stable feature of the particles (e.g., the dipole moment). Induced is a temporary feature associated with the presence of another particle. Short and long ranges refer to the distance between the particles.
- 3.
The spherical polar coordinates of the particle i make use of \(r_i\) (the module of the position vector \(\mathbf{r}_i\)) and of its orientation angles \(\Theta _i\) and \(\Phi _i\).
- 4.
Another popular formulation of the equations of motion is the Lagrange’s one
$$\begin{aligned} \frac{{\mathrm d }}{{\mathrm d }t} \frac{\partial L}{\partial \dot{r}_W} - \frac{ \partial L}{\partial r_W} = 0, \, \, \, \, \, \; \end{aligned}$$where \(L = \mathcal{T } - \mathcal{V}\) is the Lagrangian of the system.
- 5.
In fact, see Fig. 1.7, we have for the components (z, y) of \( \mathbf{r} \), \( z = -r \cos \theta \) and \( y = r \sin \theta \) (\( \theta \) \(-\pi /2 = \vartheta \)) or by differentiating with respect to time
$$\begin{aligned} v_z \equiv \frac{{ \mathrm d } z}{{ \mathrm d } t} = - \dot{r} \cos \theta + r \frac{{ \mathrm d } \theta }{ { \mathrm d } t} \sin \theta \, \; { \mathrm and } \; \; v_y \equiv \frac{{ \mathrm d } y }{ { \mathrm d } t} = \dot{r} \sin \theta + r \frac{{ \mathrm d } \theta }{ { \mathrm d } t} \cos \theta . \end{aligned}$$.
- 6.
By definition of angular momentum and vector product is in fact:
$$\begin{aligned} \mathbf{L } = \mathbf{r} \times \mathbf{p} = \mu ({ \mathbf r} \times \dot{{ \mathbf r} }) = \mu \left( z \frac{{\mathrm d}y}{{\mathrm d}t}-y\frac{{\mathrm d}z}{{\mathrm d}t} \right) . \end{aligned}$$in this case, the motion is confined to a plane yz as in our case. Then, at time \( t = 0 \), since \( { \mathrm d } y / { \mathrm d } t = 0 \), we have \( | \mathbf{L } | = y \mu { \mathrm d } z / { \mathrm d } t = \mu bv\), while at generic times t, you will have \( | \mathbf{L } | = \mu r ^ 2 \dot{\theta } \) (see next note).
- 7.
In the quantum treatment in formulating the conservation of the diatomic total angular momentum quantum number l, \(l^2\) will be replaced (apart from a constant factor) by \( l (l +1) \) which is the total angular momentum eigenvalue of the quantum operator, as we will see more forward.
- 8.
The equation state (or virial) is
$$\begin{aligned} p V_m/ RT = 1 + B (T) / V_m + C (T) / V_m ^ 2 + D (T) / V_m ^ 3 \, \, \, \, \; V_m = { \mathrm molar\,\, volume.} \end{aligned}$$ - 9.
From CMB experiments, it is impossible to distinguish between positive and negative deflections. Accordingly, the absolute value of \( \theta \) is considered.
- 10.
\( \int x e^{-x} {\mathrm d} x =-(e^{-x} + x e^{- x}) \).
- 11.
\(e^{-E_{a}/k_BT}+\frac{E_a}{k_BT}e^{-E_ a /k_BT} \simeq e^{-E_{a}/k_BT}\) provided \( 1>> E_a/k_BT\).
- 12.
\( \int (1- hx - cx ^ 2) ^ { -1 / 2 } { \mathrm d } x = - (c) ^ { -1 / 2 } \ arcsin \left[ - (cx +2 h) / (h ^ 2 +4 c) ^ { -1 / 2 } \right] \).
- 13.
\(arcsin (x_1) \pm \ arcsin (x_2) = \ arcsin \left[ x_1 (1- x_2 ^ 2) ^ { 1/2 } \pm x_2 (1- x_1 ^ 2) ^ { 1/2 } \right] \).
- 14.
The parameters for the OH molecule reported in the figure have been taken from G. Herzberg, Constant of Diatomic Molecules (Van Nostrand, 1978, New York).
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Laganà, A., Parker, G.A. (2018). From the Phenomenology of Chemical Reactions to the Study of Two-Body Collisions. In: Chemical Reactions. Theoretical Chemistry and Computational Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-62356-6_1
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DOI: https://doi.org/10.1007/978-3-319-62356-6_1
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