Vietnam part 1 CARL p1 52

This first part of the course in concerned with developing models that can account for
chemical reactivity. Early models that describe elementary reactions focused on reactions in
the gas-phase, since in this environment the time between collisions is relatively large
compared to the collision duration itself so that each collision can be considered separately
from its environment. This is not generally the case for reactions occurring in liquids and on
surfaces.
The following nine pages provide a short illustration of the potential complexity of
elementary reactions. These points was be discussed later in the course.

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In order to predict the course of a chemical reaction it is useful to know how the total
chemical energy of the reaction system changes as the two reactants (AB and CD) approach
each other. As we will see later, from these energy changes one can gain some
understanding of how the products (ABC + D) are formed and also if there are other
possible product channels.
At first sight it might seem reasonable that one could easily predict the change in energy
using atomic electronic structure information of the periodic table and estimates of the
binding energies of the valence atomic electrons. After all, we make good use of this
information to predict structures of stable molecules and the presence of double, single, and
triple bonds, for example.
On the figure above, the total chemical energy is the sum of potential energy associated
with electron-electron and nuclei-nuclei repulsion and electron-nuclei attraction. In fact it
would not be so difficult to obtain a rough estimate of this chemical energy as the two
molecules AB and CD approach each other using a simple calculator. If one did this one
would likely obtain the plot above. That is, you would not find any significant change within the uncertainties of your calculations - in chemical energy as the reaction proceeded
from reactants, AB + CD, to products, ABC + D.
The time scale for this type of chemical reaction is often in the order of pico-seconds (10-9
s).

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It is only by ‘zooming in’ by a factor of about ten thousand that one is able to discern the
energy associated with the various atomic rearrangements that give a detailed potentialenergy profile of the reaction pathway. This potential-energy profile can only be observed
when calculating the total chemical energy to the fifth or sixth significant figure.
As an analogy, if the height of a typical table was total chemical energy of reactants then
the energies associated with chemical change describing the reaction would take place
within the thickness of the varnish only! Furthermore, in order to make accurate chemical
predictions, calculations require at least ten times higher resolution than this, which is quite
a challenging task.
So predicting the rate constant (see later) and products of a chemical reaction from first
principles cannot be achieved without very high level computations.
Chemists rely heavily on of course is insight from similar types of chemical reactions as an
aid in predicting reactivity and products. Proficiency in this direction is built up over many
years of experience, but as we will see later, for most reactions it is simply not possible to
predict reactivity with sufficient accuracy to use to model complex chemical systems such
as the atmosphere or hydrocarbon combustion.

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To illustrate this point let us consider the reaction of C2H with N2O. This page shows all of
the possible product channels based on overall standard (298.15 K and 1 atm.) reaction
enthalpy.
fH(N2O )+ fH (C2H) > fH (products)
That is, all exothermic channels under standard conditions.
Here there are nineteen possible (exothermic) product channels. From a theoretical
standpoint, in order to predict which products are likely, one requires information on the
energy change as the C2H radical approaches N2O.

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Here is representation of the energy changes for the C2H + N2O reaction. Represented by
small yellow circles are local maximum (local transition states – see later) and local
minimum (quasi-stable intermediate structures) having their own set of internal energy
states. The large orange circle on the left represents the energy of the reactants and the green
circles on the right represent bi-molecular products having an energy lower than, or close to,
that of the reactants. It can be seen that the products CCNN + OH (in a slightly endothermic
reaction) are not likely to be formed since the reaction path involves transitions over large
barriers of more than 50 kcal mol-1.
1 kcal = 4.18 kJ

The four lowest-lying product channels, H + CCNNO, N2 + HOCC, CO + HCNN, HCCO
+ N2 are all accessible from atomic re-arrangements having energies below that of the
reactants except for the first step that involves a transition over transition state 1/2c. All
allowed channels have a common pathway until intermediate structure 8.
The most favourable product channel is not necessarily the one having the lowest path
(from structure 8) as one must remember that this is a quantized system so in addition to
energies one must also take into account the (vibrational and rotational) states that are
available to at least some of the local maxima (local transition states) and local minima
along the reaction pathways. These aspects will be covered later in the course. Note also ,
since total energy of the system is conserved during the passage from reactants to products
(which will have a duration of the order of tens to hundreds of pico seconds in this case) the
energy of the system does not follow the lines given in this diagram, but remains above the

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energy of the first transition state (1/2c).

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The following three pages again illustrates the potential large complexity that can be
involved in what might appear to be a fairly simple bi-molecular reaction. In this case we
list the potential exothermic products of the reaction of nitric acid (HNO3) with the C2H
radical. Here we find at least 143 exothermic reaction channels! The reaction enthalpies in
kJ mol-1 are given in the first column (the figure after the decimal point can be ignored).
After these examples one can perhaps appreciate that constructing a chemical kinetic model
of a system from first principles using elementary kinetic information can be a very
challenging task, but this is the only way that detailed chemical information as a function of
time can be ascertained for systems such as the Earth’s atmosphere and combustion. It is the
aim of this first part of the course to give you some background in how we can theoretically
treat and experimentally determine rate constants and product distributions of elementary
reactions. Later on we will consider real examples of complex chemical systems,
particularly photo-chemical processes important in the Earth’s atmosphere.

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Several broad classes of reactions can be distinguished. Even if overall energetically
favourable (due to a negative enthalpy of reaction), molecular-molecular collisions
rarely result in a chemical transformation, the barriers (given the energy EA) for
transformation are too high and thus rates of reaction are very slow. Many radicalradical reactions on the other hand proceed without a barrier and thus the fraction of
collisions that result in a reaction is high, and often remains fairy constant with
temperature. Many ion-radical also have no energy barrier and their reactivity is
governed by long-range ionic-dipole or ionic induced dipole attractive forces, again,
temperature dependencies are weak.
Whilst the latter two types of reactions are important in many chemical systems
their reactivity's are relatively (though not always) easy to predict. This might not be
the case though for product distributions.
By far the greatest range in reactivity’s occurs with interactions with radicals and
molecules. These systems are most often subject to a barrier, but unlike molecularmolecular interactions, one that may be surmountable under normal conditions of
temperature (below 1000 K). For molecular-radical interactions, the range the
barrier heights vary greatly from the occasional barrierless reaction that occurs at
nearly every collision to those that are effectively unreactive. The barrier heights for
these reactions are very difficult to predict from the properties of the isolated
reactants, without high-level quantum mechanical calculations or prior knowledge
of similar reactions. Radical-molecular reactions and radical-radical reactions
dominate atmospheric and combustion chemistry.

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Note that the average thermal energy of a gas is given by (3/2)kBT , where kB is the
Boltzmann Constant (1.38  10-23 J K-1).

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Gas-phase reactions are referred to as homogeneous reactions. They are, naturally, the most
common type of reaction that occurs in the atmosphere and in combustion. Heterogeneous
reactions (those between gases and solids or gases and liquids) also play an important role
in atmospheric chemistry, as do reactions that take place purely in the liquid phase, inside
suspended droplets. This first section will concentrate on some simple properties of
homogeneous reactions in the gas phase.
The various types of reactions are displayed above. Whether or not a reaction is possible
that leads to particular products may be determined first by simply looking up the
enthalpies of formation of the reactants and products to ascertain if the overall enthalpy
change for the reaction is positive or negative. A negative value will indicate that the
reaction is possible, provided that a significant barrier does not exists. As already
mentioned, the height of the barrier and the propensity to go to one or other of several
possible product channels is very difficult to predict based on the properties of the isolated
reactants. For this, one must nearly always rely on laboratory studies of the reactions in
isolation, or very high-level quantum calculations, or (occasionally) on prior knowledge of
the behaviour of similar reactions.
Note that termolecular reactions are generally associated with addition reactions in which
two molecules coalesce. The resulting product is then vibrationally-excited due to the initial
collision energy, requiring that some of this vibrational energy be removed by further
collisions before sufficient energy accumulates again in the initially-formed bond causing

re-dissociation, and, hence, no overall reaction. As will be seen later, the dependence on
pressure (i.e., the concentration of M) of the rate constants of termolecular reactions
changes from linear to independent over a wide enough range. Bimolecular products are
also possible from addition reactions.
Finally several photo-dissociation processes are of crucial importance in the atmosphere.

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The excess photon energy is dissipated largely as kinetic energy of the separating
fragments. These fragments are quickly thermalized via collisions with N2 or O2 resulting in
heating of the atmosphere.

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There is no reaction without a collision!

Chemical reactions were first usefully rationalised in terms of collisions of species treated
as hard spheres. After all, a bi (or ter-)-molecular reaction will not occur without a
collision. Using the Maxwell-Boltzmann expression for the distribution of molecular
speeds at a given temperature and given the molecular masses, one can derive the frequency
of collisions between two types of species. This number comprises the average relative
speeds of the species, the distance, rFG, between the centres of the two species when they
collide (expressed in terms of collision area, or cross-section), and the concentration (or
number density) product of the species. Please be sure not to confuse kB (Boltzmann’s
constant) with k (the rate constant).
If one divides collision frequency by NFNG one has units of cm3 s-1 (per molecule). These
are the same units as bi-molecular rate constant. Thus a bi-molecular rate constant can be
thought of as the volume swept out per second by a disc of radius rFG travelling at a speed

equivalent to the mean speed of a reduced mass FG at temperature T.
Not all collisions result in reaction, but if one uses this classical collision frequency and
assumes that every collision leads to reaction what (maximum) value of bi-molecular rate
constant is expected?
We can take an example of a collision between two species, say OH and CH4. If we also
assume the collision radius to be 5 Angstrom then the rate constant, assuming reaction at
every collision, should be 1.5 x 10-10 cm3 s-1. Naturally, as the reduced mass of the reacting
pairs increases so does the collision diameter. Curiously though, it turns out that for the vast
majority of radical molecule reactions that the square root of reduced mass is closely
proportional to the square of collision diameter such that a very large proportion of
reactions that occur at every collision have a (bi molecular) rate constant that lies

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between 1 and 2 x 10-10 cm3 s-1. There is only a small number of radical-molecule
reactions that have rate constants greater than 3 x 10-10 cm3 s-1.
The units of a unimolecular rate constant is s-1.
The units of a bi-molecular rate constant is cm3 s-1 (molecule -1).
The units of a ter-molecular rate constant is cm6 s-1 (molecule -2).
Note that “molecule” is not a standard unit, so it is sometimes omitted in the text. For
example, density is sometimes written with the units “molecule cm-3 “ and as “cm-3“.

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Not all collisions will result in a chemical reaction. The next step is to calculate the fraction
of collisions that result in reaction.
Those reactions that proceed over a potential-energy barrier come into this category.
The picture above shows a reaction barrier. In order for the reaction to occur, the collision

energy between reactants A and BC (that is, the relative translational energy) needs to be
greater than E, the classical barrier height. From the Maxwell-Boltzmann distribution of
molecular speeds we can calculate the distribution of collisional energies. Typical energy
distributions are shown by the blue lines for two different gas temperatures. As the
temperature increases, the fraction of total collisions with energy greater than E increases,
and therefore so does the rate of collisions that lead to reaction. This leads the multiplication
of an exponential term to the collision frequency equation which now refers to successful
reactive collisions.
The rate constant for a bi-molecular reaction, k, is then the frequency of successful
collisions divided by the product of densities of the reactants. Thus the bi-molecular rate
constant is independent of number density of reactants.

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When the distribution function for collisional energy is integrated, one arrives at a rate
constant based on simple collision theory that has an exponential dependence on the barrier
height, E. The pre-exponential factor, A(T), is the collision frequency divided by the
product of species concentrations. As just mentioned, if one inserts values for mass and
collision radius, one finds that A(T) remains nearly constant for a very large range of
reactants. Thus for reactions that occur on every collision, a bi-molecular rate constant of (1
to 3) x 10-10 cm3 s-1 is expected.
Notice that the units of a bi-molecular rate constant are volume per unit time. Is there a
visual representation then of a rate constant in terms of volume? A reasonable
representation is to imagine a surface or area r2, where r = rF + rG, that moves through
space at the average collision velocity, given by the term (…..)0.5 in A(T)coll. The volume
that is swept out every second by a typical collision cross-sectional area (r2) is of the order
of 2 x 10-10 cm3.
If there were two molecules of F (F1 and F2) and three radicals of G (say G1, G2, and
G3) per cm3, how many possibilities for collisions (per cm3) are there between F and G?

This is simply the product of the concentrations (F1-G1, F1-G2, F1-G3, F2-G1, F2-G2, F2G3) = 6. Thus it is reasonable that collision frequency requires the product of reactant
concentrations.
The rate constant as it is written above is in the form of the so-called Arrhenius equation,
which is seen in many scientific fields.

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For reactions with large barriers, the strongest temperature dependence comes from the
exponential term of the Arrhenius equation. For these reactions, a plot of ln(k) vs 1/T will
give, more or less, a straight line with an slope of approximately E/R. Indeed, this is how
experiments can obtain an approximation for the reaction barrier height provided that two or
more competitive pathways are not operative. For some reactions, the A-factor, or preexponential factor, is also a relatively strong function of temperature. In such cases, a
curvature can be observed in the Arrhenius plot and the formula for k is best described by
the so-called "modified Arrhenius" expression, for which values of A, n, and E
(sometimes referred to as activation energy, Ea) are required.
The major review publications (see later) for bi-molecular elementary reactions usually
express rate constant data in terms of A, n, and E (or Ea – activation energy, which is
equivalent)

We noted earlier that not all collisions result in a reaction and the rationalization for this was
that the reaction proceeded over a barrier and that only a fraction of collisions had sufficient
energy to overcome the barrier. But even when there is no barrier for reaction, not all
collisions may result in reaction (due, for example, to unfavourable orientations; especially
for large reactants). This is often called steric hindrance (given by a factor S) and, as a
result, the Arrhenius pre-exponential factor given above, A(T), may only be a fraction of A
that is derived from simple collision theory.

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Many reactions have barriers that are high enough to substantially reduce (sometimes to
zero effectively) the number of collisions that lead to chemical reaction. However, the
origin of barriers in chemical reactions have long been, and still are in some respects, a bit
of a mystery.
An early attempt to rationalize reaction barriers was done by Marcus (ca. 1950) who
considered two harmonic potentials, one belonging to the reactants and one to the products.
The reaction barrier in the model corresponds the crossing point of the two parabolic
potential-energy surfaces. When the systems are “weakly coupled” (this will be explained
later) the barrier height is the point of the crossing itself. For “strongly coupled” systems the
barrier is lowered by the so called coupling energy.
Here, if one considers a homologous reaction series in which the same atom is transferred
then as the enthalpy of reaction increases, this blue curve is lowered with respect to the
green and therefore so is the barrier.
This predicts that there would be a strong correlation between activation energy and
reaction enthalpy. This is indeed observed in many instances when considering H-atom
abstractions in a homologous reaction series.
But this picture is incorrect?

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Modern theories now recognize the critical role of excited electronic states in
forming the reaction barrier. For an atom or molecule having more than one
electron, there is no exact solution to the quantum-mechanical wavefunction. This
means that the energy of the molecule has to be numerically computed.
Computation time and effort is greatly reduced if one treats the motion of the atoms
separately from the motion of the electrons (due to the great disparity in their
masses). When this is done it, is often found that there is a smooth connection
between the ground electronic state of the reactants and an excited state of the

products. This approximation mostly works well, but it quite often fails in regions
(configurations) where two surfaces cross. If one considers the motions of the
electrons and the nuclei to be coupled, then the path followed appears to lead to an
avoided crossing of two surfaces.
These avoided crossing are the basis of barriers for chemical reactions. They are
very difficult to compute accurately (even on powerful mainframe computers).
Experiments (though also difficult) can, for many reactions, reasonably accurately
determine the barrier height (as will be seen later).
Because both fundamental states are neutral, their energies remain essentially
unchanged as the reactants approach until significant overlap develops between the
orbitals of the two reactants.
This picture actually predicts a strong dependence of barrier height on reactant
molecule bond strength, which is often found for a series of similar reactions.

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It has been recently proposed by Donahue that for many reactions between neutral
species ionic surfaces couple most strongly with the ground state surfaces of the
reactants and products and the molecular triplet configuration appears only as a
perturbation. Here the evolution of energies occurs over a wider distance as
described by Coulombic interaction.
Here the reaction barrier is strongly related to the difference between the
ionization potential of the molecule and the electron affinity of the radical.
Recently, Donahue and co-workers have published a series of papers on the
interpretation of H-abstraction barriers using various curve-crossing models, since
there has been some debate on whether the avoided crossing is more closely
described by either neutral-neutral or neutral-ionic interactions. For many Habstraction reactions there is evidence that the barrier height is controlled largely by
the avoided crossing of two curves each describing a transfer of a proton, XH+ + B A + BH and XH + B  X- + BH+, rather than a hydrogen atom.

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The information below is related to a study carried out at KULeuven by the kinetics group
and the group of Prof. Nguyen on a series of reaction involving H abstraction by CF2. This
radical is not important in the atmosphere but the study does illustrate the correlations
suggested by the previous slides.
The stationary points of H-atom abstraction reactions of CF2(3B1) with XHn (n = 1 – 4: X =
H, F, Cl, Br, O, S, N, P, C, and Si) were computed using UCCSD(T) methods with 6311++G(3df,2p) and aug-cc-pVTZ basis sets. Covalent surface crossing heights, calculated
using the X-H and C-H bond dissociation energies of XHn and of the CHF2 product,
correlate well with the computed classical barrier heights. Within each group of coreactants, the barrier heights increase with increasing X-H bond dissociation energy,
whereas the C-H bond lengths of the transition structures decrease. H abstractions are
energy-demanding processes for second-row X atoms, but are more facile for their thirdrow X counterparts.
It is not important here that you understand the ab initio method, UCCSD(T) and the details
of the basis sets 6-311++G(3df,2p) and aug-cc-pVTZ.

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For this series of reactions there is apparently little correlation in barrier height based on the
ionic curve-crossing model, though for OH reactions with hydrocarbons, this might be
significant. For CF3 H-abstraction, the covalent model seems to be the better picture.
Though one must remember that predictions based on these models will not necessarily
yield rates constant with sufficient accuracy for chemical modelling. Correlations such as
these are considered to be an almost last resort in the absence of experimental/ or
computationally-derived rate constants.

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