Vietnam part 1 CARL p1 52

2 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 ot

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)

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

potential-energy profile of the reaction pathway This potential-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

f H(N2O)+ f H(C2H) > f H(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

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

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 E A) for

transformation are too high and thus rates of reaction are very slow Many

radical-radical 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

molecular-molecular 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

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

The excess photon energy is dissipated largely as kinetic energy of the separating

fragments These fragments are quickly thermalized via collisions with N2or O2resulting in

heating of the atmosphere

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 N F N Gone 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 r FG 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

between 1 and 2 x 10 cm s 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 cm 3 s -1 (molecule -1)

The units of a ter-molecular rate constant is cm 6 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“

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

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, G1, G2,

F2-G3) = 6 Thus it is reasonable that collision frequency requires the product of reactant

concentrations

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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

pre-exponential 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, E a) 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 E a – 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

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

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

H-abstraction 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

6-311++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

co-reactants, 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

third-row 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

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

Statistical analysis of systems with discrete energy levels, such a molecules and atoms in the

gas phase, may be aided by consideration of partition functions

Partition functions are an expression of the distribution of energy amongst various modes

of motion (translation, vibration, rotation) and electronic energy

For the gas-phase one can make a very good approximation that the above degrees of

freedom (i.e., vibrational, translational, rotational, and electronic) are independent and that

the individual particles, N, (molecules, atoms) are independent This enables the total

partition function of a system, Q, to be expressed in terms of partition functions of

individual species and be further factored into the various degrees of freedom

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The general expression for a partition function is given above In order to calculate a

partition function one requires an expression for the energy states (or levels) of a particular

degree of freedom

For the translational partition function energy spacings are derived from the

quantum-mechanical treatment of the allowed modes in a 3-dimensional box, the derivation of which

is not a part of this course The energy spacings associated with translational motion are so

small that the translational partition function is by far the largest contributor to the total

partition function of a gaseous system A typical value is 1020 for a volume of 1 cm3

Note that, unlike the other partition function that follow, the translational partition

function depends on both temperature and volume (refer to the following page)

The small energy spacings for allowed translational energies are so small compared to the

average thermal energy of (3/2)kBT that one may replace the summation in the above

equation with an integration (note the change in lower limit) in order to arrive at a useable

expression for translational partition function

m is the mass of the molecule under consideration

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