# Reaction Dynamics

Concentrates on bimolecular reactions and Transition State Theory, and includes things like the Kinetic Isotope Effect and Temperature-dependence.

Reaction Dynamics Notes

Bimolecular Reaction Dynamics

Simple Collision Theory (SCT)

Reactions occur by collisions. Collision energy Et must exceed the barrier height Eo.

Reaction Rate: collision rate x fraction of collisions with Et > Eo.

Reaction Rate =

Collision Rate = ZABo[A][B]

ZABo, collision number, = crel σc, and σc = π d2 (collision cross section).

Thermal Rate Coefficient:

kSCT(T) =

When comparing against experiment, reset σ, as σx = Pσc, where P is the probability of reaction on collision, or the steric factor.

Collisions between two Atoms and Potential Energy Curves

Separate electronic and nuclear motions of atoms A and B, then solve the Schrodinger Equation for electronic motion:

Heψe = V(R)ψe

V(R) = dependence of electronic energy on A-B separation, R.

Hψ(R) = Eψ(R), E = hν(v+½) + …

Force acting on particles at separation R:

F(R) = -dV(R)/dR

Newton’s Laws to calculate motion.

Also, glancing collisions –

| l | = μ crel b

l is the orbital angular momentum of collision, and b is the impact parameter (distance of closest approach of the atoms in the absence of an interaction potential).

Collisions between Molecules and Potential Energy Surfaces

Consider simplest reaction:

H + H2 → H2 + H

The PE will depend on 3 coordinates, (r1, r2, θ).

Resulting potential energy function (or surface) has the following form plotted as a function of r1 and r2 at 2 fixed values of the bond angle θ:

The Potential Energy Surface (PES) provides information on the barrier height to reaction. Vibrational force constants and frequencies can also be determined.

Reaction Coordinate and Reaction Barrier

Reaction Coordinate – minimum PE pathway from reactants to products.

Reaction Barrier, V‡ - maximum PE along reaction coordinate. This is often the saddle point on the PES, and can be thought of as a bottleneck through which reactants must pass if they are to proceed to form products.

Molecular Collisions

Classical motion over the PES:

Solve classical laws of motion for a set of initial conditions yields classical trajectory. Many needed to obtain accurate k(T). The trajectory can be inelastic (E transfer) or reactive.

Types of PES

Direct reactions with barriers, e.g. F + H2 → HF + H.

Reactions with deep PE wells –

Reaction Probability, P(b,γ)

Not all collisions lead to reaction. Depends on angle of approach, γ, and impact parameter, b.

Reaction Cross-Sections

Integral of reaction probability over impact parameter –

σR = p π bmax2

P(b) is the opacity function = Nreactive(b) / Ntotal (b), where the N(b)’s are trajectories.

Unsurprisingly, head-on collisions tend to be more reactive than glancing-blows.

Cross-section dependence on Collision Energy

For a reaction with a barrier, collisions at high εt are more reactive than those at low εt.

Rate constant should therefore be obtained from reaction cross-sections by evaluating the Maxwell-Boltzmann average of crelσR(εo):

Reaction cross-sections can depend on reactant’s vibrational and rotational state. Reaction barrier is a bottleneck so motion of reactants along specific coordinates is required for reaction.

State-Specific Rate Constants

A + BC(i) → AB + C

Transition State Theory

- Quantum Mechanical calculations only possible for N ≤ 4.
- N ≥ 4 classical trajectory calculations are time consuming.
- Both require full PES.
- SCT doesn’t work very well.

Key advantage with TST is only need PES at transition state.

A + BC → ABC‡ → AB + C

Motion orthogonal to the minimum energy path (MEP) corresponds to conventional (symmetric) stretching and bending vibrations of ABC.

Motion along MEP corresponds to translational motion over barrier.

Reaction Rate Constants and Probability

The rate of reactant approach (the number per unit length per second) at relative velocities between c → c + dc.

dRate = nAnBC c f(c) dc

f(c) is the one dimensional Maxwell-Boltzmann distribution of relative velocities.

Reaction Rate is the approach rate times the probability of reaction at a given collision energy, P(εt).

dReaction Rate = nAnBCP(εt)c f(c) dc

Hence, thermally average reaction rate is:

In terms of kinetic energies:

, εt = ½ μ c2

qt is a one dimensional translational partition function (per unit length).

and represents the number of thermally accessible translational states per unit length at temperature T. Substitution into the expression for k(T):

In many dimensions, the above equation applies to each quantum state, weighted by the Boltzmann population for the molecules in these states:

where gn is the degeneracy of state n and:

qA:int = qA:el and qBC:int = qBC:rotqBC:vibqBC:el

with

(now in 3 dimensions)

More compactly written as:

where with ε = εt + εn. N(ε) is called the Cumulative Reaction Probability, and q is the total reactant partition function.

Equation is exact, providing reactants are in thermal equilibrium.

Transition State

TST approximates N(ε).

Assumptions:

- Reactants are in thermal equilibrium.
- Motion along MEP is separable from that orthogonal to it.
- Motion along reaction coordinate is classical.
- Motion along MEP is direct (no re-crossing rule).

From Assumption 2, ε = εt‡ + εn‡ + εo, where εo is the different in zero point energies between reactant and Transition State.

Assumptions 3 and 4 imply:

Pn(εt‡) = 1 if εt‡ > 0.

This gives:

where,

More simply,

Units = m3 molec-1 s-1.

Checking Validity wrt SCT:

A + B → AB‡ → products.

Applications of Transition State Theory

Estimating the Steric Factor –

A + B → AB

AB + CD → ABCD‡ → products

Proceeding via a non-linear Transition State:

Steric Factor,

P = kTST(T)/kCST(T) ~ (qvib/qrot)3

Hence, P ~ 10-2 – 10-4.

Generally, P tends to fall with increasing reactant complexity.

Rate Constant Calculation –

H + D2 → HD +D

Transition State is linear. At 300K,

εo = V‡ + εzHDD - εzDD

For the transition state,

For the reactants, I, B, qrot and qvib can be calculated easily using formulae.

For this reaction, subbing values in:

k(T) = 8.63 x 1018 cm3 molec-1 s-1.

Experimentally, obtain ~ 21 x 10-18.

Reason for difference is neglect of tunnelling in TST.

Tunnelling Corrections

Motion over barrier not always classical.

Transmission probability for a square potential energy barrier:

Tunnelling is most important for light particles passing through thin barriers, and most apparent at low temperature.

Failings of Transition State Theory

TST assumes classical motion along MEP is direct, i.e. no re-crossing. The importance of this depends on the shapes of the PES, the total energy and the masses of the species involved.

Re-crossing is a minor problem at room temperature for reactions with substantial barriers.

When it is important, TST yields an upper-bound to the classical rate constant.

Temperature Dependence

A + BC reaction via linear Transition State. Assume qvib → 1, and extract T-dependence of qrot and qvib:

In general, T-dependence of bimolecular rate constant:

From definition of Activation Energy:

i.e. Ea = Eo + nRT

Often Eo >> RT, and T-dependence is dominated by Eo.

Non-Arrhenius Behaviour

Tunnelling can lead to curvature in Arrhenius plot:

The T-dependence of the partition functions appearing in the A-factor can also cause curvature. Important when Eo ≤ nRT.

Radical Recombination reactions also Eo ~ 0.

Also arises if the rate constant displays a marked dependence on reactant quantum state:

OH + H2(v = 0,1) → H2O + H

Vibrational excitation of H2 to v=1 increases k by factor of over 100 (k1(T) >> ko(T)).

K(T) = Poko(T) + P1k1(T)

With

Pv =

Reaction barrier occurs late along reaction coordinate once the H-H bond is stretched (i.e. the PES has a “late barrier”).

Curvature for reaction is reasonably well accounted for by TST once tunnelling correction is made.

Kinetic Isotope Effects, KIEs

TST provides a reliable estimate of the kinetic isotope effect:

The last term often dominates.

Upper bound to the KIE may be obtained assuming that ZPEs of Transition State species are insensitive to isotopic substitution:

(Ea,H – Ea,D) ≈ Eo,H – Eo,D ≈ Ez,D – Ez,H

The difference in Activation Energies is then approximately the difference in the reactant zero point energies, Ez.

Thermodynamic Formulation of TST

Useful as:

- TST structure and vibrational frequencies may not have been calculated.
- For reactions in solution the partition functions are not simply defined (e.g. role of solvent).

Also, may be a simpler expression possible.

TST uses partition functions per unit volume.

, and Q = qN/N!

Hence,

Also, H = U + pV = U + NkT, and H(0) = U(0).

Use k(T) expression for A + BC reaction and multiply to NA *units change). Then,

NAεo = Δ‡Uo(0) and the above gives:

Δ‡Ho and Δ‡So are the enthalpy and entropy of activation.

Comparison with Arrhenius

RT2 = Ea

Entropy of Activation

Δ‡So = S‡o – SAo - SBCo

Represent change in entropy in going from reactants to Transition State.

For gas phase bimolecular reactions, Δ‡So < 0, reflecting the loss of translational entropy on forming single species from 2.

Applications – Thermochemical Kinetics

Unimolecular Reaction Dynamics

Relevance –

- Thermal Unimolecular Reactions.
- Radical Recombination.
- Chemical Activation.
- Photodissociation.

Thermal Unimolecular Reactions

Lindemann Mechanism:

Steady State:

Unimolecular Rate Constant:

Fall-off Curve – variation of kuni with pressure:

Limitations:

[M]1/2 is the concentration (proportional to pressure) at which kuni = k∞/2

[M]1/2calc = k∞/k1

Fix k∞ to experimental value.

SCT to calculate k1:

εo is the energy threshold to reaction (critical energy).

Unimolecular Reaction Mechanisms

Different MEPs:

Energy Dependent Lindemann Scheme –

Steady State –

Contribution to kuni at energy ε is n(ε)k2(ε).

n(ε)dε is the fraction of A molecules in the energy range ε → ε + dε

Deactivation – The Strong Collision Assumption

Deactivation tends to be efficient and vary only mildly with energy.

Strong Collisions:

Weak Collisions:

Problems to Address

High pressure limit there is a rapid pre-equilibrium.

A + M ⇌ Ao(ε) + M

g(ε) is the degeneracy (density of states).

As the pressure drops, equilibrium between A* and A is perturbed by the reaction.

The most highly excited A* molecules are preferentially depleted by reaction (bigger k2).

If n(ε) changes with p, kuni is likely to as well.

Energy Dependent Reaction Rates

Radical Recombination:

A + B → AB* ka [ recombination ]

AB* → A + B k-a [ dissociation ]

AB* + M → AB + M kb [ stabilisation ]

SSA: AB* →

- Low Pressure:
- High Pressure: krec = ka

Formal reverse of a bond fission. Kc = krec/kuni

Chemical Activation:

A + B → AB* ka

AB* → A + B k-a

AB* + M → AB + M kb

AB* → C kc

Reaction Rate:

Stabilisation Rate:

Lifetime of AB* =

Reaction Rate Constant, kl(ε)

RRK Theory

For unimolecular reactions.

Distinguish between energised molecule A*(ε) and activated molecule Am‡(ε).

Assumptions:

- Ignore A* rotation.
- Molecule A has s identical harmonic oscillators with energy ε.
- A‡(ε) also has s identical oscillators, but energy εo is localised in one vibrational mode, the reaction coordinate.
- A‡ falls apart with rate constant k‡, independent of energy.
- Vibrational Energy flows freely in energised and activated species.

Reaction Rate Constant:

RRK Theory is a statistical theory of unimolecular reactions.

If energy flow is rapid and random (free), the vibrational energy in A* will be distributed statistically.

The fraction of A* molecules with energy εo localised in one bond (reaction coordinate) is:

r‡(ε) = [(ε-εo)/ε]s-1

Thus,

k2(ε) = k‡ [(ε-εo)/ε]s-1 ε ≥ εo

Interpretation –

As ε increases, probability of localising εo in reaction coordinate also increases. Thus, k2(ε) increases.

Increasing s reduces probability that one specific oscillator (reaction coordinate) receives enough energy, εo, for reaction to occur, therefore k2(ε) falls.

How Does an Energy Molecule Behave?

Molecules close to dissociation can be observed spectroscopically.

- Direction Absorption Spectroscopy (vibrational overtone).
- Emission Spectroscopy.

IVR and Free Energy Flow

When vibrational energy is low it stays localised in the (nearly harmonic) normal modes.

As vibrational energy increases, motion becomes more chaotic (delocalises). Induced by anharmonicity of the PES.

Assumption of rapid IVR breaks down for small molecules with small reaction barriers, εo.

Does RRK Theory Work?

- k‡ and the number of oscillators s need to be adjusted to fit experiment.
- s typically takes a value s ~ ½ (3N-6).
- Assumption that all modes are harmonic and have same frequency does not work.
- Rotation can play a role.

Some can be addressed in a more sophisticated statistical theory – RRKM Theory.

Degeneracy in n(ε)

g(ε) = vibrational density of states in

How does degeneracy arise?

3 quanta to distribute in 3 identical harmonic oscillators.

In general,

RRK Expression for k1(ε)

With strong collision approximation, k-1 is the collision number, ZAMo.

εmax = (s-1)kBT

Hinshelwood Expression –

Steric Factor,

RRK Expression for k∞

In high pressure limit kni reduces approximately to:

k‡ is the limiting high pressure A factor, A∞.

TST For High Pressure Limit

A ⇌ A* → A‡ → products.

Because A* is in thermal equilibrium with A at high pressure we can use TST.

In thermodynamic form:

Conclusions

Assumptions that all modes are harmonic and have same frequency render the RRK model of little predictive use.

Statistical idea introduced in RRK Theory (i.e. energy flows freely and rapidly around A*) is an important one, and is still used in modern treatments.

Statistical theories don’t work well when reaction rate becomes competitive with rate of free energy flow.

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