Rate Constants and Kinetic Energy Releases in Unimolecular Processes,

Detailed Balance Results

Klavs HansenGöteborg University and

Chalmers University of TechnologyIgls, march 2003

Realistic theories:RRKM, treated elsewhereDetailed BalanceV. Weisskopf, Phys. Rev. 52, 295-303 (1937)

Same physicsDifferent formulae Same numbers?

Yes (if you do it right)

Physical assumptions for application of detailed balance to statistical processes

1) Time reversal,2) Statistical mixing, compound cluster/molecule: all memory of creation is forgotten at decay

General theory, requires input: Reaction cross section, Thermal properties of product and precursor

Detailed balance equation

Number of states (parent)

Evaporation rate constant

Number of states (product)

Formation rate constant

Density of state of parent, product

Detailed balance (continued)

(single atom evaporation)

Important point: Sustains thermal equilibrium,Extra benefit: Works for all types of emitted particles.

D = dissociation energy = energy needed to remove fragment,OBS, does not include reverse activation barrier. Can be incorporated (see remark on cross section later, read Weisskopf)

Ingredients1) Cross section2) Level densities of parent3) Level density of product cluster4) Level density of evaporated atom

ObservableObservableObservableKnown

Angular momentum not considered here.

Microcanonical temperatureTotal rates require integration over kinetic energy releases

Define

OBS: Tm is daughter temperature

Total rate constants, example

Geometrical cross section:

Numerical examples

Evaporated atom Au = geometric cross section = 10Å2

Evaporated atom C = geometric cross section = 10Å2

(Monomer evaporation)

g = 2

g = 1

Dimer evaporationReplace the free atom density of states with the dimer density of states (and cross section)

Integrations over vibrational and rotational degreesof freedom of dimer give rot and vib partition function:

Kinetic energy releaseGiven excitation energy, what is the distribution of the kinetic energies released in the decay?

Depends crucially on the capture cross section for the inverse process,

Stating the cross section in detailed balance theory is equivalent to specifying the transition state in RRKM

Measure or guess

Kinetic energy release

General (spherical symmetry):

Geometric cross section:

Langevin cross section:

Capture in Coulomb potential:

Simple examples:

Kinetic energy releaseSpecial cases: Motion in spherical symetric external potentials. Capture on contact.

0 1 2 3 4 5

geometriccross section

Langevin cross section

Coulombpotential

KE

R d

istri

butio

ns (a

rb. u

nits

)

Kinetic energy release (Tm)

If no reverse activation barrier,values between 1 and 2 kBTm:

Geometric cross section: 2 kBTm Langevin cross section: 3/2 kBTm

Capture in Coulomb potential: 1 kBTm

OBS: The finite size of the cluster will often change cross sections and introduce different dependences.

Average kinetic energy releases

No reverse activation barrier

Reverse activation barrier

Barriers and cross sections

Reaction coordinate Reaction coordinate

= 0 for < EB

EB

Level densitiesVibrational degrees of freedom dominates

Calculated as collection of harmonic oscillators.Typically quantum energy << evaporative activation energy

At high E/N:

(E0 = sum of zero point energies)

More precise use Beyer-Swinehart algorithm,but frequencies normally unknown

Warning: clusters may not consist of harmonic oscillators

Level densities

Examples of bulk heat capacities:

0 50 100 150 200 250 3000

1

2

3

4

5

Cp /

R

T (K)

Level DensitiesHeat capacity of bulk water

What did we forget?

Oh yes, the electronic degrees of freedom.

Not as important as the vibrational d.o.f.s but occasionally still relevant for precise numbers or special cases (electronic shells, supershells)

Easily included by convolution with vib. d.o.f.s (if levels known), or with microcanonical temperature