Transcript Rate constants and Kinetic Energy Releases in Unimolecular

Rate Constants and Kinetic Energy
Releases in Unimolecular Processes,
Detailed Balance Results
Klavs Hansen
Göteborg University and
Chalmers University of Technology
Igls, march 2003
Realistic theories:
RRKM, treated elsewhere
Detailed Balance
V. Weisskopf, Phys. Rev. 52, 295-303 (1937)
Same physics
Different 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
Evaporation rate Number of
Formation rate
states (parent) constant
states (product) constant
Density of state of parent, product
Detailed balance (continued)
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)
(single atom evaporation)
Important point: Sustains thermal equilibrium,
Extra benefit: Works for all types of emitted particles.
Ingredients
1) Cross section
2) Level densities of parent
3) Level density of product cluster
4) Level density of evaporated atom
Observable
Observable
Observable
Known
Angular momentum not considered here.
Microcanonical temperature
Total rates require integration over kinetic energy releases
Define
OBS: Tm is daughter temperature
Total rate constants, example
Geometrical cross section:
Numerical examples
(Monomer evaporation)
Evaporated atom Au
= geometric cross section = 10Å2
g=2
Evaporated atom C
= geometric cross section = 10Å2
g=1
Dimer evaporation
Replace the free atom density of states with the dimer density
of states (and cross section)
Integrations over vibrational and rotational degrees
of freedom of dimer give rot and vib partition function:
Kinetic energy release
Given 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, s(e)
Measure or guess
Stating the cross section in detailed balance theory
is equivalent to
specifying the transition state in RRKM
Kinetic energy release
Simple examples:
General (spherical symmetry):
Geometric cross section:
Langevin cross section:
Capture in Coulomb potential:
Kinetic energy release
KER distributions (arb. units)
Special cases: Motion in spherical symetric
external potentials. Capture on contact.
Coulomb
potential
Langevin
cross section
geometric
cross section
0
1
2
3
4
Kinetic energy release (Tm)
5
Average kinetic energy releases
If no reverse activation barrier,
values between 1 and 2 kBTm:
Geometric cross section:
Langevin cross section:
Capture in Coulomb potential:
2 kBTm
3/2 kBTm
1 kBTm
OBS: The finite size of the cluster will often change
cross sections and introduce different dependences.
Barriers and cross sections
No reverse
activation barrier
Reaction coordinate
Reverse activation
barrier
EB
Reaction coordinate
= 0 for
< EB
Level densities
Vibrational 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
Level densities
Warning: clusters may not consist of harmonic oscillators
Examples of bulk heat capacities:
Level Densities
Heat capacity of bulk water
5
Cp /R
4
3
2
1
0
0
50
100
150
T (K)
200
250
300
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