Advanced kinetics Solution 1 February 26, 2016

1 Chemical kinetics

1.1 For homogeneous reactions, the rate equation corresponding to the stoichiometric equa-tion 0 =

∑iνiBi is:

vc =1

νi

dcidt

(1)

where ci is the concentration of the species Bi.For heterogeneous reactions, the rate of conversion must be used:

vξ =1

νi

dnidt

(2)

where ni is the amout of the species Bi.

1.2 In some cases, the rate equation can written in the following way:

vc = k∏

i

cmii (3)

In this case, mi is the order of the reaction with respect to species Bi. The total order ofthe reaction is m =

∑imi and k is the rate constant. Not all reactions have an order. For

elementary reactions, the order is directly associated to the molecularity which indicatesthe number of reactive particles involved in the reaction. For most reactions, when anorder can be defined, it is an empirical quantity.

1.3 In the transition-state theory, reactant species have to pass a maximum energy point onthe associated hypersurface to reach the product side. They stay in quasi-equilibriumon the reactant side until this maximum point is reached. The first Eyring equationprovides an expression for unimolecular rate constants:

kuni(T ) =kBT

h

q 6=

qAexp

(− E0

kBT

)(4)

qA stands for the partition function of the reactant A. q 6= is the restricted partitionfunction for a fixed reaction coordinate refering to the top of the energy barrier and E0 isthe activation energy. The second Eyring equation provides an expression for bimolecularrate constants:

kbi (T ) =kBT

h

q 6=

qA · qBexp

(− E0

kBT

)(5)

1.4 A bimolecular reaction can be considered as a reactive collision with a reaction crosssection σ which depends on the translational energy of the reactive species A and B.Simple collision theories neglect the internal quantum state dependence of σ so thatthe rate constant k results as a thermal average over the Maxwell-Boltzmann velocitydistribution:

k(T ) = 〈vrel〉〈σ〉 (6)

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Advanced kinetics Solution 1 February 26, 2016

where 〈vrel〉 is the thermal average center-of-mass velocity:

〈vrel〉 =

√8kBT

πµ(7)

with the reduced mass µ is defined as µ−1 = m−1A +m−1B .

There are several simplified models for the reaction cross section σ (hard sphere, con-stant cross section with a threshold, with a hyperbolic threshold...) that try to representthe energy dependence of the effective reaction cross section. Some are schematicallyrepresented in Figure 1.

Figure 1: Simple models for effective collision cross section σ: hard sphere without threshold(dotted line), hard sphere with threshold (dashed line) and hyperbolic threshold(full curve). Et is the (translational) collision energy and E0 the threshold energy.σ0 is the hard sphere collision cross section (taken and modified from D. Luckhaus,M. Quack, in Encyclopedia of Chemical Physics and Physical Chemistry Vol. 1(Fundamentals), Chapter A.3.4, pages 653–682 (IOP publishing, Bristol 2001, ed.by J. H. Moore and N. D. Spencer).

1.5 Laser photolysis is a ”pump-probe” technique. First, a strong pulse (”pump”) of nanosec-ond or shorter duration excites the sample. A short time later, a second pulse (”probe”)monitors the relaxation of the sample or the reaction processes initiated by the pumppulse. Experimentally, one can use two different lasers for the pump and the probe beamsor a single laser and a combination of mirrors that introduce a time delay between thetwo pulses as indicated in Figure 2.

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Advanced kinetics Solution 1 February 26, 2016

Laser

Spiegel 1

Spiegel 2 Spiegel 3Beobachtungsblitz

Photolyseblitz Probe

Detektor

x/2

Laser

Mirror

Sample

Detector

"pump"

"probe"

Figure 2: Scheme of the experimental setup for a laser photolysis experiment with one laser.

2 Quantum mechanics

2.1 The energy of a diatomic molecule in the framework of the Born-Oppenheimer approx-imation and the harmonic oscillator-rigid rotor approximation is a sum of an electronic(Un), a vibrational (Evib) and a rotational (Erot) energy:

En,v,J = Un(Re) + Evib(v) + Erot(J) = Un(Re) + hνosc.(v + 1/2) + hcBJ(J + 1) (8)

Re is the equilibrium distance between the two atoms of the diatomic molecule. Thecorresponding energy diagram is shown in Figure 3.

68 CHAPTER 3. STRUCTURE AND SPECTRA OF DIATOMIC MOLECULES

• keeping higher terms in Equation (3.27). One then can account for the lengthening of

the average internuclear distance caused by the anharmonic vibrational motion

Bv = Be ! !e(v +1

2) + · · · , (3.33)

• taking into account centrifugal distortion (which corresponds to an elongation of the

bond as the rotational motion gets faster, i. e. at increasing J values)

E(v)rot (J) = BvJ(J + 1) ! DvJ

2(J + 1)2 + · · · . (3.34)

and thus, the rotational energy depends on the quantum number v.

Figure 3.2: Schematic of the rovibronic energy levels of a diatomic molecule.

The constants "e, "exe, "eye, Be, !e, etc. are tabulated for many electronic states of many

diatomic molecules (see Huber and Herzberg, 1979, in the literature list) and can be used to

calculate the rovibronic energies of a diatomic molecule. Nowadays e!cient ways (and good

programs) are available to solve Equation (3.25) numerically.

The harmonic oscillator (with its potential V (Q) = 12kQ2) represents a good approximation

to the vibrational motion of a molecule only in the vicinity of Re. The solution is (see Lecture

Physical Chemistry III)

E(harm.)vib (v)

hc= "e

!v +

1

2

"(3.35)

!v(Q) =1#"# v! 2v

Hv(Q)e! 12Q2

, (3.36)

PCV - Spectroscopy of atoms and molecules

Figure 3: Energy diagram for a diatomic molecule in the framework of the Born-Oppenheimerapproximation and the harmonic oscillator-rigid rotor approximation. U0(Re) standsfor the energy of the electronic ground state.

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Advanced kinetics Solution 1 February 26, 2016

2.2 The solution of the Schrodinger equation for the hydrogen atom can be expressed as:

• Eigenvalues (i. e. the energies):

En,l,m = − Z2µe4

2(4πε0)2h2

1

n2= −hcR

n2= En (9)

where R is the Rydberg constant (R ' 109700 cm−1).

• Eigenfunctions can be written as a product of a radial function Rn,l(r) and anangular function (spherical harmonics) Yl,m(θ, ϕ):

Ψn,l,m(r, θ, ϕ) = Rn,l(r)Yl,m(θ, ϕ) (10)

2.3 Particle in the box:

H = − h

2m

d2

dx2+ V (x) (11)

with (see figure 3)

V (x) =

{0, if 0 ≤ x ≤ L∞, otherwise

(12)

The solutions of the time-independent Schrodinger equation are:

En =n2h2

8mL2(13)

Ψn =

√2

Lsin(nπxL

)(14)

Figure 4: Representation of the potential describing a particle in a box of length L.

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Advanced kinetics Solution 1 February 26, 2016

3 Tunneling process

3.1 Scheme of the potential energy hypersurface

Figure 5: Vibrational term value diagram of the torsional band of phenol, including the elec-tronic Born-Oppenheimer potential (dashed) and the lowest adiabatic channel po-tential (bold), both shifted to E = 0 at the minimum (after S. Albert, P. Lerch, R.Prentner, M. Quack, Angew. Chem. Int. Ed. 52 (2013) 346-349).

3.2 The first four eigenfunctions of the time-independent Schrodinger equation are depictedin Figure 5.

3.3 The general expression for the solution of the time-dependent Schrodinger equation con-sidering the two lowest energy levels ϕ1 and ϕ2 of energies E1 and E2 respectively is:

Ψ(t) =1√2

exp

(− iE1t

h

)[ϕ1 + ϕ2 exp

(− i∆E12t

h

)](15)

with ∆E12 = E1 − E2.Assuming that ϕ1 and ϕ2 are real functions, the probability density is given by:

P (t) = Ψ(t)Ψ∗(t) =1

2

[ϕ21 + ϕ2

2 + 2ϕ1ϕ2 cos

(∆E12t

h

)](16)

which is a periodic function with the period T = 2πh/∆E12.

3.4 For the vibrational ground state: TGS = (c∆νGS)−1 = 17.5 nsFor the first torsional excited state: Ttors. = (c∆νtors.)

−1 = 370 ps

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