Download 1 Chemical kinetics

Advanced kinetics
Solution 1
February 24, 2017
1 Chemical kinetics
1.1 For homogeneous
reactions, the rate equation corresponding to the stoichiometric equaP
νi Bi is:
tion 0 =
i
vc =
1 dci
νi dt
(1)
where ci is the concentration of the species Bi .
For heterogeneous reactions, the rate of conversion must be used:
vξ =
1 dni
νi dt
(2)
where ni is the amout of the species Bi .
1.2 In some cases, the rate equation can written in the following way:
Y m
vc = k
ci i
(3)
i
In this case, mi is the
Porder of the reaction with respect to species Bi . The total order of
mi and k is the rate constant. Not all reactions have an order. For
the reaction is m =
i
elementary reactions, the order is directly associated to the molecularity which indicates
the number of reactive particles involved in the reaction. For most reactions, when an
order can be defined, it is an empirical quantity.
1.3 In the transition-state theory, reactant species have to pass a maximum energy point on
the associated hypersurface to reach the product side. They stay in quasi-equilibrium
on the reactant side until this maximum point is reached. The first Eyring equation
provides an expression for unimolecular rate constants:
kB T q 6=
E0
kuni (T ) =
exp −
(4)
h qA
kB T
qA stands for the partition function of the reactant A. q 6= is the restricted partition
function for a fixed reaction coordinate refering to the top of the energy barrier and E0 is
the activation energy. The second Eyring equation provides an expression for bimolecular
rate constants:
kB T qe6=
E0
kbi (T ) =
exp −
(5)
h qeA · qeB
kB T
1.4 A bimolecular reaction can be considered as a reactive collision with a reaction cross
section 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 that
the rate constant k results as a thermal average over the Maxwell-Boltzmann velocity
distribution:
k(T ) = hvrel ihσi
(6)
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Advanced kinetics
Solution 1
where hvrel i is the thermal average center-of-mass velocity:
s
8kB T
hvrel i =
πµ
February 24, 2017
(7)
−1
with the reduced mass µ is defined as µ−1 = m−1
A + mB .
There are several simplified models for the reaction cross section σ (hard sphere, constant cross section with a threshold, with a hyperbolic threshold...) that try to represent
the energy dependence of the effective reaction cross section. Some are schematically
represented 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).
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68
Advanced kinetics
CHAPTER 3. STRUCTURE
AND SPECTRA
OF DIATOMIC MOLECULES
Solution
1
February 24,
2017
• keeping higher terms in Equation (3.27). One then can account for the lengthening of
2 Quantumthemechanics
average internuclear distance caused by the anharmonic vibrational motion
1
· , the Born-Oppenheimer
(3.33) approxBvin
=B
αe (v + ) + · ·of
e − framework
2.1 The energy of a diatomic molecule
the
2
imation and the harmonic oscillator-rigid rotor approximation is a sum of an electronic
• taking into (E
account
centrifugal distortion (which corresponds to an elongation of the
(Un ), a vibrational
vib ) and a rotational (Erot ) energy:
bond as the rotational motion gets faster, i. e. at increasing J values)
En,v,J = Un (Re ) + Evib (v) + Erot (J) = Un (Re ) + hνosc. (v + 1/2) + hcB̃J(J + 1)
(v)
Erot (J) = Bv J(J + 1) − Dv J 2 (J + 1)2 + · · · .
(8)
(3.34)
Re is the equilibrium distance between the two atoms of the diatomic molecule. The
and thus,
the rotational
depends
on the quantum
number v.
corresponding
energy
diagramenergy
is shown
in Figure
4.
Figure 3.2: Schematic of the rovibronic energy levels of a diatomic molecule.
Figure 2: Energy diagram for a diatomic molecule in the framework of the Born-Oppenheimer
approximation
oscillator-rigid
rotor
The constants ωeand
, ωe xthe
ye , Be , αe , etc.
are tabulated for
manyapproximation.
electronic states ofUmany
0 (Re ) stands
e , ωe harmonic
fordiatomic
the energy
of the
electronic
ground1979,
state.
molecules
(see Huber
and Herzberg,
in the literature list) and can be used to
calculate the rovibronic energies of a diatomic molecule. Nowadays efficient ways (and good
programs) are available to solve Equation (3.25) numerically.
2.2 The electromagnetic
field interacts with molecules in four different regions:
The harmonic oscillator (with its potential V (Q) = 1 kQ2 ) represents a good approximation
2
• Radio
waves (frequency:
up to 900
wavelength
≈ meter),
with nuclear
to the vibrational
motion of a molecule
onlyMHz,
in the vicinity
of Re . The
solution isinterct
(see Lecture
magnetic dipoles (NMR, Nuclear Magnetic Resonances). Radio waves also affect
Physical Chemistry III)
electrons in bulk system (as metals), !
creating
electric currents that are hte physical
"
(harm.)
E
(v)
1
phenomena leading to vib
well known
as radio transmissions by
antennas.
= tecnologies
ωe v +
(3.35)
hc
2
• Microwaves (frequency: GHz, wavelength
usually interacts with ro1 ≈ centimeter),
− 12 Q2
#√
Hvbe
(Q)e
,to perturb electrons
(3.36)in metals
Ψv (Q) =but
tational levels of small molecules
can
also
used
π v! 2v
bulk system (as Wi-Fi connections usually do).
PCV - Spectroscopy of atoms and molecules
• InfraRed waves (frequency: THz, wavelength ≈ micrometer), can exchange energy
with the vibrational levels of molecules, givin informations about the kind of atoms
and bounds present. Due to quantum selection rules, only some modes can interact
direcltly with the EMF; still the others can be investigated using the Raman effect.
• Visible/UltraViolet light (frequency: PHz, wavelength ≈ 100 nanometer) promote
elctronic excitations to high energy electronic state (also called PES). Due to symmetry conditions, this electronic states can be not bounded and rapidly induce
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Advanced kinetics
Solution 1
February 24, 2017
the dissociation of molecules. Due to high quantity of energy introduced into the
molecule by a single photon, a large classes of effects (fluorescence, phosphorecence
. . . ) can be observed depending on the topology of the PES.
2.3 The solution of the Schrödinger equation for the hydrogen atom can be expressed as:
• Eigenvalues (i. e. the energies):
En,l,m = −
Z 2 µe4
1
hcR
= − 2 = En
2(4πε0 )2 ~2 n2
n
(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 an
angular function (spherical harmonics) Yl,m (θ, ϕ):
Ψn,l,m (r, θ, ϕ) = Rn,l (r) Yl,m (θ, ϕ)
(10)
2.4 Particle in the box:
Ĥ = −
~ d2
+ V (x)
2m dx2
(11)
with (see figure 4)
(
0,
V (x) =
∞,
if 0 ≤ x ≤ L
otherwise
(12)
The solutions of the time-independent Schrödinger equation are:
n 2 h2
8mL2
r
nπx 2
=
sin
L
L
En =
(13)
Ψn
(14)
2.5 Operators are well defined only when applied to a function, i.e. a vector in a Hilbert
space H; moreover, they do commute only if they have a multiplicative effect on the
vector. For a generic function f in coordinate space, we have:
1
∂
∂
1 2
∂
∂
∂2
x̂ +
x̂ −
f =
x̂ − x̂
+
x̂ − 2 f =
2
∂ x̂
∂ x̂
2
∂ x̂ ∂ x̂
∂ x̂
2
1 2
∂f
∂[x̂f ] ∂ f
1 2
∂f
∂f
∂ x̂ ∂ 2 f
x̂ f − x̂
+
− 2 =
x̂ f − x̂
+x
+f
− 2 =
2
∂ x̂
∂ x̂
∂ x̂
2
∂ x̂
∂ x̂
∂ x̂ ∂ x̂
2
2
1 2
∂ f
1 2
∂
x̂ f − 2 + f =
x̂ − 2 + 1 f ,
∀f ∈ H
2
∂ x̂
2
∂ x̂
4
Advanced kinetics
Solution 1
February 24, 2017
Figure 3: Representation of the potential describing a particle in a box of length L.
If we multiply the space derivative by the immaginary unit i, we found the definition of
ladder operators for an harmonic oscillator.
Using the definition of momentum p̂ = −i~∂/∂x we use the same approach above to
obtain
[x̂, p̂]f = x̂p̂f − p̂x̂f = −i~x̂
∂f
∂ x̂f
∂f
∂ x̂
∂f
+ i~
= − i~x̂
+ i~f
+ i~x̂
= i~f
∂x
∂x
∂x
∂x
∂x
Using the commutator property [ÂB̂, Ĉ] = (Â, [B̂, Ĉ]] + [Â, Ĉ]B̂), we find that:
1 2
[p̂ , x̂]f
2m
i~
1
p̂[p̂, x̂] + [p̂, x̂]p̂ f =
(p̂ + p̂)f
=
2m
2m
i~p̂
=
f
m
[Ĥ, x̂]f = [T̂ , x̂]f + [V̂ , x̂]f =
[Ĥ, p̂]f = [T̂ , p̂]f + [V̂ , p̂]f = [V̂ , p̂]f
= −i~ V̂
= i~
∂ V̂
f
∂ x̂
∂f
∂[V̂ f ] ∂f
∂f
∂ V̂ −
= −i~ V̂
− V̂
−f
∂ x̂
∂ x̂
∂ x̂
∂ x̂
∂ x̂
1
1
[V̂ , p̂2 ]f =
[V̂ , p̂]p̂ + p̂[V̂ , p̂] f
2m
2m
i~ ∂ V̂
∂ V̂ i~ ∂ V̂
∂[∂ V̂ /∂ x̂f ] p̂ + p̂
f =
p̂ − i~
=
2m ∂ x̂
∂ x̂
2m ∂ x̂
∂ x̂
2
i~ ∂ V̂
∂ V̂
∂ V̂ =
p̂ − i~ 2 +
p̂ f =
2m ∂ x̂
∂ x̂
∂ x̂
i~ ∂ V̂
∂ 2 V̂ =
2
p̂ − i~ 2 f
2m ∂ x̂
∂ x̂
[V̂ , T̂ ]f =
5
(15)
Advanced kinetics
Solution 1
February 24, 2017
3 Tunneling process
3.1 Scheme of the potential energy hypersurface
Figure 4: Vibrational term value diagram of the torsional band of phenol, including the electronic Born-Oppenheimer potential (dashed) and the lowest adiabatic channel potential (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 Schrödinger equation are depicted
in Figure 4.
3.3 The general expression for the solution of the time-dependent Schrödinger equation considering the two lowest energy levels ϕ1 and ϕ2 of energies E1 and E2 respectively is:
1
iE1 t
i∆E12 t
Ψ (t) = √ exp −
ϕ1 + ϕ2 exp −
(16)
~
~
2
with ∆E12 = E1 − E2 .
Assuming that ϕ1 and ϕ2 are real functions, the probability density is given by:
1 2
∆E12 t
∗
2
P (t) = Ψ (t)Ψ (t) =
ϕ + ϕ2 + 2ϕ1 ϕ2 cos
2 1
~
(17)
which is a periodic function with the period T = 2π~/∆E12 .
3.4 For the vibrational ground state: TGS = (c ∆ν̃GS )−1 = 17.5 ns
For the first torsional excited state: Ttors. = (c ∆ν̃tors. )−1 = 370 ps
3.5 In classically allowed regions E − V > 0, whereas in forbidden regions E − V < 0. The
conjunction points between them, called turning points, necessary exist when E − V = 0
and are crucial elements in semiclassiclal methods. If we take the 1D Schrödinger equation
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Advanced kinetics
Solution 1
~2 ∂ 2
ψ + V ψ = Eψ
2m ∂x2
∂2
2m
ψ = 2 (V − E)ψ
2
∂x √ ~
February 24, 2017
−
ψ ≈ ei
(18)
2m(E−V )/~2 x
Therefore, when E −V > 0, the square root in the exponetial is real and the wavefunction
have a oscillotary behaviour, due to imaginary factor (cfr Euler’s formula); in this case
the probability density |ψ|2 is greater than zero. Otherwise, the wavefunction has a
negative exponential behaviour, and its probability density go to zero with it.
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