Download 1 Chemical kinetics 2 Quantum mechanics 3 Tunneling process

Advanced kinetics
Exercise 1
February 24, 2017
The goal of this exercise is to highlight topics that have been discussed in previous lectures
and that should be known for the advanced kinetics lecture.
1 Chemical kinetics
We consider the stoichiometric equation:
0=
X
νi Bi
(1)
i
where νi is the stoichiometric coefficient of the species Bi .
1.1 Write the corresponding rate equation.
1.2 Define what the order of a reaction is. Do all reactions have an order?
1.3 Describe in your own words the transition-state theory.
1.4 Describe in your own words how collision theory is used to calculate the rate constant of
a reaction k as a function of the collision cross section σ. Discuss different simple models
to calculate σ.
2 Quantum mechanics
2.1 Draw a diagram of the energy levels of a diatomic molecule in the framework of the BornOppenheimer approximation and the harmonic oscillator-rigid rotor approximation.
2.2 Which part of the electromagnetic field interacts which the energy levels of the diagram
drawn in question 2.1?
2.3 What is the solution of the time-independent Schrödinger equation of the hydrogen atom
(eigenvalues and eigenfunctions)?
2.4 What is the Hamiltonian operator of a particle in a box? Give the expressions of the
eigenvalues and eigenfunctions.
2.5 Prove that
1
∂
∂
1 2
∂2
x̂ +
x̂ −
=
x̂ − 2 + 1
2
∂ x̂
∂ x̂
2
∂ x̂
Do you recognize what this equation represents?
(2)
2.6 Calculate the commutators:
[x̂, p̂]
[Ĥ, x̂]
[Ĥ, p̂]
[V̂ , T̂ ]
(3)
3 Tunneling process
We now consider the phenol molecule (see figure 1). The energy barrier which separates the
two equivalent planar structures is about 1200 cm−1 . The first minimum of energy will be
defined at angle τ = 0 degrees. The tunnelling between the two structures by torsion of the
OH group leads to a splitting of 56 MHz in the vibrational ground state and 0.09 cm−1 in the
first torsional excited state (ν̃tors = 309.1748 cm−1 ).
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Advanced kinetics
Exercise 1
February 24, 2017
Figure 1: Representation of the phenol molecule with torsion τ angle.
3.1 Draw a scheme collecting all information given above (potential energy, barrier, energy
levels...).
3.2 Draw the representation of the first four eigenfunctions of the one dimensional timeindependent Schrödinger equation, describing the torsional motion of the molecule.
3.3 Write the general expression of the solution of the time-dependent Schrödinger equation
considering the two lowest energy levels φ1 and φ2 of energy E1 and E2 respectively and
show that the probability density varies periodically with time.
3.4 Calculate the period of the tunneling motion for the vibrational ground state and the
first torsional excited state.
3.5 Explain in your own words the concept of classically allowed and forbidden regions. What
is the wave function behaviour in these regions?
This exercise was created by Yoann Pertot and Aaron von Conta and modified by Danilo
Calderini and Manish Thapa. For questions regarding the task, please contact
[email protected].
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