C7790 Introduction to Molecular Modelling -1C7790
Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
PS/2020 Distant Form of Teaching: Rev2
Lesson 19
Kinetics
C7790 Introduction to Molecular Modelling -2-
Context
C7790 Introduction to Molecular Modelling -3Revision:
Entropy and spontaneity
𝑑𝑆 ≥ 0
irreversible process (spontaneous)
For isolated system, the direction of the time flow
is identical with the increase of entropy.
In isolated system, entropy increases until
equilibrium is reached. Then, entropy reaches a
maximum, constant value.
int
ext
intS
Knowledge of the entropy change of the internal
system (int, system of interest) is not sufficient to
assess whether the change will take place
spontaneously. It is necessary to assess the change of
entropy of system, including its surroundings.
0int + SSext
Spontaneous process:
C7790 Introduction to Molecular Modelling -4Revision:
Free energy and process spontaneity
0−= STHG
0=−= STHG
0−= STHG
spontaneous process
non-spontaneous process
system is at equilibrium
The change of Gibbs energy indicates whether the process can occur spontaneously.
However, it does not determine how long the actual transformation will take place.
for conversion at constant temperature and pressure
entropy of the system
entropy of surroundings
C7790 Introduction to Molecular Modelling -5Thermodynamics
of chemical process
change of Gibbs
(free) energy
initial state (reactants)
final state (products)
activated complex (transition state)
aA + bB cC + dD
TS
R
P
states (reaction coordinate)
a, b, c, d - stoichiometric coefficients
C7790 Introduction to Molecular Modelling -6Thermodynamics
of chemical process
aA + bB cC + dD
TS
R
P
states (reaction coordinate)
Δ𝐺𝑟
0
standard reaction
change of Gibbs energy
Δ𝐺1
≠ Δ𝐺2
≠
standard activation
change of Gibbs energy
1
2
1 – forward reaction
2 – reverse reaction
𝜟𝑮 𝒓
𝟎
, 𝜟𝑮 𝟏
≠
, 𝜟𝑮 𝟐
≠
do not tell nothing about spontaneity of the reaction!!!!
C7790 Introduction to Molecular Modelling -7Thermodynamics
of chemical process
aA + bB cC + dD
TS
R
P
states (reaction coordinate)
0
rG
Δ𝐺1
≠ Δ𝐺2
≠
Δ𝐺𝑟
0
= Δ𝐺1
≠
− Δ𝐺2
≠
1
2
Thermodynamic cycle
Δ𝐺1
≠
− Δ𝐺2
≠
− Δ𝐺𝑟
0 = 0
C7790 Introduction to Molecular Modelling -8➢
Evolution of chemical system in time
(until equilibrium)
Kinetics
i.e., what you should already know ….
C7790 Introduction to Molecular Modelling -9-
A
B
Kinetics - summary
A B
k1

 1G
rate constant
][
][
1 Ak
dt
Ad
=−
rate of reaction,
change of substance concentration over time
TS
states (reaction coordinate)
actual concentration
of the substance
C7790 Introduction to Molecular Modelling -10-
A
B
Kinetics - summary
A B
R - universal gas constant, T - absolute temperature, h - Planck's constant, kB - Boltzmann constant
k1

 1G
][
][
1 Ak
dt
Ad
=−
Arrhenius equation (empirical)
activation energy
𝑘 = 𝐴𝑒−
𝐸 𝑎
𝑅𝑇
pre-exponential
factor
TS
states (reaction coordinate)
actual concentration
of the substance
rate constant
rate of reaction,
change of substance concentration over time
C7790 Introduction to Molecular Modelling -11-
A
B
Kinetics - summary
A B
R - universal gas constant, T - absolute temperature, h - Planck's constant, kB - Boltzmann constant
k1

 1G
][
][
1 Ak
dt
Ad
=−
RT
G
B
e
h
Tk
k


−
=
Eyring equation (theoretical model)
standard activation Gibbs energy
TS
states (reaction coordinate)
actual concentration
of the substance
rate constant
rate of reaction,
change of substance concentration over time
C7790 Introduction to Molecular Modelling -12Chemical
transformation
aA + bB cC+ dD
Substances C and D are formed by reaction of A and B.
Principle questions:
➢How fast is the reaction?
➢How is it possible to influence the rate of reaction?
C7790 Introduction to Molecular Modelling -13Reaction
rate
aA + bB cC+ dD
𝑣 =
1
𝑎
𝑑[𝐴]
𝑑𝑡
=
1
𝑏
𝑑[𝐵]
𝑑𝑡
= 𝑘 𝐴 𝛼[𝐵] 𝛽
Sign convention for ni
final state - positive value
initial state - negative value
Irreversible reaction:
Rate ​​of reaction (empirical relationship):
k
change in concentration over time
partial order of reaction
rate constant
actual concentration
𝑛 = 𝛼 + 𝛽total order of reactionpartial and total orders can be real numbers
C7790 Introduction to Molecular Modelling -14Arrhenius
equation
𝑘 = 𝐴𝑒−
𝐸 𝑎
𝑅𝑇
Arrhenius equation defines empirical relationship between the rate constant and
temperature:
activation energy
pre-exponential
factor
The pre-exponential factor and activation energy are constants characteristic for given
reaction. However, the relationship is valid only in a narrow temperature range.
Exercises:
➢ How does the rate constant change with
increasing temperature?
➢ How is the activation energy determined?
C7790 Introduction to Molecular Modelling -15Reversible
reaction
aA + bB cC+ dD
Reversible reaction: k1
k2
𝑣1 =
1
𝑎
𝑑[𝐴]
𝑑𝑡
=
1
𝑏
𝑑[𝐵]
𝑑𝑡
= 𝑘1 𝐴 𝛼
[𝐵] 𝛽
𝑣2 =
1
𝑐
𝑑[𝐶]
𝑑𝑡
=
1
𝑑
𝑑[𝐷]
𝑑𝑡
= 𝑘2 𝐶 𝛾
[𝐷] 𝛿
forward reaction
reverse reaction
At equilibrium:
𝑣1 = 𝑣2
C7790 Introduction to Molecular Modelling -16Reversible
elementary reaction
aA + bB cC+ dD
Reversible reaction: k1
𝑣1=𝑘1 𝐴 𝑎
[𝐵] 𝑏
= 𝑣2= 𝑘2 𝐶 𝑐
[𝐷] 𝑑
At equilibrium:
𝑘1
𝑘2
=
𝐶 𝑐[𝐷] 𝑑
𝐴 𝑎[𝐵] 𝑏
= 𝐾
k2
Δ𝐺𝑟
0 = Δ𝐺1
≠
− Δ𝐺2
≠
Exercises:
➢ Using the equations for the equilibrium
constant and rate constant (Eyring equation)
prove the equivalence between the provided
relations.
➢ Under what conditions does equivalence
apply?
➢ What does the comparison show?
always valid, G - is a state function
it is valid only in a limited extent because concentrations are
not activities
C7790 Introduction to Molecular Modelling -17Elementary
reaction
R
P
Elementary reaction is the transformation of reactants and products separated by just
one transition state.
TS
states (reaction coordinate)
➢ partial orders of the reaction are
stoichiometric coefficients
➢ overall order of the reaction determines
molecularity of the process
➢ molecularity of elementary reaction is
typically 1 (monomolecular) or 2
(bimolecular)
C7790 Introduction to Molecular Modelling -18For
consideration
➢ What is characteristic of an irreversible process?
➢ What is a kinetically controlled process?
➢ What is a thermodynamically controlled process?
R
P1
TS2
states (reaction coordinate)
TS1
P2
R
P1
TS2
states (reaction coordinate)
TS1
P2
C7790 Introduction to Molecular Modelling -19Theory
of activated complex
The theory of the activated complex (also transition state theory - TST) describes the
kinetics of the elementary reaction:
A
B
TS
states (reaction coordinate)
pseudo-
equillibrium
TS decay
aA + bB cC+ dD
𝐾≠
𝑘≠
TS
Assumptions:
a) activated complex is in pseudoequilibrium
with the initial state
b) activated complex decomposes into
products and reactants
c) apparatus of statistical thermodynamics
is used for derivation
𝑘
C7790 Introduction to Molecular Modelling -20Theory
of activated complex
aA + bB cC+ dD
𝐾≠
𝑘≠
TS
𝑘
ad a)
𝐾≠
=
[𝑇𝑆]
𝐴 𝑎[𝐵] 𝑏
[𝑇𝑆] = 𝐾≠
𝐴 𝑎
[𝐵] 𝑏
ad b)
𝑣 =
1
𝑎
𝑑[𝐴]
𝑑𝑡
=
1
𝑏
𝑑[𝐵]
𝑑𝑡
= 𝑘≠[𝑇𝑆]
𝑣 =
1
𝑎
𝑑[𝐴]
𝑑𝑡
=
1
𝑏
𝑑[𝐵]
𝑑𝑡
= 𝑘≠ 𝐾≠ 𝐴 𝑎[𝐵] 𝑏= 𝑘 𝐴 𝑎[𝐵] 𝑏
The resulting relationship
𝑘 = 𝑘≠ 𝐾≠
C7790 Introduction to Molecular Modelling -21Theory
of activated complex
➢ Activation Gibbs energy is always a positive number, yet TS is
present in the reaction mixture, see equilibrium for G0
r > 0.
➢ Activation Gibbs energy corresponds to the change in which the
reactants are quantitatively converted to the transition state. This
is a hypothetical process that does not actually occur.
aA + bB
𝐾≠
TS
Δ𝐺≠ = −𝑅𝑇𝑙𝑛𝐾≠
𝐾≠ = 𝑒−
Δ𝐺≠
𝑅𝑇
C7790 Introduction to Molecular Modelling -22Theory
of activated complex, cont.
cC+ dD
𝑘≠
TS
𝑘≠ = 𝜐
rate constant is proportional to the
frequency of vibration at which the TS
decays into products or reactants
(imaginary number = not observable)
original approach (via translational
partition function)
modern approach (via vibrational partition
function)
𝑘≠ =
1
𝜏
𝜏 =
𝛿
ത𝑢
average TS lifetime
width of activated state
(not well defined)
average velocity along
reaction path
Both approches lead to the same result (not well-defined properties fortunately cancel out).
𝑘≠ =
𝑘 𝐵 𝑇
ℎ
it depends only on temperature but not
on molecular structure of TS
C7790 Introduction to Molecular Modelling -23Eyring
equation
aA + bB cC+ dD
𝐾≠
𝑘≠
TS
𝑘
RT
G
B
e
h
Tk
k


−
=
𝑘 = 𝑘≠ 𝐾≠
R - universal gas constant, T - absolute temperature, h - Planck's constant, kB - Boltzmann constant
Eyring equation
transmission coefficient, correction term
C7790 Introduction to Molecular Modelling -24the
activation free energy can be
obtained by modelling
Summary
➢ Description of the equilibrium and kinetics of chemical processes is important in several
applications (Which?).
➢ Equilibrium and kinetics can be quantified using one thermodynamic quantity, namely
changes in the free energy, which can be determined either experimentally or
calculated using computational chemistry methods.
𝑘 = 𝜅
𝑘 𝐵 𝑇
ℎ
𝑒−
Δ𝐺≠
𝑅𝑇
Eyring equation
TST is based on several oversimplification.
Thus, TST can fail in several cases: (they can be sometimes corrected by 𝜅):
➢ fate of products is not considered in TST (backward/inverse reaction is not
considered)
➢ tunneling (light atoms and low barriers), typically proton transfers
➢ electronic state change (change from one to another potential energy surface)
➢ others …
transmission coefficient,
correction term
C7790 Introduction to Molecular Modelling -25Recommended
readings
➢ Helrich, C. S. Modern Thermodynamics with Statistical Mechanics; Springer: Berlin,
2009.
➢ Dill, K. A.; Bromberg, S. Molecular Driving Forces: Statistical Thermodynamics in Biology,
Chemistry, Physics, and Nanoscience, 2nd ed.; Garland Science: London ; New York,
2011.
C7790 Introduction to Molecular Modelling -26-
Homework
C7790 Introduction to Molecular Modelling -27Homework
exercise I
1. Determine how many times the reaction below slows down if the activation
Gibbs energy increases by 0.25; 0.5; 1.0; 2.5; 5.0 and 10 kcal/mol. Consider
standard conditions. Discuss the results.
A B
use a spreadsheet to
solve the exercise