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/2022 Present Form of Teaching: Rev5
Lesson 5
Phenomenological thermodynamics (equilibrium)
C7790 Introduction to Molecular Modelling -2-
Overview
C7790 Introduction to Molecular Modelling -3Chemical
process
Or what you should already know….
C7790 Introduction to Molecular Modelling -4Thermodynamics
of chemical process
Gibbs (free)
energy change
initial state (reactants)
end 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 -5Thermodynamics
of chemical process
aA + bB cC + dD
TS
R
P
states (reaction coordinate)
0
rG
standard reaction Gibbs energy
Δ𝐺1
≠ Δ𝐺2
≠
standard activation Gibbs energy
1
2
1 - forward reaction
2 - backward reaction
C7790 Introduction to Molecular Modelling -6Thermodynamics
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 -7-
Equilibrium
Or what you should already know….
C7790 Introduction to Molecular Modelling -8Equilibrium
- summary
KRTGr ln0
−=
standard reaction Gibbs energy
equilibrium constant0
rG
  
   
   
   b
r
a
r
d
r
c
r
b
r
a
r
d
r
c
r
BA
DC
BA
DC
K =
activities concentration
(r) equilibrium
R
P
aA + bB cC + dD
states (reaction coordinate)
C7790 Introduction to Molecular Modelling -9Equilibrium
- summary
   
   
   
   
 ==
i
rib
r
a
r
d
r
c
r
b
r
a
r
d
r
c
r i
X
BA
DC
BA
DC
K

The standard reaction free energy can be calculated, for example,
from standard formation (f) Gibbs energies.
aA + bB cC + dD
KRTGr ln0
−=
Δ𝐺𝑟
0 = 𝑐Δ𝐺𝑓,𝐶
0
+ 𝑑Δ𝐺𝑓,𝐷
0
− 𝑎Δ𝐺𝑓,𝐴
0
+ 𝑏Δ𝐺𝑓,𝐵
0
= ෍
𝑖
𝜐𝑖 𝐺 𝑓,𝑖
0
C7790 Introduction to Molecular Modelling -10Chemical
transformation
Reaction of A and B provides C and D and vice versa. Both processes continues until the
rate of both forward and backward reactions is the same and equilibrium is established.
Principle questions:
➢ What is the composition of the reaction mixture in equilibrium and how can it
be determined?
➢ How is it possible to influence the composition of the reaction mixture in
equilibrium?
aA + bB cC + dD
C7790 Introduction to Molecular Modelling -11Gibbs
energy of reaction mixture
= 








=
N
i
i
nnTpi
N dn
n
G
nnndG
ij
1 ,,
21 ),...,,(
ij nnTpi
i
n
G









=
,,

ni is the molar amount of substance i
The Gibbs energy of reaction mixture G is a function of the composition of the reaction
mixture.
At the constant temperature and pressure, it is possible to write the Gibbs energy as total
differential in the following form:
=
=
N
i
iiN dnnnndG
1
21 ),...,,( 
Derivative of the Gibbs energy with respect to molar amount is a very useful quantity
called chemical potential  :
N - number of reacting compounds
C7790 Introduction to Molecular Modelling -12Chemical
potential
ij nnTpi
i
n
G









=
,,

Chemical potential expresses the effort of the substance:
• to react with another substance
• to change its status
• to change its spatial distribution
Value of chemical potential:
• is related to the very nature of the substance
• is related to the environment (temperature, pressure, concentration, ...)
• however, it is not related to the nature of the substances with which it reacts or is
transformed to
iii aRT ln0
+= 
Relationship between chemical potential i and activity ai of substance:
The chemical potential is a state function:
C7790 Introduction to Molecular Modelling -13-
Activity
00
p
p
p
f
a ii
i =
00
c
c
c
c
a ii
ii = 
gas mixtures
solutions
a mixture of ideal gases
ideal solution (diluted solution)
gas mixture
solution
f - fugacity (effective pressure)
p - partial pressure
c - molar concentration
 - activity coefficient
Activity expresses the effective amount of a substance relative to a standard state.
It is a dimensionless quantity.
The reason for introducing an activity coefficient (or fugacity)
is to maintain a simple relationship between activity and
chemical potential. The relationship for chemical potential
can therefore be taken as the definition of the activity:
RT
i
ii
ea
0
 −
=
C7790 Introduction to Molecular Modelling -14Standard
chemical potential
Standard chemical potential is the change in Gibbs energy that is associated with the
formation of one mole of matter in the standard state.
It is most often expressed in the form of the standard formation (f) or combustion (c)
Gibbs energy.
𝜇𝑖
0
= Δ𝐺𝑓,𝑖
0
Standard formation Gibbs energy is the change of Gibbs energy that corresponds to the
formation of one mole of matter from chemical elements in the standard state.
Chemical elements in the standard state have zero formation Gibbs energy (this is the
definition of the reference state).
Standard state (IUPAC):
p0 = 100 kPa
c0 = 1 mol dm-3 = 1 M
1=ia
The activity of pure substances in condensed
phases (solid or liquids) is normally taken as
unity (the number 1)
𝜇𝑖
0
= Δ𝐺 𝑐,𝑖
0
C7790 Introduction to Molecular Modelling -15Gibbs
energy of reaction mixture
It is better to express the Gibbs energy change using
the extent of the reaction:=
=
N
i
iiN dnnnndG
1
21 ),...,,( 
too many parameters
C7790 Introduction to Molecular Modelling -16Extent
of reaction
Extent of reaction x is defined as the molar change of a substance in relation to its
stoichiometric coefficient:
i
in

x

=
Sign convention for i
products (end state) - positive value
reactants (initial state) - negative value
d
n
c
n
b
nn
a
nn DCBBAA
==
−
−
=
−
−
= ,0,0
x
Example: initial state: n0, A and n0, B
The reaction progress can be described by the extent of reaction, which considers the
stoichiometry of the transformation.
aA + bB cC + dD
C7790 Introduction to Molecular Modelling -17Gibbs
energy of reaction mixture
It is better to express the Gibbs energy change using
the extent of the reaction:
i
in

x

= x ddn ii = =
=
N
i
ii ddG
1
x =
=
N
i
ii
d
dG
1

x
=
=
N
i
iiN dnnnndG
1
21 ),...,,( 
  = ==
+==
N
i
N
i
iii
N
i
ii
i
aRT
d
dG
1 1
0
1
ln 

x
QRTG
d
dG
r ln0
+=
x
The chemical potential of the individual substances depends on their effective amount
relative to the standard state, i.e., on the composition of the reaction mixture.
The change is therefore proportional to the composition of the reaction mixture and the
standard chemical potential of the individual reactants:
reaction quotient
standard Gibbs reaction energy
C7790 Introduction to Molecular Modelling -18Change
of G during reaction
A B
only for a given reaction and n0, A = 1.0 mol
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
0
rG
from integration dG/dx
C7790 Introduction to Molecular Modelling -19Change
of G during reaction
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
change of Gibbs energy due to the reaction (this is the Gibbs energy of individual
substances in the standard state in the amount determined by the extent of the reaction)
A B
C7790 Introduction to Molecular Modelling -20Change
of G during reaction
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
mixing Gibbs energy (Gibbs energy that is released as a result of mixing substances in the
standard state in an amount determined by the extent of the reaction)
A B
C7790 Introduction to Molecular Modelling -21Change
of G during reaction
A B
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
local extreme (minimum) determines the composition of the reaction mixture in equilibrium
C7790 Introduction to Molecular Modelling -22Qualitative
conclusions
▪ The change in Gibbs energy consists of two contributions due to:
a) reaction
b) mixing (entropy)
▪ The change of Gibbs energy from the initial or final state to equilibrium is always
negative, so it is a spontaneous process. Even if the standard Gibbs reaction energy is
zero or positive.
▪ There is only one local extreme (minimum) of reaction Gibbs energy and it corresponds
to the equilibrium state.
C7790 Introduction to Molecular Modelling -23Finding
the extreme
0ln0
=+= rr QRTG
d
dG
x
KRTQRTG rr lnln0
−=−=
At the local extreme, the derivative of the function takes zero value:
Equilibrium constant K is a dimensionless quantity that corresponds to the reaction
quotient in the equilibrium state. Equilibrium constant depends only on the nature of the
reaction, the temperature and the definition of the standard state, but it does not
depend on the initial composition of the reaction mixture.
=
=
N
i
ir
i
aK
1
,

Sign convention for and
end state - positive value
default state - negative value
at equilibrium (r)
C7790 Introduction to Molecular Modelling -24-
Example
aA + bB cC+ dD
=
=
N
i
ir
i
aK
1
,

  
   
   
   b
r
a
r
d
r
c
r
b
r
a
r
d
r
c
r
b
Br
a
Ar
d
Dr
c
Crd
Dr
c
Cr
b
Br
a
Ar
BA
DC
BA
DC
aa
aa
aaaaK === −−
,,
,,
,,,,
dimensionless
!!! it has a size !!!
K dimension is (mol dm-3)n (or Mn), where n is the sum of
stoichiometric coefficients
This follows from the definition of the standard state for solution.
at equilibrium (r)
C7790 Introduction to Molecular Modelling -25-
Conclusion
▪ At the given temperature and definition of the standard state, the equilibrium constant
is determined only by the standard reaction Gibbs energy:
▪ The standard reaction Gibbs energy corresponds to the conversion of the initial state to
the final state, which is a hypothetical process that does not actually occur.
▪ When equilibrium is established either from the initial or final state, the change of
reaction Gibbs energy is always negative, regardless of whether the standard reaction
Gibbs energy is zero or positive.
▪ Thus, the reactions always proceed spontaneously from the initial or final state to
equilibrium.
Δ𝐺𝑟
0
= −𝑅𝑇 ln 𝐾
Δ𝐺𝑟
0
<> Δ𝐺(𝜉)
it determines spontaneity of the processit determines composition of the reaction
mixture but it does not say anything about
spontaneity of the reaction
𝜉 is the extent of reaction (≠ reaction
coordinate)
C7790 Introduction to Molecular Modelling -26Recommended
Literature
• Atkins, P. W. Physical Chemistry, 5. ed., repr. (with correct.).; Oxford Univ. Press: Oxford,
1994.
• 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 -27-
Homework
C7790 Introduction to Molecular Modelling -28-
Homework
1. What is pH of acetic acid solution with c0(CH3COOH)=10-3 M?
2. Determine the equilibrium composition of the reaction mixture under
standard conditions for the reaction below, provided that the standard
reaction Gibbs energy is 0.5; 1.0; 2.5; 5.0 and 10.0 kcal/mol. The starting
amount of substance A is 0.001 mol. The volume of the reaction mixture,
which is constant during the reaction, is 1 liter. Next, determine the extent of
the reaction and the ratio of the concentrations of substance B to substance A.
A B
C7790 Introduction to Molecular Modelling -29Homework
- results
1. c0=10-3 M (pKa = 4.756), pH=3.9
2. Solution for constant volume 1L:
T= 298.15 K Gr
0
K -logK [A] [B] x [B]/[A]
c0= 0.001 M (kcal/mol) (M) (M) (mol)
0.50 4.300E-01 0.4 6.993E-04 3.007E-04 3.007E-04 4.300E-01
1.00 1.849E-01 0.7 8.439E-04 1.561E-04 1.561E-04 1.849E-01
2.50 1.470E-02 1.8 9.855E-04 1.449E-05 1.449E-05 1.470E-02
5.00 2.162E-04 3.7 9.998E-04 2.161E-07 2.161E-07 2.162E-04
10.00 4.672E-08 7.3 1.000E-03 4.672E-11 4.672E-11 4.672E-08