C8863 Free Energy Calculations -1-3. Chemical Equilibrium
C8863 Free Energy Calculations
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
JS/2023: Rev1
Lesson 3
Equilibrium
C8863 Free Energy Calculations -2-3. Chemical Equilibrium
Overview
C8863 Free Energy Calculations -3-3. Chemical Equilibrium
Equilibrium
Or what you should already know….
C8863 Free Energy Calculations -4-3. Chemical Equilibrium
Chemical 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
C8863 Free Energy Calculations -5-3. Chemical Equilibrium
Extent 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 ni
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
C8863 Free Energy Calculations -6-3. Chemical Equilibrium
Gibbs energy of reaction mixture
𝑑𝐺(𝑛1, 𝑛2, . . . , 𝑛 𝑁) = ෍
𝑖=1
𝑁
𝜕𝐺
𝜕𝑛𝑖 𝑝,𝑇,𝑛 𝑗≠𝑛 𝑖
𝑑𝑛𝑖
𝜇𝑖 =
𝜕𝐺
𝜕𝑛𝑖 𝑝,𝑇,𝑛 𝑗≠𝑛 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:
𝑑𝐺(𝑛1, 𝑛2, . . . , 𝑛 𝑁) = ෍
𝑖=1
𝑁
𝜇𝑖 𝑑𝑛𝑖
Employing chemical potential m , the relation can be simplified:
N - number of reacting compounds, ni is the molar amount of substance i
C8863 Free Energy Calculations -7-3. Chemical Equilibrium
Gibbs 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m =
=
N
i
ii
d
dG
1
m
x
=
=
N
i
iiN dnnnndG
1
21 ),...,,( m
  = ==
+==
N
i
N
i
iii
N
i
ii
i
aRT
d
dG
1 1
0
1
ln 
mm
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
C8863 Free Energy Calculations -8-3. Chemical Equilibrium
Change 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
C8863 Free Energy Calculations -9-3. Chemical Equilibrium
Change 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
C8863 Free Energy Calculations -10-3. Chemical Equilibrium
Change 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
C8863 Free Energy Calculations -11-3. Chemical Equilibrium
Change 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
C8863 Free Energy Calculations -12-3. Chemical Equilibrium
Qualitative 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.
C8863 Free Energy Calculations -13-3. Chemical Equilibrium
Finding 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 nand
end state - positive value
default state - negative value
at equilibrium (r)
C8863 Free Energy Calculations -14-3. Chemical Equilibrium
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 dimension !!!
the 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)
C8863 Free Energy Calculations -15-3. Chemical Equilibrium
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 the
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)
C8863 Free Energy Calculations -16-3. Chemical Equilibrium
Recommended 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.