C8863 Free Energy Calculations -1-
C8863
Free Energy Calculations
2. Chemical Equilibrium
Petr Kulhánek
kulhanek@chemi.muni.cz
NCBR - National Centre for Biomolecular Research
&
CEITEC - Central Institute of Technology,
Masaryk University,
Kamenice 5, 625 00 Brno
C8863 Free Energy Calculations -2Basic
Overview(repetition or what you already know)
C8863 Free Energy Calculations -3Thermodynamics
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 – stochiometric coefficients
C8863 Free Energy Calculations -4-
Equilibrium
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 concentrations
at equilibrium (r)
R
P
aA + bB cC + dD
C8863 Free Energy Calculations -5-
Chemical
Equlibrium
C8863 Free Energy Calculations -6Chemical
Change
aA + bB cC+ dD
The reaction of substances A and B gives rise to substances C and D and vice versa, that is,
the reaction of substances C and D produces substances A and B. Both processes (forward
and reverse reactions) continue until the changes in both directions are balanced and
equilibrium is reached .
Principal questions:
➢ What is the composition of the reaction mixture in equilibrium and how it is
determined?
➢ How is it possible to influence the composition of the reaction mixture in
equilibrium?
C8863 Free Energy Calculations -7Progress
of Reaction
Extent of reaction x is a quantity defined as ratio of molar amount change of a compound i
to its stoichiometry coefficient:
i
in

x

=
aA + bB cC+ dD
Sign convention forni
final state – positive value
initial state – negative value
d
n
c
n
b
nn
a
nn DCBBAA
==
−
−
=
−
−
= ,0,0
x
Example: initial state: n0,A; n0,B
Progress of reaction can be quantified by extent of reaction, which takes into account
stoichiometry of the change.
C8863 Free Energy Calculations -8Gibbs
Energy of Reaction Mixture
= 








=
N
i
i
nnTpi
N dn
n
G
nnndG
ij
1 ,,
21 ),...,,(
ij nnTpi
i
n
G









=
,,

ni is molar amount of i
The Gibbs energy of the reaction mixture is a function of the composition of the reaction
mixture. Under constant temperature and pressure, the total differential of the Gibbs
energy of the reaction mixture can be written in the following form:
=
=
N
i
iiN dnnnndG
1
21 ),...,,( 
The derivation of the Gibbs energy according to the molar amount of a substance is a very
useful quantity called chemical potential i :
C8863 Free Energy Calculations -9Chemical
Potential
ij nnTpi
i
n
G









=
,,

Chemical potential expresses the ability of the substance to:
• react with another substance
• change its status
• change its spatial arrangement
Value of chemical potential:
• is related to the amount of the substance
• is related to the environment conditions (temperature, pressure, ...)
• 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 of compound ai:
Chemical potential is a state function:
C8863 Free Energy Calculations -10Dependence
of i on Activity ai
aRT ln0
+= 
For ideal gas (a system of non-interacting molecules), the change in Gibbs energy is equal
to volume work when the pressure changes:
dp
p
nRT
VdpdG == 0
00
ln
0 p
p
nRTGGdp
p
nRT
G
p
p
+=+= 
The chemical potential can then be written as:
kde:
dn
dG
= 0
p
p
a =
The value of the chemical potential can only be
expressed in relation to a precisely defined state.
The activity of the substance then expresses the effective
amount of the substance against the standard state.
C8863 Free Energy Calculations -11-
Activity
00
p
p
p
f
a ii
i =
00
c
c
c
c
a ii
ii = 
gaseous mixtures
solutions
mixture of ideal gases
ideal solution (diluted solution)
mixture of gasees
solution
f – fugacity
p – partial pressure
c – molar concentrations
 – activity coefficient
Standard state conditions
(IUPAC):
p0 = 100 kPa
c0 = 1 mol dm-3 = 1 Msolid a liquid substances at the standard state: 1=ia
Activity expresses the effective amount of a substance against a standard state. It is a
dimensionless quantity.
The reason for introducing of an activity coefficient (or fugacity) is to maintain a simple
relationship between activity and chemical potential. Therefore, a relationship can be taken
as definition of activity:
RT
i
ii
ea
0
 −
=
C8863 Free Energy Calculations -12Standard
Chemical Potential
The standard chemical potential is a change in the Gibbs energy, which is associated with
the formation of one mole of the substance in the standard state. The change of Gibbs
energy is most often expressed in the form of standard formation Gibbs energy.
0
,
0
ifi G=
The standard Gibbs formation energy is a change in the Gibbs energy, which corresponds
to the formation of one mole of substance from the individual chemical elements in the
standard state. Chemical elements in the standard state have a zero Gibbs energy (this is
definition of the reference state).
Standard state conditions (IUPAC):
p0 = 100 kPa
c0 = 1 mol dm-3 = 1 M
C8863 Free Energy Calculations -13Gibbs
Energy of Reaction Mixture
It is suitable to express Gibbs energy using the extent of 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 ),...,,( 
Derivation of the Gibbs energy according to the extent of the reaction can be used to
quantify the change in the Gibbs energy that occurs during the reaction:
=−=
x
x
x
x
0
)0()( d
d
dG
GGG
integration
C8863 Free Energy Calculations -14Gibbs
Energy of Reaction Mixture
=
=
N
i
ii
d
dG
1

x =−=
x
x
x
x
0
)0()( d
d
dG
GGG
integration
What is the function of G?
  = ==
+==
N
i
N
i
iii
N
i
ii
i
aRT
d
dG
1 1
0
1
ln 

x
QRTG
d
dG
r ln0
+=
x
It is necessary to take into account the fact that the chemical potential of the individual
substances depends on their effective amount relative to the standard state, i.e., the
composition of the reaction mixture.
reaction quotientstandard Gibbs reaction energy
C8863 Free Energy Calculations -15Change
of G During Reaction
A B
11
,0 BAA nnn
=
−
−
=x
 
  x
x
−
==
AnA
B
Q
,0
x
x
x −
+=+=
A
rr
n
RTGQRTG
d
dG
,0
00
lnln
( ) ( )  0
,0,0,0,0
0
0
lnlnln)0()( AAAAAr GnnnnRTGGd
d
dG
G +−−−−+=+=  xxxxxx
x
x
x
Example:
- constant volume
- activity coefficients are
1 (ideal solution)
Result:
C8863 Free Energy Calculations -16Change
of G During Reaction
A B
only for given reaction and n0,A = 1.0 mol
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
0
rG
C8863 Free Energy Calculations -17Change
of G During Reaction
A B
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
change of Gibbs energy as a result of reaction, i.e., response to change of reaction
mixture composition (sum of G of individual substances in the standard state in the
amount determined by the extent of reaction)
C8863 Free Energy Calculations -18Change
of G During Reaction
A B
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
mixing Gibbs energy (The Gibbs energy released as a result of mixing the substances in
the standard state in the amount determined by the extent of reaction)
C8863 Free Energy Calculations -19Change
of G During Reaction
A B
( ) ( )  0
,0,0,0,0
0
lnlnln)( AAAAAr GnnnnRTGG +−−−−+= xxxxxx
local extreme (minimum) – it determines the
composition of the reaction at the equilibrium
C8863 Free Energy Calculations -20Qualitative
Conclusions
▪ Change of the Gibbs energy is result of two contributions:
a) “reaction"
b) “mixing"
▪ Changing the Gibbs energy from the initial or final state to the equilibrium is always
negative. Even if the standard Gibbs reaction energy is zero or positive.
▪ There is only one local extreme (minimum) of the Gibbs energy function to the extent of
the reaction that corresponds to the state of equilibrium.
C8863 Free Energy Calculations -21Finding
the Extreme
0ln0
=+= rr QRTG
d
dG
x
KRTQRTG rr lnln0
−=−=
In the local extreme, the derivative takes on a zero value:
The equilibrium constant K is a dimensionless quantity that corresponds to the reaction
quotient in the equilibrium. The value of the equilibrium constant depends only on the
nature of the reaction, the temperature and the definition of the standard state, but does
not depend on the starting composition of the reaction mixture
=
=
N
i
ir
i
aK
1
,

at equilibrium (r)
Sign convention forni
final state – positive value
initial state – negative value
C8863 Free Energy Calculations -22-
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 === −−
,,
,,
,,,,
unitless
!!! it has a unit !!!
only the values given in (mol dm-3)n, where n
is the sum of the stoichiometric coefficients
(the sign convention) are compatible with
unitless version of K
at equilibrium (r)
C8863 Free Energy Calculations -23-
Summary
C8863 Free Energy Calculations -24-
Conclusions
▪ The equilibrium constant at a given temperature and definition of the standard state is
determined only by the standard Gibbs reaction energy:
▪ The standard reaction Gibbs energy corresponds to the conversion of the initial state to
the final, which is a hypothetical event that does not actually occur.
▪ When establishing the equilibrium from the initial or final state, the change in Gibbs
energy is always negative regardless of the standard Gibbs energy is zero or positive.
▪ Thus, the reactions always proceed spontaneously to the equilibrium from the initial or
final state.
KRTGr ln0
−=