1. 1
Gibbs-Duhem equation
)
,
,
,
( 2
1 n
n
P
T
G
G
j
i n
n
P
T
i
i
n
G
,
,
2
,
,
2
1
,
,
1
,
,
,
, 1
2
2
1
2
1
dn
n
G
dn
n
G
dP
P
G
dT
T
G
G
n
T
P
n
T
P
n
n
T
n
n
P
S
T
G
n
n
P
2
1,
,
V
P
G
n
n
T
2
1,
,
N
i
i
idn
VdP
SdT
dn
dn
VdP
SdT
dG
1
2
2
1
1
2. 2
• The change in the Gibbs free energy and the partial-molar Gibbs
energy (the Chemical Potential ()) can be written as follows:
where:
j
n
P
T
i
i
n
G
,
,
• If Pressure and Temperature are held constant (which is the case
in most metallurgical processes), then we have the following:
....
3
3
2
2
1
1
1
dn
dn
dn
dn
dG
N
i
i
i
N
i
i
idn
Vdp
sdT
dG
1
3. • Integrating the above equation, we have
....
3
3
2
2
1
1
1
n
n
n
n
G
N
i
i
i
....
2
2
2
2
1
1
1
1
1
1
d
n
dn
d
n
dn
d
n
dn
dG i
N
i
i
N
i
i
i
• Taking the differential of the above equation, we have
• The two forms of dG must be equal
N
i
i
idn
dG
1
....
2
2
2
2
1
1
1
1
1
1
d
n
dn
d
n
dn
d
n
dn
dG i
N
i
i
N
i
i
i
at constant P and T
4. 4
• Then we will obtain the following
0
1
i
N
i
id
n 0
3
3
2
2
1
1
d
n
d
n
d
n
• This equation shows that if the chemical potential of one
component increases, that of the other component must
decrease; e.g.
Gibbs-Duhem equation
• For example, in a binary metallic solutions (components A and B),
one has:
0
B
B
A
A d
n
d
n
A
B
A
B d
n
n
d
5. 5
• Thermodynamic equations for calculation of excess free
energy(Gxs) and integral molar free energy of a solution need
activity coefficient and activity of all the components of the
solution
• However, experimental techniques viz. chemical equilibria, vapor
pressure and electrochemical can measure activity of only one
component
• In order to get activity of the second component in a binary
solution we must couple activity and atom/mole fractions of both
the components with the aid of Gibbs-Duhem equation as
follows:
0
1
i
N
i
i Q
d
n Q is any extensive property
GIBBS – DUHEM INTEGRATION
6. 6
• Since activity of a component is related to the partial molar free
energy, we can write Gibbs-Duhem equation as under:
)
1
(
0
B
B
A
A G
d
x
G
d
x
)
2
(
0
ln
ln
B
B
A
A a
d
x
a
d
x
)
3
(
ln
ln B
A
B
A a
d
x
x
a
d
A
A
A
A
A
x
x
x
B
A
B
x
x
A a
d
x
x
a
1
ln
|
ln
• In terms of activity, this can be written as
• By integrating the above equation from xA = 1 to xA = xA
8. 8
• Instead, the activity coefficient is measured first, from which
activity can easily be measured knowing that k
k
k x
a
• Due to the uncertainity in the tails of the above curve, activity of
components in the solution is not measured directly
1
B
A x
x )
4
(
0
B
A dx
dx
• Equation (4) can be mathematically manipulated as follows:
)
5
(
0
B
B
B
A
A
A
x
dx
x
x
dx
x
)
6
(
0
ln
ln
B
B
A
A x
d
x
x
d
x
• Subtracting Equation (6) from equation (2), we have the following:
9. 9
)
7
(
0
ln
ln
B
B
A
A d
x
d
x
0
ln
,
0
,
0
A
B
B
B
B
B
x
x
for
finite
also
is
and
finite
is
a
x
as
10. 10
2
1
ln
i
i
i
x
,
ln
2
B
A
A
x
2
ln
and
A
B
B
x
B
A
A
A
B
B d
x
dx
x
d
2
2
ln
• As a further aid to the integration of the Gibbs Duhem equation,
let’s introduce an alpha-function as follows:
is always finite because
1
1
i
i x
as
• For the components of a binary solution
2
ln A
B
B x
• αB is known as a function of composition:
• On differentiation of the composition function, we get
substituting
this into this
B
A
B
A d
x
x
d
ln
ln
11. 11
B
A
B
A
B
B
B
A
A
B
A
A
B
A
B
A d
x
x
dx
x
d
x
x
x
dx
x
x
x
d
.
2
.
.
2
ln 2
(8)
2
ln
1 1
A
A
A
A
A
B
A
B
x
x
x
x
x
at
x
at
B
A
B
A
B
B
A d
x
x
dx
x
• We have the following
• On integration
A
A
A
A
A
A
A
B
x
x
A
A
B
x
x
A
B
B
x
x
x
at
B
A
B x
d
x
x
x
d
x
x
1
1
1
)
2
1
(
• Then, replacing into equation (8), we have the following
12. 12
(8)
-
ln
1
A
A
x
x
A
B
A
B
B
A x
d
x
x
• Thus ln γA at xA=xA is obtained as – xBxAαB minus the area under the
plot of αB vs xA from xA=xA to xA = 1.
• Since αB is everywhere finite, this integration does not involve a tail
to infinity.
2
1
ln
i
i
i
x
is always finite because
1
1
i
i x
as
14. 14
Thermodynamics of slags
What is metallurgical slag?
• Slag is a melt (after solidifying, a rocklike or glasslike substance)
usually covering the surface of molten metal in metallurgical
processes usually for the refining of metallic materials
• Slags are alloys of oxides of various compositions
• The major components of slags are:
o the acidic oxide SiO2
o the basic oxides CaO, FeO, and MgO
o the neutral Al2O3 and (less often) ZnO
15. 15
• Slag covers the metallic bath due to their immiscibility and lighter
than the metallic phase
• The specific gravity of slag ranges between 3–4, as compared to
7–8 for iron and steel
• The slag cover protects the metal from oxidation and prevents
heat losses due to its poor thermal conductivity
• Slags play a very important role in carrying out a number of
physical and chemical functions
• In primary extraction, slags accept gangue( e.g rock material ,dust
,soil, sand, earthy particle ,limestone, mica) and unreduced
oxides, whereas in refining they act as a reservoir of chemical
reactant(s) and absorber of extracted impurities
16. 16
• In order to achieve the above mentioned objectives, slag must
possess a certain optimum level of:
physical properties such as:
o low melting point
o low viscosity
o low surface tension, and
o high diffusivity
chemical properties such as:
o basicity
o oxidation potential
o thermodynamic properties
• The required properties of slag are controlled by its composition
and structure
Reading assignment: Structure of slag materials
17. 17
• In secondary refining of steelmaking, two slag systems are
commonly used
The first slag system is
featured with low basicity
and low Al2O3 content to
target inclusions in the low
temperature region in
adjacent to compounds of
CaO Al2O3∙2SiO2, CaO∙SiO2
and tridymite in CaO-SiO2-
Al2O3 ternary system –
region A (≤1350 °C)
• usually used in production of motor engine, valve spring steel and
tyre cord steel
18. 18
There exists another
lower temperature
region in CaO-SiO2- Al2O3
system and CaO-MgO-
Al2O3 system, i.e. the
region in adjacent to:
o 3CaO∙Al2O3
o 5CaO ∙3Al2O3
o 12CaO ∙ 7Al2O3 and
o CaO ∙ Al2O3 – region B
• The melting temperature
is ≤ 1500 °C
19. 19
• Taking the inclusions of simple oxides of CaO, MgO, Al2O3 for
instance, in secondary refining of steel, the chemical reactions
between [Ca], [Mg], [Al], [O] and the slag can be written as:
represents the
dissolved Ca, Mg,
Al, etc in liquid
steel
dissolved
oxygen in liquid
steel
formed reaction product
such as CaO, MgO, Al2O3,
etc in the slag
• When reaction between the slag and steel reaches equilibrium,
the free energy change of the reaction is written as
20. 20
• Inside the liquid steel, reactions between [Ca], [Mg], [Al], etc. and
the inclusions take place, which can be expressed as
represents products such as CaO, MgO,
Al2O3, etc in the inclusions
• When the reaction between the inclusions and Ca, Mg, Al, etc. in
liquid steel approaches to equilibrium, the free energy change of
the above reaction can be written as:
21. 21
• If same activity standard is adopted for components in slag and
inclusions, the standard free energy change of both reactions
should have the same value, i.e.
• By combining both equations, the following two equations are
obtained
=
=
22. 22
State of Oxidation of Slag
• The oxidizing or reducing power of a slag refers to its capacity to
participate in the transfer of oxygen to and from the metallic
bath
• In iron and steelmaking the metallic bath contains iron as the
principal component and other elements that are more or less
noble than iron
• More noble elements will oxidize less readily than iron during
the refining of pig iron and, consequently, will be found in the
metallic phase
• Less noble elements are easier to oxidize as compared to iron
and will be found to a large extent in the slag as oxides.
23. 23
• However, these oxides may be so stable that they are not able to
supply oxygen to the metallic bath and, hence, the oxidizing
power of the slag depends on the activity of the iron oxide
present in the slag
• The equilibrium between oxygen dissolved in the metal and iron
oxide dissolved in the slag is given as:
represents the constituent/ion dissolved in the slag
Equilibrium constant, K
24. 24
• Thus, the activity of oxygen in the melt, [a0] is proportional to the
activity of FeO in the slag (aFeO)
• At 1600 °C liquid iron saturated with pure FeO dissolves 0.23% O,
that is, when aFeO = 1
• In the presence of slag forming oxides, solubility of oxygen
decreases and [% O] is accordingly modified as:
25. 25
Basicity and refining ability of slags
• Chemical composition of molten slags often becomes the key in the
optimization of each refining process
• Among others, basicity is known to be a major factor in the redox
reactions during metal refining
• In the refining and alloying processes of metals, the distribution
ratios of the impurity or additive elements between slags and
metals will naturally control their removal fractions or alloying yields
• Distribution of the element M between slags and metals can be
categorized into the following three types of reactions depending
on the form of M in the slag phase:
26. 26
1.
2.
3.
On the other hand, M can be removed by oxidation and exist as
either
In reaction 1, M in the metal phase will be removed via a reducing
reaction
Sulfur is a typical impurity in hot metal which is generally removed
into slags as the S2- ion
Typical examples are the removal of Mn and P from hot metal by
the respective slag–metal reactions
27. 27
• In aqueous solutions, the base accepts one or more protons
whereas the acid provides proton(s)
• In slag systems, an acid oxide forms a complex anion by accepting
one or more O2- anions whereas a basic oxide generates O2− anion:
• As can be seen from the effect of O2- ion in each reaction, more
basic slags are preferable for the removal of M, such as S and P
• Thus, basicity is a very important factor for most of the slag refining
reactions in metallurgical processes
• For example, SiO2 (and P2O5, CO2, SO3 etc) is an acid oxide
because it accepts O2− anions to form a complex silicate anion as
per the reaction:
28. 28
• The amphoteric oxides like Al2O3, Cr2O3 Fe2O3 behave as bases in
the presence of acid(s) or as acids in the presence of base(s):
• On the other hand, basic oxides like CaO, Na2O,MnO and so on
generate O2− anions:
29. 29
• Different methods have been suggested to express basicity index
• Ionic theory expresses basicity as the excess of O2− anions in 100 g
of slag:
• A slag containing 33.3 mol% SiO2, corresponding to the
composition: 2CaO∙SiO2 in the CaO─SiO2 system, is neutral
one molecule of SiO2 is neutralized by
two molecules of CaO by forming two
cations of calcium (Ca2+) and one
complex silicate anion (SiO4
−4) which in
turn forms one molecule of calcium
orthosilicate (2CaO∙SiO2)
30. 30
• The molecular theory assumes ideal behavior for all the
molecules present in the slag
• Without forming ions, simple oxides such as CaO, MnO, FeO,
Fe2O3, Al2O3, SiO2 either associate to form complex molecules like
Ca2SiO4, CaAl2O4, and Ca4P2O9 or remain as free compounds
• This makes it necessary to consider all possible molecules existing
in the slag system
• For example, in a ternary slag system of CaO─MnO─SiO2 eight
molecules MnO, CaO, CaSiO3, Ca2SiO4, MnSiO3, Mn2SiO4, Ca2Si2O6,
and Ca4Si2O8 may exist.
• This requires a set of the following eight equations to calculate the
mole fraction of each compound as under:
32. 32
• From the mass balance the number of moles of CaO, MnO, and
SiO2 can be expressed as:
• Basicity index, in general, is defined as the ratio of the total
weight percent of the basic oxides to the total weight percent of
the acid oxides
• In a binary slag, for example, CaO─SiO2 the basicity index (B) is
given as:
33. 33
Slag capacity
• The slag capacity is the ability of a molten slag to absorb species
such as sulfur, phosphorus, nitrogen, and carbon dioxide
desulfurization reaction
34. 34
dephosphorization reaction
• Each of these reactions can be converted to the dissolution of
sulfur and phosphorus from the gas phase into the molten slag
dissolution
of sulfur
dissolution of
phosphorus
• From sulfur dissolution reaction, the equilibrium constant can
be written as follows:
35. 35
• The sulfide capacity of the slag, i.e., the ability of the slag to absorb
the respective sulfide gas, can be defined in terms of measurable
parameters as shown in the following equation:
)
(
)
(
2
2
2
S
O
S
x
K
2
/
1
2
/
1
2
2
2
2
2
2
2
2
2
2
)
(
)
(
)
(
)
(
S
O
O
S
S
S
O
O
S
S
p
p
x
x
p
p
a
a
K
• From phosphorus dissolution reaction, the equilibrium constant
can be written as follows
4
/
5
2
/
1
4
/
3
2
2
2
2
4
2
1
)
(
)
(
O
P
O
PO
P
p
p
a
a
K
36. 36
)
(
)
(
2
4
2
2
2
/
3
PO
O
P
a
K
• For practical applications, each capacity can be converted to a
distribution ratio of a particular element between slag and metal,
when the thermodynamic property of the element dissolved in
the metal is of interest
• The phosphide capacity of the slag, i.e., the ability of the slag to
absorb the respective gases:
• For example, in the case of sulfur and phosphorus, their
dissolution reactions from the gas phase into molten iron can be
expressed by
37. 37
• the corresponding Gibbs energies of reaction can be expressed by
the following equation
For sulfur dissolution reaction
2
S
K 2
P
K
38. 38
• The distribution ratios of sulfur and phosphorus between slag
and molten iron, LS and LP, can be
1/2
O
S
S
2
2
2
p
K
C
S
For phosphorus dissolution reaction
2
2
-
3
4
P
5/4
O
PO
P
32
.
0
K
p
C
