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A method for the prediction of the temperature dependence of ternary interaction parameters
CALPHAD
Printed
Vo1.7,
No.1,
in the USA.
pp.
37-40,
A METHOD
0364-5916/a3/010037-04$03.00/0 (c) 1983 Pergamon Press Ltd.
1983
FOR THE PREDICTION
DEPENDENCE
OF
TERNARY
T. Saario'
OF THE TEMPERATURE
INTERACTION
and
PARAMETERS
R.O.Toivanen
Laboratory of Physical Metallurgy, Helsinki University of Vuarimiehentie ZA, 02150 Espoo 15, FINLAND. Technology, 'present address: Metals Laboratory, Technical Research Centre of Finland, Metallimiehenkuja 6, 02150 Espoo 15, FINLAND,
INTRODUCTION Measured
activity
coefficients
can be expressed k
In fi = In fp + The Gibbs
free
be divided
energy
higher.order
+
jzzE?lxj
interaction
into enthalpy
by polynomials
(11,
.
parameters
ni and entropy
terms
(I)
E; can, according
o?, interaction
totheirnature,
parameters
(21,
c; = [l/R)(++). Both
the parameters
03 and
(21
03
depend
on the metallic
alloy
system
and on
temperature. METHOD We use the Jacob-Alcock the ratio
of enthalpy
The model
Z-k-l.
modification interaction
gives
that
dominating,
parameters
in two alloy
model
systems
to predict
l-i-j and
for this purpose = -n{(f~rl)/f~(jl)l'"*
si(modell Assuming
(3) of the quasichemical
the enthalpy
part
of the Gibbs
,fg,ll,a-ll. free
energy
(31 in this
equation
is
we obtain +T')/+T)
Eq. 4 can be utilized arbitary temperature,
= {T*~;[T' .model)}/~TE~(T,model)~~ in at least two ways. and the enthalpy
Firstly,
interaction
(41
if E: is known
parameter
at some
for the reference
nf, is known, we can solve the enthalpy interaction parameter T-I; J (from eq. 41, the entropy interaction parameter ok (from eq. 21, and thus the
system
l-i-j,
temperature
dependence
of the parameter
interaction
parameters
are usually
which
to a linear
leads
dependence
ok.
Secondly,
the enthalpy and entropy
taken
to be independent of temperature. -1 It is easy to show of ek on T (eq. 2).
that E;(T) Received
= +T')
+
T / {+)/R?}dT.
T' ______-____----__-______________________~_~~_--~~~~~~_~_-~-~~~~____ 30 September 1982.
37
(51
T. Saario and R.O. Toivanen
38
When
n~(i’l
range eq.
is
near 5 to
wider
known
T’)
one
give
an
from
can
experimental
calculate
estimate
temperature
results
(valid
n~[T)/n~lT’I
of
the
from
temperature
for eq.
a
narrow
4 and
dependence
temperature
then
of
ck
of
ni/$
integrate
over
a much
range. APPLICATIONS
The
comparison
lead
(A,B
are
given
table
in
2.
average the
by
25%
1.
from
eq.
linear.
predicted
shown
values
from
eq.
Data
used
taken
from
(41
(with
of
table
narrow,
the
sk
-
predicted (51,
T
1. -1
ei
sE”
Obviously,
zero
Fig.
in the
shows
lead,
as
experimental
range the
on 2
liquid
this has
calculations given
sincethe in
dependence
approaching
in
liquid
deviate
results.
dependence
temperature with
in are
n=4,cr=0.5)
of
in
form
is
very
expected
asymptotically
as
the
increases.
-12 r
-16
I1
-8 I1
/
0
/
/
/
4/
---6 ---8
/
---IO
FIG. Comparison liquid
4
1.
based on experimental Sn Em and Fig. 3 that
of
basis
values
Fig.
values
data
The
in
experimental
5 using is
a theoretical
and
obtained
the
range
temperature
T‘he
dependence
from
to
141 is
predictions
temperature
temperature
from
experimental
table
The
calculated
close
of
= Ag,Bi,Cu,Ni,In,Snl
lead
of
experimental CA,3
= Ag,
(ref. Bi,
Cu,
41 Ni,
and In,
1 predicted
Sn).
(eq.
4)
values
of
T$/$
in
PREDICTION
OF TEMPERATURE DEPENDENCE
39
T (K) 1700
1300
1100
900
TASKINEN (4) JACOB 8 JEFFES
(15) i
-60
10
6 T-' :K%04
FIG. Comparison in
liquid
of
predi
cted
(eq.
5)
and
2
experimental
temperature
dependence
of
Sn E,,
lead. T (K) 1700
1300
1100
900
Pb-0-cu --PREDICTED 0 TASKINEN (4) 0
I
I
6
8
JACOB .3 JEFFES (16) ’
i
10
T-' (K-').104 FIG. Comparison in
liquid
of lead.
predicted
(eq.
5)
and
3
experimental
temperature
dependence
of
cOCU
40
T. Saario
TABLE Gibbs Me
energyof
and
of
in
TABLE
Pb
AG;(Me) -
ref.
Ag cu Sn In Ei Ni Pb
- 13084+42.962*T - 74509+37.656.T -192807+71.77 *T -177600+52.09 lT -101103+53.944*T - 71002+28.886*T -116717+50,989.T
6,7 8 9 IO 11 14 4
method
capacity
dependence
of The
systems. enthalpy were
of
case
the
model Gibbs
the
free
same
predicted
the
models
1.
C.
Wagner,
2.
C.H.P.
3.
K.T.
4.
A,
5.
C.H.P.
6.
C. Oiaz, C.R. 183 (1966).
7.
E.H.
8.
P.
9.
W.A.
Lupis
and
Taskinen,
Baker
be
J.F.
will
and
Z.
range -
1203 1172 1204 1273 1173 1273
be
use
of
the used
to
predict
parameters very
reference
Pb)
in
accurate, and
Although
in
for
of
the
other
the
Pb-Q-X
-
when
predicted
the systems
illustrative
reference the
the
temperature
liquid
present
allow
to
of
the
especially
the
both
predictive
the
applicability
two
systems,
and
the
solvents
to
be
different
REFEKENC~S
of
Alloys, Acta
Acta
51,
Ch
Trans.
Arch.
146
(1983).
236,
Ch
Min.
Eisenh.,
(19661.
Inst.
Min.
MetalI.,
C77,
128
C75, (1968).
(1981). 37,
(1966).
S.
Osuka
8.
Isecke,
12.
C.R.
13.
0. Kubachewski Press, Oxford
14.
G.K. Sigworth, J.F. Elliott, G. Waughn and G.H. Geiger, Annual Volume Featuring Molybdenum IProc. Conf.1,1977. Canadian Inst. Min. Met., Montreal, Quebec (1973).
15.
K.T.
Jacob
and
J.H.E.
Jeffes,
Trans.
Inst.
Min.
Metall..,
CEO,
79
[197'l).
16.
K.T.
Jacob
end
J.H.E.
Jeffes.
Trans.
Inst.
Min.
Metall.,
CBO,
32
(19711.
Cavanaugh
and
J.F.
and C.8, (1979).
Univ.
Elliott, Alcock,
119
679
Ii.
Tech,
IIB,
(1967).
Metall..,
145
(1952).
(19721.
130
Trans.
Mass.
IO.
Dissertation,
Trans.,
256
221
Inst.
Stand.,
Ackermann,
15,
20,
AIME,
Richardson,
Talukdar,
Addison-Wesley,
Metall.,
Metall.,
Trans.
F.D.
Met.
p,
Stand.,
Elliott,
Kozuka,
the
was
method
Polytechnica W.
on
illustrate (3)
the
used.
and
based
nevertheless
Polytechnica
M.I.
and
1132 1101 1131 1131 1102 1035
the
Elliott,
J.F.
Acta
Fischer
-14.7+ 20.43*109/T 5.8- ll.60~103~ 75.5-129.1 l 1O3/T 29.5- 61.9~103/T - 9.8+ 11.78*103/T 8.6- 31.41*103/T
11.
Alcock,
Masson
and
Taskinen,
Ag Cu Sn In Bi Ni
(liquid
applying
C.B.
and
13 13 13 13 13 12
to
of
method
Acta
Lupis
Tmp.
interaction
same
can
and
Me EO
To
found
Thermodynamics
Jacob
~~t~rr~4*TinK).
liquid lead
MB
is
(Fig.
the
when
Also,
thermodynamic
was
magnitude was
of ~!~in
dependence
ref.
Alcock
parameters
systems,
different.
and energy
prediction
solvent
work
models.
Jacob
interaction
tQm~eratur8
Experimental
in 10.
2
SUMRARY in this
of
J/moI,
AGh(Pbl
thermodynamic
the
method,
(in
in liquid
'l1715-3.314*T 27624~5.581.T 6276+9.979*T 4000-3.456.T - 3515-l.OBB*T 24995+2.355-T
presented of
oxygen
liquid
Me
The
Toivanen
1
solutionof
Me
and R.O.
Berlin,
Trans.
(19801. (19771.
AIME,
230,
Metallurgical
633
(19641.
Thermochemistry,
Pergamon
Net. Sot. GIN, p. 104, The
Printed
Vo1.7,
No.1,
in the USA.
pp.
37-40,
A METHOD
0364-5916/a3/010037-04$03.00/0 (c) 1983 Pergamon Press Ltd.
1983
FOR THE PREDICTION
DEPENDENCE
OF
TERNARY
T. Saario'
OF THE TEMPERATURE
INTERACTION
and
PARAMETERS
R.O.Toivanen
Laboratory of Physical Metallurgy, Helsinki University of Vuarimiehentie ZA, 02150 Espoo 15, FINLAND. Technology, 'present address: Metals Laboratory, Technical Research Centre of Finland, Metallimiehenkuja 6, 02150 Espoo 15, FINLAND,
INTRODUCTION Measured
activity
coefficients
can be expressed k
In fi = In fp + The Gibbs
free
be divided
energy
higher.order
+
jzzE?lxj
interaction
into enthalpy
by polynomials
(11,
.
parameters
ni and entropy
terms
(I)
E; can, according
o?, interaction
totheirnature,
parameters
(21,
c; = [l/R)(++). Both
the parameters
03 and
(21
03
depend
on the metallic
alloy
system
and on
temperature. METHOD We use the Jacob-Alcock the ratio
of enthalpy
The model
Z-k-l.
modification interaction
gives
that
dominating,
parameters
in two alloy
model
systems
to predict
l-i-j and
for this purpose = -n{(f~rl)/f~(jl)l'"*
si(modell Assuming
(3) of the quasichemical
the enthalpy
part
of the Gibbs
,fg,ll,a-ll. free
energy
(31 in this
equation
is
we obtain +T')/+T)
Eq. 4 can be utilized arbitary temperature,
= {T*~;[T' .model)}/~TE~(T,model)~~ in at least two ways. and the enthalpy
Firstly,
interaction
(41
if E: is known
parameter
at some
for the reference
nf, is known, we can solve the enthalpy interaction parameter T-I; J (from eq. 41, the entropy interaction parameter ok (from eq. 21, and thus the
system
l-i-j,
temperature
dependence
of the parameter
interaction
parameters
are usually
which
to a linear
leads
dependence
ok.
Secondly,
the enthalpy and entropy
taken
to be independent of temperature. -1 It is easy to show of ek on T (eq. 2).
that E;(T) Received
= +T')
+
T / {+)/R?}dT.
T' ______-____----__-______________________~_~~_--~~~~~~_~_-~-~~~~____ 30 September 1982.
37
(51
T. Saario and R.O. Toivanen
38
When
n~(i’l
range eq.
is
near 5 to
wider
known
T’)
one
give
an
from
can
experimental
calculate
estimate
temperature
results
(valid
n~[T)/n~lT’I
of
the
from
temperature
for eq.
a
narrow
4 and
dependence
temperature
then
of
ck
of
ni/$
integrate
over
a much
range. APPLICATIONS
The
comparison
lead
(A,B
are
given
table
in
2.
average the
by
25%
1.
from
eq.
linear.
predicted
shown
values
from
eq.
Data
used
taken
from
(41
(with
of
table
narrow,
the
sk
-
predicted (51,
T
1. -1
ei
sE”
Obviously,
zero
Fig.
in the
shows
lead,
as
experimental
range the
on 2
liquid
this has
calculations given
sincethe in
dependence
approaching
in
liquid
deviate
results.
dependence
temperature with
in are
n=4,cr=0.5)
of
in
form
is
very
expected
asymptotically
as
the
increases.
-12 r
-16
I1
-8 I1
/
0
/
/
/
4/
---6 ---8
/
---IO
FIG. Comparison liquid
4
1.
based on experimental Sn Em and Fig. 3 that
of
basis
values
Fig.
values
data
The
in
experimental
5 using is
a theoretical
and
obtained
the
range
temperature
T‘he
dependence
from
to
141 is
predictions
temperature
temperature
from
experimental
table
The
calculated
close
of
= Ag,Bi,Cu,Ni,In,Snl
lead
of
experimental CA,3
= Ag,
(ref. Bi,
Cu,
41 Ni,
and In,
1 predicted
Sn).
(eq.
4)
values
of
T$/$
in
PREDICTION
OF TEMPERATURE DEPENDENCE
39
T (K) 1700
1300
1100
900
TASKINEN (4) JACOB 8 JEFFES
(15) i
-60
10
6 T-' :K%04
FIG. Comparison in
liquid
of
predi
cted
(eq.
5)
and
2
experimental
temperature
dependence
of
Sn E,,
lead. T (K) 1700
1300
1100
900
Pb-0-cu --PREDICTED 0 TASKINEN (4) 0
I
I
6
8
JACOB .3 JEFFES (16) ’
i
10
T-' (K-').104 FIG. Comparison in
liquid
of lead.
predicted
(eq.
5)
and
3
experimental
temperature
dependence
of
cOCU
40
T. Saario
TABLE Gibbs Me
energyof
and
of
in
TABLE
Pb
AG;(Me) -
ref.
Ag cu Sn In Ei Ni Pb
- 13084+42.962*T - 74509+37.656.T -192807+71.77 *T -177600+52.09 lT -101103+53.944*T - 71002+28.886*T -116717+50,989.T
6,7 8 9 IO 11 14 4
method
capacity
dependence
of The
systems. enthalpy were
of
case
the
model Gibbs
the
free
same
predicted
the
models
1.
C.
Wagner,
2.
C.H.P.
3.
K.T.
4.
A,
5.
C.H.P.
6.
C. Oiaz, C.R. 183 (1966).
7.
E.H.
8.
P.
9.
W.A.
Lupis
and
Taskinen,
Baker
be
J.F.
will
and
Z.
range -
1203 1172 1204 1273 1173 1273
be
use
of
the used
to
predict
parameters very
reference
Pb)
in
accurate, and
Although
in
for
of
the
other
the
Pb-Q-X
-
when
predicted
the systems
illustrative
reference the
the
temperature
liquid
present
allow
to
of
the
especially
the
both
predictive
the
applicability
two
systems,
and
the
solvents
to
be
different
REFEKENC~S
of
Alloys, Acta
Acta
51,
Ch
Trans.
Arch.
146
(1983).
236,
Ch
Min.
Eisenh.,
(19661.
Inst.
Min.
MetalI.,
C77,
128
C75, (1968).
(1981). 37,
(1966).
S.
Osuka
8.
Isecke,
12.
C.R.
13.
0. Kubachewski Press, Oxford
14.
G.K. Sigworth, J.F. Elliott, G. Waughn and G.H. Geiger, Annual Volume Featuring Molybdenum IProc. Conf.1,1977. Canadian Inst. Min. Met., Montreal, Quebec (1973).
15.
K.T.
Jacob
and
J.H.E.
Jeffes,
Trans.
Inst.
Min.
Metall..,
CEO,
79
[197'l).
16.
K.T.
Jacob
end
J.H.E.
Jeffes.
Trans.
Inst.
Min.
Metall.,
CBO,
32
(19711.
Cavanaugh
and
J.F.
and C.8, (1979).
Univ.
Elliott, Alcock,
119
679
Ii.
Tech,
IIB,
(1967).
Metall..,
145
(1952).
(19721.
130
Trans.
Mass.
IO.
Dissertation,
Trans.,
256
221
Inst.
Stand.,
Ackermann,
15,
20,
AIME,
Richardson,
Talukdar,
Addison-Wesley,
Metall.,
Metall.,
Trans.
F.D.
Met.
p,
Stand.,
Elliott,
Kozuka,
the
was
method
Polytechnica W.
on
illustrate (3)
the
used.
and
based
nevertheless
Polytechnica
M.I.
and
1132 1101 1131 1131 1102 1035
the
Elliott,
J.F.
Acta
Fischer
-14.7+ 20.43*109/T 5.8- ll.60~103~ 75.5-129.1 l 1O3/T 29.5- 61.9~103/T - 9.8+ 11.78*103/T 8.6- 31.41*103/T
11.
Alcock,
Masson
and
Taskinen,
Ag Cu Sn In Bi Ni
(liquid
applying
C.B.
and
13 13 13 13 13 12
to
of
method
Acta
Lupis
Tmp.
interaction
same
can
and
Me EO
To
found
Thermodynamics
Jacob
~~t~rr~4*TinK).
liquid lead
MB
is
(Fig.
the
when
Also,
thermodynamic
was
magnitude was
of ~!~in
dependence
ref.
Alcock
parameters
systems,
different.
and energy
prediction
solvent
work
models.
Jacob
interaction
tQm~eratur8
Experimental
in 10.
2
SUMRARY in this
of
J/moI,
AGh(Pbl
thermodynamic
the
method,
(in
in liquid
'l1715-3.314*T 27624~5.581.T 6276+9.979*T 4000-3.456.T - 3515-l.OBB*T 24995+2.355-T
presented of
oxygen
liquid
Me
The
Toivanen
1
solutionof
Me
and R.O.
Berlin,
Trans.
(19801. (19771.
AIME,
230,
Metallurgical
633
(19641.
Thermochemistry,
Pergamon
Net. Sot. GIN, p. 104, The