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Extension of a subregular equation for multicomponent systems to a “subsubregular” one
CALPHAD
Vol.8,
Printed
the
in
No.1, USA.
EXTENSION
67-68,
pp.
1984
0364-5916/84
(c)
OF A SUBREGULAR TO
A
EQUATION
FOR
MULTlCOMPONENT
” SUBSUBRECULAR”
Rarner
1984
Pergamon
$3.00 + .oo Press Ltd.
SYSTEMS
ONE
Schmid
lnstitut fur Ersenhuttenkunde und Grefiererwesen Technische Universitat Clausthal D - 3392 Clausthal-Zellerfeld, West Germany
ABSTACT.
A subregular equation for the description of mrxmg effects from binary parameters is extended to m&de an additional, This formula is much more useful for practical applications suffix polynomial equation.
For the description of asymmetric types of polynomial representations been given by Kellogg (I):
A
G”
= (l/2)
Rr In
= $
Xp
-5
t
t{w,j
j=l
i=l
quantrtres in a binary or multicomponent solution many A very practical formula for a n-component system has
+ (w
{ Wij/2
‘1
( (WIP + wp,)/2
2
]=I
excess exrst.
In a multicomponent system optional binary parameter. than the correspondmg four-
wjl)
‘11’1
+ (WIP - wp,)
+ (w,,
- Wj,)
x,}
’
(xp
(I)
I
- Xi/Z))
x,
(2)
xi xi
i=l
IS the excess Gibbs energy of mixing, R the gas constant, T the absolute temperature, v where AC” the activity coefficient of component p - the standard state bemg pure component p in the same p&se and w are interaction parameters of the binary system and x. the mole fraction of component I. The w i - j. if a temperature dependence like wij = a :‘b/T 1s gssumed, the heat and excess entropy of mixrng are given by formulas analogous to Eqs. (I) and (2). However, there are cases where the complexrty of the system and the amount of experrmental data requrre an additional parameter for the descriptron of at least some of the brnaries. Such an extension of the subregular Eqs. (1) and (2) to a “subsubregular” case IS given below: A
Gxs
r
In
yp
= (l/2)
= g
-2 j=l
fj
{ (wip
f{Wij/2 I=1
s
fWi,
+ wp,)/2
+ (Wij - Wji) xj - 2 v,, xi Xi),,
+ (wip
+ (Wij
- wp,)
- w,J
(xp
- xi/2)
(3)
xj
- 4 vIp xi xp)
x,
xj - 3 Vij xi Xj} xi xj
The v.. is an additional interaction parameter of the binary system i - j. By defmition, v.. = v . to the subregular formulas for the case v.. 0, whrch may b$ further reEqs. (Y, and (4) are reduced a very condensed form of duced to regular formulas for the case w. = w... Eqs. (3) and (4) consyitute the algebraically equivalent four - suffix’bolynd’mial equations given by Benedict et al. (2). q
Received
May
20,
1983
67
R.
68
This is advantageous the condensed form.
for practical In addition,
applications, the parameters
SCHMID
because much less computational work must be done with can be represented in a 2 - dimensional matrix instead
of a 4 - dimensional one. An explicit but arbitrary setting of the parameters is illustrated in a simple FORTRAN program, Table I, for the calculation of activities and activity coefficients according to Eq. (4). It is not recommended to extend this “subsubregular” equation further, because higher order polynomials usually tend to oscillate around the experimental data points - at least up to a managable polynomial order. If the accuracy is still unsufficient one should consider specific interactions of the components like in an associated solution or a sublattice model. The Gibbs - Duhem consistency of Eqs. (3) and (4) can be proved as outlined below. The mole fractions in Eq. (3) are substituted by the mole numbers of component i, ni, and the total mole number, n o, according to xi = ni/no. Then, by definition of the partial quantity, we have
In Xp
=z
at n.,
(noAGXS/RT)
= const,
i + p
(5)
P With the use of an /an double summation,“coll&t
= I and %./an terms, iniert
= 0 or 1, if i j= p or i = p, we can carry out parts of the ?he mole fractions again and we end up with Eq. (4). References
1.
Kellogg, H. H.: Physical Chemistry in Metallurgy, eds., p. 49, U. 5. Steel Research Lab., Monroeville,
2.
Benedict,
M;
Johnson,
C.
A.;
Solomon,
E.;
Rubin, Table
R. M. Fischer, PA (1976) L. C.:
Trans.
R.
Am.
A. Oriani
Inst.
Chem.
and E. T. Turkdogan,
Eng.
41,
371/81
1
FORTRAN - program for the calculation of activities and activity coefficients according which illustrates an arbitray setting of interaction parameters in 2 - dimensional matrices
to Eq. (4) W and V.
(1945)
Vol.8,
Printed
the
in
No.1, USA.
EXTENSION
67-68,
pp.
1984
0364-5916/84
(c)
OF A SUBREGULAR TO
A
EQUATION
FOR
MULTlCOMPONENT
” SUBSUBRECULAR”
Rarner
1984
Pergamon
$3.00 + .oo Press Ltd.
SYSTEMS
ONE
Schmid
lnstitut fur Ersenhuttenkunde und Grefiererwesen Technische Universitat Clausthal D - 3392 Clausthal-Zellerfeld, West Germany
ABSTACT.
A subregular equation for the description of mrxmg effects from binary parameters is extended to m&de an additional, This formula is much more useful for practical applications suffix polynomial equation.
For the description of asymmetric types of polynomial representations been given by Kellogg (I):
A
G”
= (l/2)
Rr In
= $
Xp
-5
t
t{w,j
j=l
i=l
quantrtres in a binary or multicomponent solution many A very practical formula for a n-component system has
+ (w
{ Wij/2
‘1
( (WIP + wp,)/2
2
]=I
excess exrst.
In a multicomponent system optional binary parameter. than the correspondmg four-
wjl)
‘11’1
+ (WIP - wp,)
+ (w,,
- Wj,)
x,}
’
(xp
(I)
I
- Xi/Z))
x,
(2)
xi xi
i=l
IS the excess Gibbs energy of mixing, R the gas constant, T the absolute temperature, v where AC” the activity coefficient of component p - the standard state bemg pure component p in the same p&se and w are interaction parameters of the binary system and x. the mole fraction of component I. The w i - j. if a temperature dependence like wij = a :‘b/T 1s gssumed, the heat and excess entropy of mixrng are given by formulas analogous to Eqs. (I) and (2). However, there are cases where the complexrty of the system and the amount of experrmental data requrre an additional parameter for the descriptron of at least some of the brnaries. Such an extension of the subregular Eqs. (1) and (2) to a “subsubregular” case IS given below: A
Gxs
r
In
yp
= (l/2)
= g
-2 j=l
fj
{ (wip
f{Wij/2 I=1
s
fWi,
+ wp,)/2
+ (Wij - Wji) xj - 2 v,, xi Xi),,
+ (wip
+ (Wij
- wp,)
- w,J
(xp
- xi/2)
(3)
xj
- 4 vIp xi xp)
x,
xj - 3 Vij xi Xj} xi xj
The v.. is an additional interaction parameter of the binary system i - j. By defmition, v.. = v . to the subregular formulas for the case v.. 0, whrch may b$ further reEqs. (Y, and (4) are reduced a very condensed form of duced to regular formulas for the case w. = w... Eqs. (3) and (4) consyitute the algebraically equivalent four - suffix’bolynd’mial equations given by Benedict et al. (2). q
Received
May
20,
1983
67
R.
68
This is advantageous the condensed form.
for practical In addition,
applications, the parameters
SCHMID
because much less computational work must be done with can be represented in a 2 - dimensional matrix instead
of a 4 - dimensional one. An explicit but arbitrary setting of the parameters is illustrated in a simple FORTRAN program, Table I, for the calculation of activities and activity coefficients according to Eq. (4). It is not recommended to extend this “subsubregular” equation further, because higher order polynomials usually tend to oscillate around the experimental data points - at least up to a managable polynomial order. If the accuracy is still unsufficient one should consider specific interactions of the components like in an associated solution or a sublattice model. The Gibbs - Duhem consistency of Eqs. (3) and (4) can be proved as outlined below. The mole fractions in Eq. (3) are substituted by the mole numbers of component i, ni, and the total mole number, n o, according to xi = ni/no. Then, by definition of the partial quantity, we have
In Xp
=z
at n.,
(noAGXS/RT)
= const,
i + p
(5)
P With the use of an /an double summation,“coll&t
= I and %./an terms, iniert
= 0 or 1, if i j= p or i = p, we can carry out parts of the ?he mole fractions again and we end up with Eq. (4). References
1.
Kellogg, H. H.: Physical Chemistry in Metallurgy, eds., p. 49, U. 5. Steel Research Lab., Monroeville,
2.
Benedict,
M;
Johnson,
C.
A.;
Solomon,
E.;
Rubin, Table
R. M. Fischer, PA (1976) L. C.:
Trans.
R.
Am.
A. Oriani
Inst.
Chem.
and E. T. Turkdogan,
Eng.
41,
371/81
1
FORTRAN - program for the calculation of activities and activity coefficients according which illustrates an arbitray setting of interaction parameters in 2 - dimensional matrices
to Eq. (4) W and V.
(1945)