Thermodynamics of binary systems using interaction parameters
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I.
INTRODUCTION
VARIOUS analytical equations have been proposed to interpret properties as a function of composition. Since important thermodynamic parameters, such as infinite dilution constants and first- and higher order interaction parameters, m are basically extrapolated quantities at infinite dilution, analytical equations are important in determining them. The proposed functions must, however, satisfy closely the experimental data for determination of the above parameters. These parameters are important in interpreting properties of the binary system, and their use may be extended to high-order systems. The interaction parameters are normally determined from the Maclaurin infinite series which is expressed in terms of the partial property in the neighborhood of a solvent component of a system, t2] Since a partial function of a binary system may readily be determined from the other by the Gibbs-Duhem relation, the choice of an integral or a partial property for expressing the Maclaurin series does not in any way restrict the thermodynamic consistency due to truncation of the series. The truncation, however, limits the applicability of the function beyond the dilute solution range. Such a function does not fulfill the necess .ary boundary conditions unless further assumptions are made with regard to the interaction parameters. These boundary conditions are expressed as
AG xs ---* 0
as Xl --~ 1.0
AG xs-* O asX2--~ 1.0 In order to develop a consistent methodology so that the present method may be extended to ternaries and higher order systems, the Maclaurin infinite series is expressed based on the integral property of the system. Furthermore, the infinite series is expressed separately in the neighborhood of each of the pure components and subjected to appropriate boundary conditions. The derivatives of the above series are then converted to the J.P. HAJRA, Associate Professor, and B. MAZUMDAR, Research Scholar, are with the Department of Metallurgy, Indian Institute of Science, Bangalore 560 012, India. Manuscript submitted June 1, 1990.
METALLURGICALTRANSACTIONSB
corresponding infinite dilution constants and interaction parameters, as shown in the text, with the development of a new function. This article investigates the applicability of the present form of function to the experimental data of relatively weakly interacting binary systems and attempts to determine the above parameters. II.
THEORETICAL CONSIDERATIONS
The application of the Maclaurin infinite series as X~ ~ 1.0 and X2 ~ 1.0 of a binary system based on integral function may be expressed, respectively, as
Acxs= (a.%:.,.o
+
+-
2
1 [~3AGXS~
3
_/__._Z77_.3 /
+ 3! \
6X2 /x,--,lo
X2 J r . . . .
[1]
and
SAGXS~ A G XS = (AGXS)x2...+,.o -l- ~ ] x 2 _ _ _ + l . O
f
1 [t$2AGXS~
2
+ -/-'--Z77-,2 /
X~
2 \
6Xl
]x2-~l.o
1 [t~3AGXS~ -- I----gUT-,3/
+ 3! \
x'
3 X 1
+ ...
fiX1 Jx2-~l.O Imposing the boundary condition AG xs ~ 0 as X~
[2]
1.0, infinite series [1] reduces to
1 (rrAGXS'~ --
E,=, r,
=
\~]x,--,,o
0
Similarly,
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