Legendre polynomial expansions of thermodynamic properties of binary solutions
- PDF / 550,199 Bytes
- 7 Pages / 594 x 774 pts Page_size
- 46 Downloads / 351 Views
I.
INTRODUCTION
SEVERAL years ago, in this journal we proposed the use of orthogonal Legendre polynomials for the representation of excess thermodynamic properties of binary solutions. ~ Although Legendre series have since been adopted by a number of authors, their widespread use has been hindered by the fact that some of the mathematical relationships given in the original article were unnecessarily complex. In particular, it was not shown how partial excess properties could easily be expressed with the same set of coefficients as was used for the integral excess property. In the present article, the advantages of Legendre series are reiterated. In particular, it is shown how the use of Legendre series can aid in the search for empirical correlations among the thermodynamic properties of groups of similar systems. The Legendre series are reformulated, and simple relationships permitting partial and integral properties to be calculated from one single set of coefficients are presented. The calculation of Legendre polynomials via a simple recursion relationship is also described in detail. Finally, explicit relationships permitting conversion from simple power series or Redlich-Kister series to Legendre series are given.
II. ADVANTAGES OF L E G E N D R E POLYNOMIALS In a binary system with components A and B, let w e be any integral excess thermodynamic property (such as excess Gibbs energy, G e, excess enthalpy, H E, or excess entropy, Se). It is customary to express coe as a simple empirical power series expansion in the mole fractions XA and XB: A. Simple Power Series w E = X a X s ( q o + qIXB + q2X 2 + . . . )
B. Redlich-Kister Expansion = X,~XB(ko + klzB + k2z~ + k3z~ + . . . )
[2]
where: [1]
The disadvantage of a simple power series representation is the interdependence (correlation) of the coefficients q,. In Eq. [1], all the terms q, Xg have a maximum absolute value ARTHUR D. PELTON and CHRISTOPHER W. BALE are with Centre de Recherche en Calcul Thermochimique, Ecole Polytechnique de Montrral, P.O. Box 6079, Station A, Montreal, PQ H3C 3A7, Canada. Manuscript submitted October 2, 1985. METALLURGICALTRANSACTIONSA
at X~ = 1.0 and all are zero at XB = 0. Therefore, as n becomes progressively larger, the terms q,X~ become very small near XB = 0, but are many orders of magnitude larger in absolute value near X8 = 1.0. The result is that, as n increases, the absolute value of q, must become very large in order that this term can contribute to the total summation when XB is close to zero. As the total number of coefficients is increased, the absolute values of q, increase dramatically. This necessitates the storage and manipulation of variables to a large number of significant digits, since the total summation involves taking small differences between very large numbers. It is also clear that no significance, either mathematical or physical, can be attached to the numerical values of the coefficients. Because of the high interdependence (correlation) between coefficients, simply adding one more term to the
Data Loading...
