The polynomial representation of thermodynamic properties in dilute solutions
- PDF / 235,697 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 26 Downloads / 257 Views
TRODUCTION
THE well-known first-order interaction parameter formalism of Wagner[1] for a dilute solution of components 1-2-3 . . . -N, where component 1 is the solvent, may be written as ln gi 5 ln g i0 1 «i 2 X2 1 «i 3 X3 1 ... 1 «i N XN (i ≥ 2)
[1]
where gi is the activity coefficient of solute i defined as
gi 5 ai /Xi
[2]
where ai is the activity of i and Xi is the mole fraction of i defined as Xi 5 ni /(n1 1 n2 1 ... 1 nN )
[3]
where ni is the number of moles of i in solution. The activity coefficient at infinite dilution (where X1 → 1) is denoted by g 0i . The parameters εij are called ‘‘first-order interaction parameters.’’ From the Gibbs–Duhem equation,
Σ X d ln g 5 0 i
[4]
i
it can be shown that in the limit at infinite dilution, «i j 5 «j i
[5]
and 1 ln g1 5 2 2
ΣΣ« N
N
i52 j52
ij
Xi Xj
[6]
The Wagner formalism is very simple and useful but has frequently been misunderstood or misapplied. Much of the confusion has arisen because Eqs. [1] and [6] only obey the Gibbs–Duhem equation in the limit of infinite dilution, and because the formalism does not obey the following
ARTHUR D. PELTON, Professor, is with the Centre de Recherche en Calcul Thermochimique (Center for Research in Computational Thermochemistry), E´cole Polytechnique de Montre´al, Montre´al, PQ, Canada H3C 3A7. Manuscript submitted November 21, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
necessary thermodynamic relationship except at infinite dilution: (] ln gi /]nj ) 5 (] ln gj /]ni )
[7]
This problem was simply resolved many years ago for the case of the first-order formalism for 2- and 3-component systems by Darken[2,3] in two articles, which have not received sufficient attention. Darken’s Quadratic Formalism In a binary system 1-2 (where 1 is the solvent), Darken proposed a ‘‘quadratic formalism’’: gE/RT 5 a12 X1 X2 1 C2 X2
[8]
where gE is the excess molar Gibbs energy:
Σ X ln g N
gE/RT 5
i51
i
i
[9]
and where aij and C2 are constants. By differentiation, ln g2 5 a12 X 12 1 C2 5 (a12 1 C2) 2 2a12 X2 1 a12 X 22 ln g1 5 a12 X 22
[10] [11]
When C2 5 0, the quadratic formalism reduces to the regular solution equations. It has been observed that many simple binary solutions conform quite closely to the quadratic formalism up to relatively large values of X2. This observation can be rationalized by a simple regular solution model. Assume a random distribution of atoms or molecules of components 1 and 2 on a quasilattice. Let Eij be the bond energy of one mole of i-j nearest-neighbor pairs. The probabilities that a given pair is a 1-1, a 2-2, or a 1-2 pair are X21, X22, and 2 X1 X2, respectively. The excess Gibbs energy is then equal to the sum of the energies of the pair bonds in 1 mole of solution, less the energy of the 1-1 bonds in X1 moles of pure component 1 and the energy of the 2-2 bonds in 1 mole of pure component 2: VOLUME 28B, OCTOBER 1997—869
gE 5 (Z/2)(2X1 X2 E12 1 X 12 E11 1 X 22 E22) 2 (Z/2) X1 E11 2 (Z 0/2) X2 E 022 5 (Z/2)(2E12 2 E11 2 E22) X1 X2 1 X2 (ZE22 2 Z 0 E 022)/2
[12]
where Z is t
Data Loading...
