Page 87 - Separation process principles 2

52 Chapter 2 Thermodynamics of Separation Operations

Table 2.9 Empirical and Semitheoretical Equations for Correlating Liquid-Phase Activity Coefficients of Binary Pairs
Name Equation for Species 1 Equation for Species 2

(1) Margules log yl = AX; log y2 = AX:
(2) Margules (two-constant) log yl = x;[AI2 + 2xl(Az1 - AI2)1 1% = x?[A21 + 2~2(A12 - A2111
,,.\ 7 , A 12 1. A21
(3) van Laar (two-constant) lnyl = 1ny2 =
[I + (xIAIz)~(xzAzI)I~ [1 + (xzAzI)~(xIAI~)I~
(4) Wilson (two-constant) 111 y1 = - ln(xl + A 12x2) 1ny.r = - ln(x2 + 1221x1)
+x2 ( 1212 - 1\12 -

xi + A12xz xz + A21xi XI + A12~2 ~2 + AZIXI
x;nl~:, + x:.~IzG~z x:.r12Gh + xi721 GZI
(5) NRTL (three-constant) lnyl = 1ny2 =
(XI
(xi + x2G21I2 (x2 $11 G1d2 (XZ + XIG~~)~ +x2G21I2
Gij = exp(-aij.rij) Gij = exp(-aij.rij)

1 01 2 02
(6) UNIQUAC (two-constant) In yl = In - + -91 In - Inn = In - + -q21n -
XI 2 q1 12 2 qz
- 92 ln(02 + 01T12)
T12 -
+ 'lq2 (02 + 01 T12 01 + 02T21

Margules Equations
constants, shown as AI2 and Azl in (3) of Table 2.9, are best
The Margules equations (1) and (2) in Table 2.9 date back back-calculated from experimental data. These constants
to 1895, and the two-constant form is still in common use are, in theory, constant only for a particular binary pair at a
because of its simplicity. These equations result from power- given temperature. In practice, they are frequently computed
series expansions in mole fractions for jf and conversion to from isobaric data covering a range of temperature. The van
activity coefficients by means of (2-59). The one-constant Laar theory expresses the temperature dependence of Aij as
form is equivalent to symmetrical activity-coefficient A!
curves, which are rarely observed experimentally. A,. - '.' (2-70)
" - RT
van Laar Equation
Regular-solution theory and the van Laar equation are
Because of its flexibility, simplicity, and ability to fit many equivalent for a binary solution if
systems well, the van Laar equation is widely used. It was
derived from the van der Waals equation of state, but the A. - --(& V~L - 6.)2
1 J (2-7 1)
The van Laar equation can fit activity coefficient-
Table 2.10 Classical Partial Molar Excess Functions of composition curves corresponding to both positive and
Thermodynamics negative deviations from Raoult's law, but cannot fit curves

Excess volume: that exhibit minima or maxima such as those in Figure 2.1%.
When data are isothermal, or isobaric over only a narrow
(I) (ztL -n)p) =ifL RT(+) range of temperature, determination of van Laar constants is
=
T,x
Excess enthalpy: conducted in a straightforward manner. The most accurate
procedure is a nonlinear regression to obtain the best fit to
(2) (hlL - iip) E ipL = -RT~ -
(a p,x the data over the entire range of binary composition, subject
Excess entropy: to minimization of some objective function. A less accurate,
but extremely rapid, manual-calculation procedure can be
(3) (5 - = iFL = - R T - used when experimental data can be extrapolated to infinite-
i;:)
+ 1nYtL]
[ (a ;;L),,~
dilution conditions. Modem experimental techniques are
ID = ideal mixture; E = excess because of nonideality. available for accurately and rapidly determining activity