## On Jjpj

where zi can be xi or yi and (aa)ij = (1 — kjj V(aa)i(aa)j. kij is a binary interaction parameter that is usually obtained by fitting experimental VLE data.

The fugacity coefficient can be obtained by substituting eqns [31]-[44] into eqn [6] leading to:

A number of expressions have been proposed for gE as a function of composition. Some of the more popular of these are outlined below.

Margules Equation

The Margules equation is one of the simplest expressions for the molar excess Gibbs energy. For a binary solution:

where A12 and A21 are binary parameters dependent on temperature, but not on the composition. The Margules activity coefficients in a binary mixture are obtained by differentiation of eqn [47] and are given by:

A12 and A21 are generally obtained by fitting VLE data. Note that the value of the activity coefficient of each component tends to unity as the mole fraction of that component goes to unity. This behaviour is inherent in all gE models. The Margules equation works well for binary systems in which the two components are very similar in size, shape and chemical nature. Margules parameters for a large number of systems are tabulated in dechema books on VLE data.

Van Laar Equation

The van Laar equation for the excess Gibbs energy may be written as:

1 A12 X1

A21 X2

A21 x2 A12 X1

the molar liquid volume v of the pure component i, and the energies of interaction between the molecules i and j as follows:

The expression for the liquid activity coefficients are:

where q, and q2 are the effective volumes of the two molecules and a12 is an interaction parameter. Differentiation according to eqn [46] leads to the following expressions for the activity coefficients:

The Wilson equation has two parameters A12 and A21 (or equivalently, i12 — i11 and i21 — i22) and is able to correlate VLE data for a wide variety of miscible systems, including those containing polar or associating components in nonpolar solvents. However, the equation is incapable of predicting liquid-liquid immiscibility in a system.

For multicomponent mixtures, the Wilson equation is written as follows:

where A,2 = 2q1a12 and A2, = q2a,2. As in the case of the Margules equation, the two parameters A,2 and A2, are obtained by fitting VLE data. The van Laar equations have been shown to work well for a number of binary systems where the size, shape and chemical nature of the components are dissimilar, and parameters for many binary systems have been tabulated in the dechema data books.

### Wilson Equation

The Margules equation is based on the assumption that the ratio of species ! to species 2 molecules surrounding any molecule is the same as the ratio of the mole fractions of species , and 2. A different class of g models has been proposed based on the assumption that, around each molecule, there is a local composition that is different from the bulk composition. The Wilson equation is such a local composition model and the Wilson excess Gibbs energy has the following form for a binary system:

—— = x, ln(x, + A,2x2) — x2 ln(x2 + A^x,) [53] RT

where A,2 and A2, are parameters specific to the binary pair. These parameters are defined in terms of ln 7k =— ln

X xiAkj

xiAik

X xiAii

Note that only binary parameters Aij are required to evaluate activity coefficients in multicomponent systems. These parameters are obtained by fitting VLE data for the binary pairs, and many of these parameters have been tabulated in the dechema data books. Moreover, because a temperature dependence is included in eqn [54], the same binary parameters may be used over a range of temperatures (although no more that about 50 K).

### NRTL Equation

The NRTL (non-tandom two-liquid theory) equation is also based on a local composition model for the excess Gibbs energy. However, it is applicable to miscible as well as partially miscible systems due to the inclusion of a third binary parameter in the model. The expression for the molar excess Gibbs energy is:

12,02

gij gjj

y T12G1

G21x2 ( x2#G12x1)

x2#G12x1 (x1#G21x2)

X TjiGjiXj m x G

^(combinatorial) $2

RT x1 x2

gij describes the energy of interaction between component i and / and Cij (=j is a nonrandomness parameter which is often set equal to 0.3. Thus, only two parameters tij and x)l (or, equivalently, gij — g and g — gii) are required per binary pair.

The activity coefficients expressions are as follows:

^(residual)

where:

xiri y 1

j xiqi y xiqi tji = expi

A major advantage of the NRTL equation lies in its ability to represent highly nonideal systems, particularly partially miscible systems.

For multicomponent mixtures, the liquid-phase activity coefficients are expressed as:

In eqn [65] z is a coordination number ( = 10 usually), are volume fractions, and 6i are surface area fractions for component i. The volume and surface area parameters r and qi can be evaluated from pure component molecular structure information and are tabulated in the dechema data books. Thus, there are two binary parameters a^ and afl in the uniquac model and these are found by fitting binary VLE data. The activity coefficient expressions become:

As with the Wilson equation, only binary data are needed to calculate activity coefficients in multicom-ponent systems, and these parameters have been tabulated in the dechema data books for many systems. Furthermore, because of the inclusion of the temperature in eqn [60] the parameters obtained by fitting VLE data at one temperature may be used to calculated VLE at other temperatures (within a range of about 50 K).

uniquac Equation

The uniquac (universal quasi-chemical theory) equation expresses the molar excess Gibbs energy as a sum of a combinatorial part and residual part.

The combinatorial part accounts for differences in the size and shape of the molecules, whereas the residual contribution accounts for energetic interactions.

where:

The uniquac equation is applicable to a wide range of systems, including partially miscible systems.

For multicomponent systems, the uniquac equation becomes:

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