CHEE 311 Lecture 17 1

Excess Gibbs Energy Models

Purpose of this lecture:

To introduce some popular empirical models (Margules, van Laar) that can be used toward the calculation of activity coefficients in binary mixtures

Highlights

• The Margules and van Laar (see Lecture 18) are two-parameter excess Gibbs energy models of empirical nature• They are derived by expressing GE/RT x1 x2 in a power series form • They are applicable to binary mixtures• They neglect the effect of pressure; this limits their applicability in the range of low to moderate P

Reading assignment: Sections 12.1 and 12.2

CHEE 311 Lecture 17 2

Excess Gibbs Energy ModelsPracticing engineers find most of the liquid-phase information needed for equilibrium calculations in the form of excess Gibbs Energy models. These models:

reduce vast quantities of experimental data into a few empirical parameters,provide information an equation format that can be used in thermodynamic simulation packages (Provision)

“Simple” empirical modelsSymmetric, Margules, van LaarNo fundamental basis but easy to useParameters apply to a given temperature, and the models usually cannot be extended beyond binary systems.

Local composition modelsWilson, NRTL, UniquacSome fundamental basisParameters are temperature dependent, and multi-component behaviour can be predicted from binary data.

CHEE 311 Lecture 17 3

Excess Gibbs Energy ModelsOur objectives are to learn how to fit Excess Gibbs Energy models to experimental data, and to learn how to use these models to calculate activity coefficients.

⎟⎟⎠

⎞⎜⎜⎝

⎛=γ sat

11

11 Px

Pylnln

⎟⎟⎠

⎞⎜⎜⎝

⎛=γ sat

22

22 Px

Pylnln

2211E lnxlnxRT/G γ+γ=

CHEE 311 Lecture 17 4

Margules’ EquationsWhile the simplest Redlich/Kister-type expansion is the Symmetric Equation, a more accurate model is the Margule’s expression:

(12.9a)

Note that as x1 goes to zero,

and from L’hopital’s rule we know:

therefore,

and similarly

21212121

ExAxA

xRTxG

+=

∞

→γ= 1

210xln

xRTxG

limE

1

120x21

E

AxRTx

G

1

=→

∞γ= 112 lnA ∞γ= 221 lnA

CHEE 311 Lecture 17 5

Margules’ EquationsIf you have Margules parameters, the activity coefficients are easily derived from the excess Gibbs energy expression:

(12.9a)

to yield:

(12.10ab)

These empirical equations are widely used to describe binary solutions. A knowledge of A12 and A21 at the given T is all we require to calculate activity coefficients for a given solution composition.

21212121

ExAxA

xRTxG

+=

]x)AA(2A[xln 1122112221 −+=γ

]x)AA(2A[xln 2211221212 −+=γ

CHEE 311 Lecture 17 6

Example 1You desire to separate an equimolar binary mixture of n-pentane (1) and acetone (2) by feeding it into a flash drum that operates at T=24 oC and P=50 kPa. Using information provided below, determine whether or not separation of the mixture can be accomplished under these operating conditions.

- DewP = 45 kPa (at T= 24 oC) - P1

sat(24 oC )=65.0 kPa; P2sat(24 oC )=31.0 kPa

- Reduced experimental P-x-y data for this mixture (GE/RTx1x2 vs. x1) are given in Figure 1. - The activity coefficients can be calculated from the Margules model - Due to low pressures involved, you can assume here that all fugacity coefficients and Poynting

factors are equal to one.

P-x-y data for the mixture n-pentane (1)/acetone (2)