7. Liquid Phase Properties from VLE Data (11.1)

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7. Liquid Phase Properties from VLE Data (11.1)

• The fugacity of non-ideal liquid solutions is
  defined as
• (10.42)
• from which we derive the concept of an activity
  coefficient
• (10.89)
• that is a measure of the departure of the
  component behaviour from an ideal solution.
• Using the activity coefficient, equation 10.42
  becomes
• How do we calculate/measure these properties?

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Liquid Phase Properties from VLE Data

• Suppose we conduct VLE experiments on our system
  of interest.
• At a given temperature, we vary the system
  pressure by changing the cell volume.
• Wait until equilibrium is established (usually
  hours)
• Measure the compositions of the liquid and vapour

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Liquid Solution Fugacity from VLE Data

• Our understanding of molecular dynamics does not
  permit us to predict non-ideal solution fugacity,
  fil . We must measure them by experiment, often
  by studies of vapour-liquid equilibria.
• Suppose we need liquid solution fugacity data for
  a binary mixture of AB at P,T. At equilibrium,
• The vapour mixture fugacity for component i is
  given by,
• (10.47)
• If we conduct VLE experiments at low pressure,
  but at the required temperature, we can use the
  perfect gas mixture model,
• by assuming that ?iv 1.

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Liquid Solution Fugacity from Low P VLE Data

• Since our experimental measurements are taken at
  equilibrium,
• according to the perfect gas
  mixture model
• What we need is VLE data at various pressures
  (all relatively low)

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Activity Coefficients from Low P VLE Data

• With a knowledge of the liquid solution fugacity,
  we can derive activity coefficients. Actual
  fugacity
• Ideal solution fugacity
• Our low pressure vapour fugacity simplifies fil
  to
• and if P is close to Pisat
• leaving us with

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Activity Coefficients from Low P VLE Data

• Our low pressure VLE data can now be processed to
  yield experimental activity coefficient data

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Activity Coefficients from Low P VLE Data
8
7. Correlation of Liquid Phase Data

• The complexity of molecular interactions in
  non-ideal systems makes prediction of liquid
  phase properties very difficult.
• Experimentation on the system of interest at the
  conditions (P,T,composition) of interest is
  needed.
• Previously, we discussed the use of low-pressure
  VLE data for the calculation of liquid phase
  activity coefficients.
• As practicing engineers, you will rarely have the
  time to conduct your own experiments.
• You must rely on correlations of data developed
  by other researchers.
• These correlations are empirical models (with
  limited fundamental basis) that reduce
  experimental data to a mathematical equation.
• In CHEE 311, we examine BOTH the development of
  empirical models (thermodynamicists) and their
  applications (engineering practice).

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Correlation of Liquid Phase Data

• Recall our development of activity coefficients
  on the basis of the partial excess Gibbs energy
• where the partial molar Gibbs energy of the
  non-ideal model is provided by equation 10.42
• and the ideal solution chemical potential is
• Leaving us with the partial excess Gibbs energy
• (10.90)

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Correlation of Liquid Phase Data

• The partial excess Gibbs energy is defined by
• In terms of the activity coefficient,
• (10.94)
• Therefore, if as practicing engineers we have GE
  as a function of P,T, xn (usually in the form of
  a model equation) we can derive ?i.
• Conversely, if thermodynamicists measure ?i, they
  can calculate GE using the summability
  relationship for partial properties.
• (10.97)
• With this information, they can generate model
  equations that practicing engineers apply
  routinely.

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Correlation of Liquid Phase Data

• We can now process this our MEK/toluene data one
  step further to give the excess Gibbs energy,
• GE/RT x1ln?1 x2ln?2

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Correlation of Liquid Phase Data

• Note that GE/(RTx1x2) is reasonably represented
  by a linear function of x1 for this system. This
  is the foundation for correlating experimental
  activity coefficient data

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Correlation of Liquid Phase Data

• The chloroform/1,4-dioxane system exhibits a
  negative deviation from Raoults Law.
• This low pressure VLE data can be processed in
  the same manner as the MEK/toluene system to
  yield both activity coefficients and the excess
  Gibbs energy of the overall system.

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Correlation of Liquid Phase Data

• Note that in this example, the activity
  coefficients are less than one, and the excess
  Gibbs energy is negative.
• In spite of the obvious difference from the
  MEK/toluene system behaviour, the plot of
  GE/x1x2RT is well approximated by a line.

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8.4 Models for the Excess Gibbs Energy

• Models that represent the excess Gibbs energy
  have several purposes
• they reduce experimental data down to a few
  parameters
• they facilitate computerized calculation of
  liquid phase properties by providing equations
  from tabulated data
• In some cases, we can use binary data (A-B, A-C,
  B-C) to calculate the properties of
  multi-component mixtures (A,B,C)
• A series of GE equations is derived from the
  Redlich/Kister expansion
• Equations of this form fit excess Gibbs energy
  data quite well. However, they are empirical and
  cannot be generalized for multi-component (3)
  mixtures or temperature.

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Symmetric Equation for Binary Mixtures

• The simplest Redlich/Kister expansion results
  from CD0
• To calculate activity coefficients, we express GE
  in terms of moles n1 and n2.
• And through differentiation,
• we find

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7. Excess Gibbs Energy Models

• Practicing 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 models
• Symmetric, Margules, vanLaar
• No fundamental basis but easy to use
• Parameters apply to a given temperature, and the
  models usually cannot be extended beyond binary
  systems.
• Local composition models
• Wilsons, NRTL, Uniquac
• Some fundamental basis
• Parameters are temperature dependent, and
  multi-component behaviour can be predicted from
  binary data.

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Excess Gibbs Energy Models

• Our 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.

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Margules Equations

• While the simplest Redlich/Kister-type expansion
  is the Symmetric Equation, a more accurate model
  is the Margules expression
• (11.7a)
• Note that as x1 goes to zero,
• and from Lhopitals rule we know
• therefore,
• and similarly

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Margules Equations

• If you have Margules parameters, the activity
  coefficients are easily derived from the excess
  Gibbs energy expression
• (11.7a)
• to yield
• (11.8ab)
• 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.

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van Laar Equations

• Another two-parameter excess Gibbs energy model
  is developed from an expansion of (RTx1x2)/GE
  instead of GE/RTx1x2. The end results are
• (11.13)
• for the excess Gibbs energy and
• (11.14)
• (11.15)
• for the activity coefficients.
• Note that as x1?0, ln?1? ? A12
• and as x2 ? 0, ln?2? ? A21

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Local Composition Models

• Unfortunately, the previous approach cannot be
  extended to systems of 3 or more components. For
  these cases, local composition models are used to
  represent multi-component systems.
• Wilsons Theory
• Non-Random-Two-Liquid Theory (NRTL)
• Universal Quasichemical Theory (Uniquac)
• While more complex, these models have two
  advantages
• the model parameters are temperature dependent
• the activity coefficients of species in
  multi-component liquids can be calculated from
  binary data.
• A,B,C A,B A,C B,C
• tertiary mixture binary binary binary

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Wilsons Equations for Binary Solution Activity

• A versatile and reasonably accurate model of
  excess Gibbs Energy was developed by Wilson in
  1964. For a binary system, GE is provided by
• (11.16)
• where
• (11.24)
• Vi is the molar volume at T of the pure
  component i.
• aij is determined from experimental data.
• The notation varies greatly between publications.
  This includes,
• a12 (?12 - ?11), a12 (?21 - ?22) that you
  will encounter in Holmes, M.J. and M.V. Winkle
  (1970) Ind. Eng. Chem. 62, 21-21.

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Wilsons Equations for Binary Solution Activity

• Activity coefficients are derived from the excess
  Gibbs energy using the definition of a partial
  molar property
• When applied to equation 11.16, we obtain
• (11.17)
• (11.18)

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Wilsons Equations for Multi-Component Mixtures

• The strength of Wilsons approach resides in its
  ability to describe multi-component (3) mixtures
  using binary data.
• Experimental data of the mixture of interest (ie.
  acetone, ethanol, benzene) is not required
• We only need data (or parameters) for
  acetone-ethanol, acetone-benzene and
  ethanol-benzene mixtures
• The excess Gibbs energy is written
• (11.22)
• and the activity coefficients become
• (11.23)
• where ?ij 1 for ij. Summations are over all
  species.

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Wilsons Equations for 3-Component Mixtures

• For three component systems, activity
  coefficients can be calculated from the following
  relationship
• Model coefficients are defined as (?ij 1 for
  ij)

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Comparison of Liquid Solution Models
Activity coefficients of 2-methyl-2-butene
n-methylpyrollidone. Comparison of experimental
values with those obtained from several equations
whose parameters are found from the
infinite-dilution activity coefficients. (1)
Experimental data. (2) Margules equation. (3)
van Laar equation. (4) Scatchard-Hamer equation.
(5) Wilson equation.