8. Solute (1) / Solvent (2) Systems 12.7 SVNA

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8. Solute (1) / Solvent (2) Systems 12.7 SVNA

• Until now, all the components we have considered
  in VLE calculations have been below their
  critical temperature. Their pure component
  liquid fugacity is calculated using
• Our VLE equation that describes the distribution
  of each component between liquid and vapour has
  the form
• How do we deal with components that, at the
  temperature of interest, are above Tc and no
  longer have a Pisat?

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VLE Above the Critical Point of Pure Components
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Pure Species Fugacity of a Solute

• The difficulty in handling a component that is
  above its critical temperature or simply unstable
  as a pure liquid is to define a pure component
  fugacity for the purpose of VLE calculations.
• While this component must
• have a liquid solution
• fugacity, f1l, it does not have
• a pure liquid fugacity, f1l at
• x1 1.
• The tangent line at x10 is
• the Henrys constant, k1.
• It is useful for predicting the
• mixture fugacity of a dilute
• component, but it cannot
• be extrapolated to x11
• with any degree of accuracy.

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Pure Component Fugacity of a Solute

• The pure component fugacity of a solute is
  calculated from a combination of Henrys Law and
  an activity coefficient model.
• Recall that Henrys Law may be used to represent
  the mixture fugacity of a minor (xilt0.02)
  component in a liquid.
• defines the Henrys
• constant
• and
• is accurate as long
• as x1 lt 0.02
• Unfortunately, we cannot extrapolate the above
  equation to x1 1 to give us the pure component
  f1.
• An activity coefficient model can refine this
  approach

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Pure Component Fugacity of a Solute

• Recall that the activity coefficient is the ratio
  of the mixture fugacity of a component to its
  ideal solution fugacity
• At infinite dilution (x1?0), the activity
  coefficient becomes
• Since the pure component fugacity is a constant
  at a given T, we can write this expression as
• Using the definition of the Henrys Constant, ki,
  we have
• or 12.34

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Pure Component Fugacity of a Solute

• Equation 12.34 is a rigorous thermodynamic
  equation,
• 12.34
• for the fugacity of a pure solute. However, it
  is evaluated at P2sat (where x1 0) and its use
  requires us to assume that pressure has an
  insignificant influence on the solutes fugacity.
• To apply 12.34, we require a Henrys constant for
  the system at the temperature of interest, ki(T),
  and an excess Gibbs energy model for the system,
  also at the T of interest.

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VLE Relationship for a Supercritical Component

• Consider a system where one component is above Tc
  (species 1) and the other component is below Tc
  (species 2).
• The equilibrium relationship for component 2 is
  unchanged
• or
• However, component 1 is handled differently,
  using a Henrys constant (k1) and the infinite
  dilution activity coefficient (?1?). Both are
  properties specific to this mixture.
• 12.36

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Solute (1) / Solvent (2) Systems Example
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9. Phase Stability and Liquid-Liquid Equilibria

• Throughout the course we have developed methods
  of calculating the thermodynamic properties of
  different systems
• Gibbs energy of pure vapours and liquids
• Gibbs energy of ideal and real mixtures
• Definition of vapour liquid equilibrium
  conditions
• As we apply these methods, we assume that the
  phases are stable.
• Recall our calculation of the Gibbs energy of a
  hypothetical liquid while developing Raoults
  law.
• In our flash calculations that we calculated Pdew
  and Pbubble before assuming that two phases exist
• A slight extension of the thermodynamic theory
  covered in CHEE 311 provides us with a means of
  assessing the stability of a phase.
• Answers the question Will the system actually
  exist in the state I have chosen?

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Phase Stability

• A system at equilibrium has minimized the total
  Gibbs energy.
• Under some conditions (relatively low P, high T)
  it assumes a vapour state
• Under others (relatively high P, low T) the
  system exists as a liquid
• Mixtures at specific temperatures and pressures
  exist as a liquid and vapour in equilibrium
• Consider the mixing of two, pure liquids. We can
  observe two behaviours
• complete miscibility which creates a single
  liquid phase
• partial miscibility which creates two liquid
  phases
• in the extreme case, these phases may be
  considered completely immiscible.

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Stability and the Gibbs Energy of Mixing

• We have already discussed the property changes of
  mixing, in particular the Gibbs energy of mixing.
• Before After
• GA GB G
• nA moles nB moles nA nB moles
• liquid A liquid B of mixture
• The Gibbs energy of mixing is defined as
• which in terms of mole fractions becomes

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Stability and the Gibbs Energy of Mixing

• The mixing of liquids changes the Gibbs energy of
  the system by
• Clearly, this quantity must
• be negative if mixing is to
• occur, meaning that the
• mixed state is lower in
• Gibbs energy than the
• unmixed state.

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Stability Criterion Based on ?Gmix

• If the system can lower its Gibbs energy by
  splitting a single liquid phase into two liquids,
  it will proceed towards this multiphase state.
• A criterion for single phase stability can be
  derived from a knowledge of the composition
  dependence of ?Gmix.
• For a single phase to be stable at a given
  temperature, pressure and composition
• ?Gmix and its first and second derivatives must
  be continuous functions of x1
• The second derivative of ?Gmix must satisfy
• 14.5

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Phase Stability Example Phenol-Water
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Phase Stability Example Phenol-Water (25C)
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Liquid-Liquid Equilibrium Phenol-Water
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9. Liquid Stability 14.1 SVNA

• Whether a multi-component liquid system exists as
  a single liquid or two liquid phases is
  determined by the stability criterion
• 14.5
• If this condition holds, the liquid is stable.
  If not, it will split into two (or more) phases.
• Substituting ?Gmix for a binary system,
• (A)
• we derive an alternate stability criteria based
  on component 1
• (A) into 14.5
• This quantity must be positive for a liquid to be
  stable.

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Wilsons Equation and Liquid Stability

• Given our phase stability criterion
• what we require to gauge liquid stability is an
  activity coefficient model. Wilsons equation
  for component 1 of a binary system gives
• 11.17
• Applying our stability criterion to 11.17 yields
• Given that all Wilsons coefficients Lij are
  positive, all quantities on the right hand side
  are greater than zero for all compositions
• Wilsons equation cannot predict liquid
  instability, and cannot be used for LLE modeling.

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Liquid-Liquid Equilibrium (LLE) SVNA 14.2

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Liquid-Liquid Equilibrium Phenol-Water
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Liquid-Liquid Equilibrium Relationships

• Two liquid phases (a,b) at equilibrium must have
  equivalent component mixture fugacities
• In terms of activity coefficients
• If each component can exist as a liquid at the
  given T,P, the pure component fugacities cancel,
  leaving us with
• 14.10
• Note that the same activity coefficient
  expression applies to each phase.

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Liquid-Liquid Equilibrium-NRTL

• Consider a binary liquid-liquid system described
  by the NRTL excess Gibbs energy model.
• For phase a, we have
• 11.20
• 11.21
• For phase b, we have
• 11.20
• 11.21
• The activity coefficients of the two liquids are
  distinguished solely by the mole fractions of the
  phases to which they apply.

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Liquid-Liquid Equilibrium Calculations

• In CHEE 311, we will consider only binary
  liquid-liquid systems at conditions where the
  excess Gibbs energy is not influenced by
  pressure.
• The phase rule tells us F2-pC
• 2-22 2 degrees of freedom
• If T and P are specified, all intensive variables
  are fixed
• For this two-component system we can write the
  following equilibrium relationships
• and 14.10
• The latter can be stated in terms of component 1,
  to yield
• and 14.12
• The activity coefficients gia and gib are
  functions of xia and xib.

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Liquid-Liquid Equilibrium Calculations

• Once we establish that a LLE condition exists, we
  are interested to know the composition of the two
  phases.
• Given T (and P), find the composition of the two
  liquids.
• Start with our equilibrium relationships.
• Component 1
• Component 2
• The natural logarithm is usually simpler to
  solve
• We have two equations, and two
• unknowns xia and xib.