9. LLE Calculations

1
9. LLE Calculations

• For two liquid phases at equilibrium
• the fugacity of each component in
• the phases must be equal.
• For the binary case shown
• are the two relationships that
• govern the partitioning of species
• 1 and 2 between the two phases.

2
Binary LLE Separations

• The equivalent of a VLE flash calculation can be
  carried out on liquid-liquid systems.
• Given T, P and the overall composition of the
  system
• F, z1, z2
• Find La, x1a, x2a
• Lb, x1b, x2b

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Binary LLE Separations - Governing Eqn

• Solving these problems requires a series of
  material balances
• Using a unit feed as our basis, an overall
  material balance yields
• (A)
• A material balance on component 1 give us
• (B)
• Substituting for Lb from A into equation B
• (C)
• An analogous material balance on component 2,
  yields
• (D)
• We have two equations (C,D) and three unknowns
  (La, x1a and x1b).
• We need an equilibrium relationship between xia
  and xib

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Binary LLE Separations - Governing Eqn

• Our LLE expression is
• (14.10)
• or
• and (E)
• The governing equation we require to solve the
  problem is generated from a final material
  balance on one of the liquid phases
• (F)
• Substituting equations C, D, E into the material
  balance F gives us the final equation

5
Solving Binary LLE Separation Problems

• Given T, P,F, z1, z2 Find La, x1a, x2a
• Lb, x1b, x2b
• The solution procedure follows that of binary VLE
  flash calculations very closely.
• You can immediately solve for x1a and x1b using
  the LLE relationships
• Or
• You can solve the governing equation by
  iteration, starting with estimates of x1a and x1b
  to determine activity coefficients, and refining
  these estimates and La by successive substitution.

6
Vapour-Liquid-Liquid Equilibrium (VLLE)

• In some cases we observe
• VLLE, in which three
• phases exist at
• equilibrium.
• F 2 - p C
• 2 - 3 2 1
• Therefore, at a given P,
• all intensive variables
• are fixed, and we have
• a single point on a binary
• Tx,x,y diagram

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Vapour-Liquid-Liquid Equilibrium (VLLE)

• At a given T, we can
• create Px,x,y diagrams
• if we have a good
• activity coefficient
• model.
• Note the weak
• dependence of the
• liquid phase
• compositions on the
• system pressure.

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10. Chemical Reaction Equilibrium SVNA 15

• If sufficient data exists, we can describe the
  equilibrium state of a reacting system.
• If the system is able to lower its Gibbs energy
  through a change in its composition, this
  reaction is favourable.
• However, this does not imply that the reaction
  will occur in a finite period of time. This is a
  question of reaction kinetics.
• There are several industrially important
  reactions that are both rapid and equilibrium
  limited.
• Synthesis gas reaction
• production of methyl-t-butyl ether (MBTE)
• In these processes, it is necessary to know the
  thermodynamic limit of the reaction extent under
  given conditions.

9
Reaction Extent

• Given a feed composition for a reactive system,
  we are most interested in the degree of
  conversion of reactants into products.
• A concise measure is the reaction extent, e.
• Consider the following reaction
• In terms of stoichiometric coefficients
• where, nCH4 -1, nH20 -1, nCO 1, nH2 3
• For any change in composition due to this
  reaction,
• 15.2
• where de is called the differential extent of
  reaction.

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Reaction Extent

• Another form of the reaction extent is
• (i1,2,,N) 15.3
• The second part of our definition of reaction
  extent is that it equals zero prior to the
  reaction.
• Given that we are interested in the reaction
  extent, and not its differential, we integrate
  15.3 from the initial, unreacted state to any
  reacted state of interest
• or
• 15.4

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Reaction Extent and Mole Fractions

• Translating the reaction extent into mole
  fractions is accomplished by calculating the
  total number of moles in the system at the given
  state.
• Where,
• Mole fractions for all species are derived from
• 15.5

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Multiple Reactions and the Reaction Extent

• The reaction extent approach can be generalized
  to accommodate two or more independent,
  simultaneous reactions.
• For j reactions of N components
• (i1,2,,N)
• and the number of moles of each component for
  given reaction extents is
• 15.6
• and the total number of moles in the system
  becomes
• where we can write

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Chemical Reaction Equilibrium Criteria

• To determine the state of a
• reactive system at equilibrium,
• we need to relate the reaction
• extent to the total Gibbs
• energy, GT.
• We have seen that GT of a
• closed system at T,P
• reaches a minimum at
• an equilibrium state
• Eq. 14.4

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Reaction Extent and Gibbs Energy

• For the time being, consider a single phase
  system in which chemical reactions are possible.
• The changes in Gibbs energy resulting from shifts
  in temperature, pressure and composition are
  described by the fundamental equation
• At constant temperature and pressure, this
  reduces to
• and the only means the
  system has to lower the Gibbs
• energy is to alter the number of moles of
  individual
• components.
• What remains is to translate changes in moles to
  the reaction extent.

15
Criterion for Chemical Equilibrium

• For a single chemical reaction, we can apply
  equation 15.3 which relates the reaction extent
  to the changes in the number of moles
• 15.3
• Substituting for dni in the fundamental equation
  yields
• At equilibrium, we know that d(nG)T,P, 0.
  Therefore, for the above equation to hold at any
  reaction extent, we require that
• 15.8

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Reaction Equilibrium and Chemical Potential

• We have developed a criterion for chemical
  equilibrium in terms the chemical potentials of
  components.
• 15.8
• While this criterion is complete, it is not in a
  useable form.
• Recall our definition of fugacity which applies
  to any species in any phase (vapour, liquid,
  solid)
• In dealing with reaction equilibria, we need to
  pay particular attention to the reference state,
  Gi(T). We can assign a standard state, Gio, as

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Standard States 4.4 SVNA

• For our purposes, the Gibbs energy at standard
  conditions is of greatest interest.
• This is the molar Gibbs energy of
• pure component i
• at the reaction temperature
• in a user-defined phase
• at a user-defined pressure (often 1 bar)
• A great deal of thermodynamic data are published
  as the standard properties of formation at STP
  (Table C.4 of the text)
• DGfo is standard Gibbs energy of formation per
  mole of the compound when formed from its
  elements in its standard state at 25oC.
• Gases pure, ideal gas at 1 bar
• Liquids pure substance at 1 bar

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Chemical Potential and Activity

• Substituting our standard Gibbs energy (Gio) in
  the place of Gi(T), the chemical potential of
  component i in our system becomes
• 15.9
• We define a new parameter, activity, to simplify
  this expression
• 15.11
• where,
• The activity of a component is the ratio of its
  mixture fugacity to its pure component fugacity
  at the standard state.

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Reaction Equilibrium and Activity

• When a reactive system reaches an equilibrium
  state, we know that the equilibrium criterion is
  satisfied. Recall that chemical reaction
  equilibrium requires
• where ni is the stoichiometric coefficient of
  component i and mi is the chemical potential of
  component i at the given P,T, and composition.
• Substituting our expression for chemical
  equilibrium into the above equation gives us
• Or,

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The Equilibrium Constant

• Our equilibrium expression for reactive systems
  can be expressed concisely in the form
• 15.12
• where P signifies the product over all species.
• The right hand side of equation 15.12 is a
  function of pure component properties alone, and
  is therefore constant at a given temperature.
• The equilibrium constant, K, for the reaction is
  defined as
• 15.13
• K is calculated from the standard Gibbs energies
  of the pure components and the stoichiometric
  coefficients of the reaction.

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Standard Gibbs Energy Change of Reaction

• The conventional means of representing the
  equilibrium constant uses DGo, the standard Gibbs
  energy change the reaction.
• Using this notation, our equilibrium constant
  assumes the familiar form
• 15.14
• When calculating an equilibrium constant (or
  interpreting a literature value), pay attention
  to standard state conditions.
• Each Gio must represent the pure component at the
  temperature of interest
• The state of the component and the pressure are
  arbitrary, but they must correspond with fio used
  to calculate the activity of the component in the
  mixture.

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Temp. Dependence of Reaction Equilibrium

• Defined by the following relationship,
• the equilibrium constant is a function of
  temperature.
• Recall that DGo represents the standard Gibbs
  energy of reaction at the specified temperature.
• We know that 15.15
• From which we can derive the temperature
  dependence of K
• 15.16
• If we assume that DHo is independent of
  temperature, we can integrate 15.16 directly to
  yield
• 15.17

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K vs Temperature

• Equation 15.17 predicts that ln K
• versus 1/T is linear. This is based on
• the assumption that DHo is a weak
• function of temperature over the
• range of interest.
• This is true for a number of
• reactions, including those
• depicted by Figure15.2.
• A rigorous development of
• temperature dependence
• of K may be found in the text
• (Equation 15.20)

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Equilibrium State of a Reactive System

• Given that an equilibrium constant for a reaction
  can be derived from the standard state Gibbs
  energies of the pure components, we can define
  the composition of the system at equilibrium.
• 15.13
• Consider the gas phase reaction
• The equilibrium constant gives us
• Or