Fugacity of Non-Ideal Mixtures (SVNA 11.6 and 11.7)

1
Fugacity of Non-Ideal Mixtures (SVNA 11.6 and
11.7)

• In our attempt to describe the Gibbs energy of
  real gas and liquid mixtures, we examine two
  sources of non-ideal behaviour
• Pure component non-ideality due to PVT behaviour
• concept of fugacity
• Non-ideality in mixtures
• partial molar properties
• mixture fugacity and residual properties
• We will begin our treatment of non-ideality in
  mixtures by considering gas behaviour.
• Start with an ideal gas mixture expression.
• Modify this expression for cases where pure
  component non-ideality is observed.
• Further modify this expression for cases in which
  non-ideal mixing effects occur.

2
Ideal Gas Mixtures

• In an ideal gas mixture
• all molecules have negligible volume
• interactions between molecules of any type are
  negligible.
• Based on this model, the chemical potential of
  any component in a perfect gas mixture is
• where the reference state, Giig(T,P) is the pure
  component Gibbs energy at the given P,T.
• For a pure ideal gas, we derived

3
Ideal Gas Mixtures

• Substitute to get
• (11.29)
• which is the chemical potential of component i in
  an ideal gas mixture at T,P.
• The total Gibbs energy of the ideal gas mixture
  is the sum of the contributions from the
  individual components
• (11.11)
• (11.30)

4
Ideal Mixtures of Real Gases

• Consider an ideal solution of real gases.
• The molecules have finite volume and interact,
  but assume these interactions are equivalent
  between components
• The appropriate model is that of an ideal
  solution
• where Gi(T,P) is the Gibbs energy of the real
  pure gas
• (11.31)
• Our ideal solution model applied to real gases is
  therefore

5
Non-Ideal Mixtures of Real Gases

• In cases where molecular interactions differ
  between the components (polar/non-polar mixtures)
  the ideal solution model does not apply
• Our knowledge of pure component fugacity is of
  little use in predicting the mixture properties
• We require experimental data or correlations
  pertaining to the specific mixture of interest
• To cope with highly non-ideal gas mixtures, we
  define a mixture fugacity
• (11.47)
• where fi is the fugacity of species i in
  solution, which replaces the product yiP in the
  perfect gas model, and yifi of the ideal solution
  model.

6
Non-Ideal Mixtures of Real Gases

• To describe non-ideal gas mixtures, we define the
  solution fugacity
• and the fugacity coefficient for species i in
  solution
• (11.52)
• In terms of the solution fugacity coefficient
• Notation
• fi, ?i - fugacity and fugacity coefficient for
  pure species i
• fi, ?i - fugacity and fugacity coefficient for
  species i in solution

7
Calculating ?iv from Compressibility Data

• Consider a two-component vapour of known
  composition at a given pressure and temperature
• If we wish to know the chemical potential of each
  component, we must calculate their respective
  fugacity coefficients
• In the laboratory, we could prepare mixtures of
  various composition and perform PVT experiments
  on each.
• For each mixture, the compressibility (Z) of the
  gas can be measured from zero pressure to the
  given pressure.
• For each mixture, an overall fugacity coefficient
  can be derived at the given P,T
• How do we use this overall fugacity coefficient
  to derive the fugacity coefficients of each
  component in the mixture?

8
Calculating ?iv from Compressibility Data

• The mixture fugacity coefficients are partial
  molar properties of the residual Gibbs energy,
  and hence partial molar properties of the overall
  fugacity coefficient
• In terms of our measured compressibility

9
Calculating ?iv from the Virial EOS

• We know how to use the virial equation of state
  to calculate the fugacity and fugacity
  coefficient of pure, non-polar gases at moderate
  pressures.
• The virial equation can be generalized to
  describe the calculation of mixture properties.
• The truncated virial equation is the simplest
  alternative
• where B is a function of temperature and
  composition according to
• (11.61)
• Bij characterizes binary interactions between i
  and j BijBji

10
Calculating ?iv from the Virial EOS

• Pure component coefficients (B11 B1, B22
  B2,etc) are calculated as previously and cross
  coefficients are found from
• (11.69b)
• where,
• and
• (11.70-73
• Bo and B1 for the binary pairs are calculated
  using the standard equations 3.65 and 3.66 using
  TrT/Tcij.

11
Calculating ?iv from the Virial EOS

• We now have an equation of state that represents
  non-ideal PVT behaviour of mixtures
• or
• We are equipped to calculate mixture fugacity
  coefficients from equation 11.60

12
Calculating ?iv from the Virial EOS

• The result of differentiation is
• (11.64)
• with the auxiliary functions defined as
• In the binary case, we have
• (11.63a)
• (11.63b)

13
6. Calculating ?iv from the Virial EOS

• Method for calculating mixture fugacity
  coefficients
• 1. For each component in the mixture, look up
• Tc, Pc, Vc, Zc, ?
• 2. For each component, calculate the virial
  coefficient, B
• 3. For each pair of components, calculate
• Tcij, Pcij, Vcij, Zcij, ?ij
• and
• using Tcij, Pcij for Bo,B1
• 4. Calculate ?ik, ?ij and the fugacity
  coefficients from