6' NonIdeal Mixtures

1
6. Non-Ideal Mixtures

• 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
• 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 the perfect gas mixture model derived
  earlier.
• 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
Perfect Gas Mixtures

• We examined perfect gas mixtures in lecture 9.
  The assumptions made in developing an expression
  for the chemical potential of species i in a
  perfect gas mixture were
• 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.
• We can choose a more convenient reference
  pressure that is standard for all fluids, that is
  Punit pressure (1 bar,1 psi,etc)
• In this case the pure component Gibbs energy
  becomes

3
Perfect Gas Mixtures

• Substituting for our new reference state yields
• (11.29)
• which is the chemical potential of component i in
  a perfect gas mixture at T,P.
• The total Gibbs energy of the perfect gas mixture
  is provided by the summability relation
• (11.11)
• (11.30)

4
Ideal Mixtures of Real Gases

• One source of mixture non-ideality resides within
  the pure components. Consider an ideal solution
  that is composed of real gases.
• In this case, we acknowledge that 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 solution 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

• It can be shown that 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 compressiblity

9
Calculating ?iv from the Virial EOS

• We have used the virial equation of state to
  calculate the fugacity and fugacity coefficient
  of pure, non-polar gases at moderate pressures.
• Under these conditions, it represents non-ideal
  PVT behaviour of pure gases quite accurately
• 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