6. Coping with Non-Ideality SVNA 11.3

1
6. Coping with Non-Ideality SVNA 11.3

• Up until now, we have considered only ideal
  mixtures that (unfortunately) do not represent
  many cases experienced in practice.
• Non-ideality can take two forms
• Deviations in pure component behaviour e.g. pure
  gases at high pressure
• Deviations in mixture behaviour e.g. V ? ? xi Vi
• So far, our treatment of non-ideality has
  involved
• the development of a method for describing
  non-ideal, single-component, gas behaviour
• the extension of this treatment to the
  description of pure liquids
• What remains is to revise our treatment of
  perfect gas mixtures and ideal solutions to
  account for non-ideal mixing effects.

2
Partial Properties Thought Experiment

• Suppose we add a drop of water
• to pure acetone.
• What change in volume
• would result?
• If the resulting water/acetone mixture is ideal
  (recall definition of ideal) the volume increase
  is simply that of the volume of the water
  droplet.
• If the mixture behaves non-ideally, the volume
  increase will not equal the volume of the water
  droplet. The effect may, in fact, be quite
  different.
• Non-ideality in mixtures results from complex
  intermolecular interactions that we cannot
  predict.
• We still have to solve engineering problems
  (separations, property calculations, ) using
  these non-ideal systems.

3
Partial Properties Thought Experiment

• If the volume change of the acetone-water mixture
  does not equal the volume of the water droplet,
  then the properties of pure water are irrelevant.
• We like to assign values or contributions to
  each component in non-ideal mixtures to account
  for the variation of a property with respect to
  composition.
• This leads us to define partial molar properties,
  which in our thought experiment gives us the
  partial molar volume for water in an
  acetone-water solution.
• This quantity represents change in solution
  volume as the number of moles of water is varied
  at a given P,T, and nacetone.

4
Partial Molar Quantities

• We prefer to think of mixtures in terms of their
  components
• Overall property has a contribution from each
  component in the mixture.
• In non-ideal systems, the properties of the pure
  components have little meaning, forcing us to
  find an alternate way of defining molar
  quantities.
• If nM represents the total thermodynamic property
  of interest
• (11.7)
• where is a partial molar property, also a
  function of (T,P, nj)
• A partial molar property is specific to the P,T
  and composition from which it is derived by
  equation 10.7
• It is difficult to predict, but can be measured
  experimentally.

5
Total Properties of Non-Ideal Mixtures

• Ideal mixtures result from a lack of molecular
  interactions (ideal gas) or equivalent molecular
  interactions (ideal solution). In these cases, a
  total thermodynamic property (nM) for a mixture
  is
• nM ? ni Mi where Mi represents the pure
• component property of i.
• Non-ideal systems do not obey this simple
  formula, as cross-component molecular
  interactions differ from pure component
  interactions.
• nM ? ni where represents the
  partial molar
• property of component i.
• In terms of mole fractions (dividing by n)
• M ? xi (11.11)
• If we know the partial properties of the
  components of the mixture (from experimental
  data) we can derive its total property.
• This is summability relation, which is opposite
  to 10.7 that defines a partial property

6
The Gibbs-Duhem Equation

• An important question we need to answer is
• how do the partial molar properties of a mixture
  relate?
• Start with the definition of total Gibbs Energy
  at T,P
• nG ? ni Gi ? ni ?i
• If we change the composition of the system at
  constant T,P, the Gibbs energy responds
  accordingly
• dnG ? d(ni ?i)
• ? ni d?i ? ?i dni
• But we know that the total change in Gibbs energy
  is defined by
• d(nG) nV dP - nS dT ? ?i dni 11.2
• which at constant P,T (dP0, dT0), is
• d(nG) ? ?i dni

7
The Gibbs-Duhem Equation

• d(nG) provided by these two relations must be
  equal. Therefore,
• d(nG) ? ni d?i ? ?i dni
• ? ?i dni
• For this to be true,
• ? ni d?i 0 (11.14)
• This is the Gibbs-Duhem equation applied to
  chemical potential
• Why is it useful?
• It states that partial molar properties cannot
  change independently, if one partial property
  increases, others must decrease
• Estimates of partial molar properties (from
  experimental data, correlations) can be checked
  for consistency

8
Calculating Partial Molar Props. - Binary Systems

• Partial molar properties (Mi) can always be
  derived from an equation for the solution
  property (M) as a function of composition
• (11.7)
• Comparatively simple relations exist for binary
  systems
• (11.11)
• therefore
• The Gibbs-Duhem equation tells us that
• (11.14)
• Leaving us with

9
Calculating Partial Molar Props. - Binary Systems

• We left off with
• Since x1x21, dx1 - dx2 and
• which we can write as
• Substituting the total solution property
• (11.15, 11.16)
• What we need to calculate a partial molar
  property is an expression for the total molar
  property (M) as a function of composition.

10
Notation for the Course

• Our superscripts and subscripts are propagating
  rapidly, so lets revisit our definitions
• S entropy of one mole of the mixture
• Si entropy of one mole of pure i
• Si entropy of one mole of pure i in the
  mixture
• partial molar entropy
• Suggested Readings
• Examples 11.1, 11.2, 11.3, 11.4 - Partial Molar
  Properties