Chapter 8 Multicomponent Homogeneous Nonreacting Systems: Solutions

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Chapter 8Multicomponent Homogeneous Nonreacting
Systems Solutions

• Notes on
• Thermodynamics in Materials Science
• by
• Robert T. DeHoff
• (McGraw-Hill, 1993).

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Extensive Quantities of theState Functions F/,
G/, H/, S/, U/, V/

• Using G/ as example

A differential form of G/
and at constant T P
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Partial Molal Quantities of theState Functions
.

• Using as example

A differential form of G/ (use definition of
)
and at constant T P
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Gibbs-Duhem Equation

• The contributions of the components sum to the
  whole

Differentiating the products on the right yields
Inspection yields the Gibbs-Duhem equation
For a binary system
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Reference States F/O, G/O, H/O, S/O, U/O, V/O

• T, S, V, P have absolute values.
• F, G, H, U have relative values.
• The difference in values between states is
  unique.
• To compare values, use the same reference state.
• Superscript O refers to the reference state.

Rule of mixtures.

• Preferably, for a solution use the pure
  components in the same phase as the solution as
  the reference state.

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Rule of Mixtures
For binary
Extensive
Molar
For binary
Rule of mixtures
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Mixing Values for SolutionsDF/mix, DG/mix,
DH/mix, DS/mix, DU/mix, DV/mix

• For solution Gibbs free energy of mixing.

For component k Change experienced when 1 mole
of k is transferred from its reference state to
the given solution.
Contributions of the components add to the whole
and
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Mixing Values
Rule of mixtures
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Mixing values for SolutionsDF/mix, DG/mix,
DH/mix, DS/mix, DU/mix, DV/mix

• Differential form

0
Gibbs-Duhem
0
Total derivative
Gibbs-Duhem for mixing
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Graphical Evaluation of Partial Molal Values

• Consider a binary system (alloy)

Note
Substitute rearrange
and
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Derivation Graphical Evaluationof Partial Molal
Values
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Integration of the Gibbs-Duhem Equation(s)

• For a binary system (alloy)

and
Integrating the left side from X2 0 to X2
0
Now integrate right side
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Molar Values of the State Functions

• Note

Then,
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Chemical Potential of (Open) Multicomponent
Systems
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Chemical Potential of (Open) Multicomponent
Systems
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Activities and Activity Coefficients

• Definition of activity, ak (dimensionless)

Definition of activity coefficient, gk
(dimensionless)
If gklt1 akltXk k is less apparent than its
mole fraction. gk1 akXk k is as apparent as
its mole fraction. gkgt1 akgtXk k is more
apparent than its mole fraction.
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Ideal Solution

• No heat of mixing.

No volume change.
No change in internal energy.
Entropy increases.
Helmholtz free energy decreases.
Gibbs free energy decreases.
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Ideal Solution
T
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Ideal Solution

• All plots (e.g. DGmix vs. Xk) are symmetrical
  with composition.
• Slopes of plots of DSmix, DFmix, DGmix are
  infinite at Xk 0 Xk 1.
• Entropy of mixing is independent of temperature.

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Ideal SolutionActivity is the same as mole
fraction. Activity coefficient is one.
1
a2
a1
Slope 1
0
X2
0
1
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Dilute SolutionsRaoult Henrys Laws

• Raoults Law for the solvent in dilute solutions

Henrys Law for the solute in dilute solutions
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Real Solutions Relation of Activity Coefficient
to Free Energy

• Ideal partial molal free energy of mixing

Excess partial molal free energy of mixing
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Regular Solution

• Heat of mixing is a function of composition, only.

Entropy is the same as for ideal solution.
Helmholtz free energy decreases.
Gibbs free energy decreases.
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Regular Solution
T
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Regular Solution
T
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Regular Solution
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Regular Solution
T
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Regular Solution
T
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Problem 8.6 DeHoff
Find
Given
Rewrite in general form
Differentiate
Substitute dX1-dX2
Substitute X1X21
Gibbs-Duhem