CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 620–621
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CALPHAD: Computer Coupling of Phase Diagrams and
Thermochemistry
journal homepage: www.elsevier.com/locate/calphad
Short communication
A remark on the configurational entropy of a liquid solution with a weak
tendency to association
Dmitri V. Malakhov
Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4L7, Canada
article
info
Article history:
Received 14 May 2011
Received in revised form
12 August 2011
Accepted 18 August 2011
Available online 22 September 2011
abstract
A weak tendency to association may significantly increase the molar configurational entropy of a liquid
solution. This circumstance should not be overlooked if one intends to use a positive excess entropy to
describe experimental data within the framework of the CALPHAD method.
© 2011 Elsevier Ltd. All rights reserved.
Keywords:
Liquid solutions
Associated solutions
Configurational entropy
Positive excess entropy
1. Introduction
Let us consider a box (a lattice) divided into cells (lattice
sites). Each cell can be occupied by one ball (a mixing species,
an atom), and none can be empty (no vacancies). If there are 5
indistinguishable blue balls and 5 identical yellow balls, then the
entropy1 S = k ln(10!/(5!5!)) ≈ 5.53k. If 1 blue and 1 yellow ball
react forming 1 green ball (a new mixing species, a molecule), then
the size of the box measured in the number of cells decreases from
10 to 9, and S = k ln(9!/(4!4!1!)) ≈ 6.45k. If 4 blue and 4 yellow
balls react forming 4 green balls, then the size of the box is equal
to 6, and S = k ln(6!/(1!1!4!)) ≈ 3.40k. If all blue and yellow balls
react, then S = k ln(5!/(0!0!5!)) = 0.
This trivial combinatorial result points to two counteracting
effects. Firstly, a shrinking lattice makes the entropy smaller.
Secondly, an emergence of new mixing species makes it larger.
In the case of a strong tendency to association (almost all balls
are green), the former effect dominates, and the configurational
entropy of the solution with associates is less than that of the
cluster-free solution. In the case of a weak tendency to association
(a few green balls), the configurational entropy may exceed that of
the solution without complex molecules.
In the practice of thermodynamic modeling, the associated
solution model [1–6] is usually employed to describe the Gibbs
energies of liquids in which the tendency to association is so
strong that it manifests itself in V -shaped molar enthalpies of
mixing and M-shaped molar entropies of mixing. In contrast, in this
short communication, the case of a weak tendency to association
is touched on. If such a behavior is overlooked, then one will
likely introduce a positive excess entropy of mixing to find a
sufficiently accurate description, which might be questionable
from the methodological viewpoint.
2. Analysis
Let us take a binary liquid solution containing 1 − x moles of
A and x moles of B. Let us assume that there are no other mixing
species except A1 and B1 , and that they mix randomly. The entropy
of the solution (numerically equal to its molar entropy, indeed) is:
S
R
= −nA1 ln (1 − x) − nB1 ln x = − (1 − x) ln (1 − x) − x ln x.
Now let us imagine that ε moles of A1 and ε moles of B1 react
forming ε moles of the new species AB randomly mixing with A1
and B1 . Apparently, 0 ≤ ε ≤ min (1 − x, x), which, in particular,
means that ε can never exceed 1/2. The entropy of the liquid
solution with the associates is:
Sa
R
= −nA1 ln xA1 − nB1 ln xB1 − nAB ln xAB
= − (1 − x − ε) ln
E-mail address: [email protected].
1 Since the combinatorial entropy is only the one we are concerned with,
‘‘entropy’’ always means ‘‘configurational entropy’’.
0364-5916/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.calphad.2011.08.007
− (x − ε) ln
1−x−ε
x−ε
1−ε
1−ε
− ε ln
ε
1−ε
.
(1)
D.V. Malakhov / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 620–621
621
and attains the uppermost position at K = 1. Another special value
of the equilibrium constant is K = 8, at which Sa (x = 1/2) =
S (x = 1/2). Despite the identical values at x = 1/2, the shapes
of Sa (x) and S (x) are noticeably different.
It should be emphasized that the mathematical analysis is based
on the assumption that the formation of the associates entails a
decrease in the number of lattice sites, i.e., a physical shrinkage
of the liquid quasi-lattice. In other words, the formation of AB
molecules is not treated here within the framework of the quasichemical formalism [7] in which a number of lattice sites is fixed,
and in which an affinity between A and B results in a number of
A–B nearest neighbors exceeding that in a random mixture.
3. Conclusion
Fig. 1. Concentration dependencies of the configurational entropy corresponding
to different values of the equilibrium constant K of the reaction A1 + B1 AB. The
bold curve corresponds to the case K = 0 (no associates).
Since a molar property is by definition an extensive property per
number of moles of components, the molar entropy is numerically
equal to the entropy given by (1), because Sa should be divided not
by 1 − ε (the total number of moles of species), but by the total
number of components, which is always equal to 1.
Assuming that the mixing species A1 , B1 and AB are in
equilibrium, let us define the equilibrium constant of the reaction
A1 + B1 AB as:
K =
=
xAB
xA1 xB1
ε
  x−ε 
=  1−x−ε1−ε
1−ε
Acknowledgment
1−ε
ε (1 − ε)
.
(1 − x − ε) (x − ε)
(2)
A rearrangement of (2) gives the quadratic equation ε 2 (K + 1)−
ε (K + 1) + Kx (1 − x) = 0 whose roots are:
ε=
1
2


1±
It may happen during a thermodynamic assessment that it
seems necessary to introduce a positive excess molar entropy of
mixing for a liquid phase in order to obtain a good match between
experimental and calculated quantities. Before pursuing this
option, it might be reasonable to invoke chemical and other nonthermodynamic arguments to inquire whether this phase behaves
as a solution with a weak tendency to association. Such a behavior
results in an increase of the molar configurational entropy, i.e., in
what the excess term is supposed to address. Consequently, this
term may not be necessarily needed. It should be admitted that
since it might be difficult to find strong experimental or theoretical
arguments supporting the actual existence of complex molecules
in a liquid phase, the suggestion put forward in this work has a
limited scope.
1−
4K
K +1

x (1 − x) .
(3)
The root with + in ± is inadmissible, because the condition ε ≤
1/2 is not fulfilled. Once a value of K is fixed, ε is found from
(3) and then is used in (1) to calculate the entropy Sa for a given
mole fraction x. Results of such calculations are shown in Fig. 1.
As expected, a pronounced tendency to association (large values
of the equilibrium constant) results in M-shaped concentration
dependencies of the molar entropy Sa (x) located below the S (x)
curve. If K slightly deviates from zero, then the entropy increases
Financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) is gratefully acknowledged.
References
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functions of liquid alloys I. Basic concepts, Z. Metallkde 73 (1982) 72–76.
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77–86.
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[4] R. Lück, U. Gerling, B. Predel, An entropy paradox of the association model, Z.
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