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A statistical mechanical theory for activity coefficients of a dilute solute in a binary solvent

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Abstract

A statistical mechanical calculation of the activity coefficients of a dilute solute,C, in a binary solvent,A-B, was made using a model for the interactions of a coordination cluster consisting of a solute atom and its neighboring solvent atoms. The derived equations are applicable to a variety of types of ternary solutions including substitutional and interstitial alloys as well as additive and reciprocal molten salts. The theory takes into account the interactions between solute and solvent atoms (ions) as well as changes in interactions of the solvent atoms (ions) which are neighbors of solute atoms (ions). Prior theories such as those of Wagner and the quasi-chemical theories of Alcock and Richardson and Jacob and Alcock can all be shown to be special cases of the present theory. The dependence of the activity coefficients of a solute on the solvent composition is calculated from a knowledge of the activity coefficients of the solvent components, solute activity coefficients in the two pure solvent components, a coordination number, a geometric factor which depends upon the type of solution, and a term which represents the nonadditivity of pair bond interactions within the cluster of a solute atom (ion) and its neighboring solvent atoms (ions). In the model, the thermodynamic properties of the solute are related to the relative concentrations of the different coordination clusters as well as to the thermodynamic properties of the solvent.

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Abbreviations

k :

Boltzmann constant

T :

Temperature of the system

Z :

Coordination number of dissolved solute atoms

Z′ :

Coordination number of solvent atoms

N A, NB, Nc :

Number ofA, B, C atoms in solution withN c ≪ (N A +N B)

N :

Total number of atoms in solution:N =N A +N B +N C

N i :

Number ofC atoms which havei or moreB neighboring atoms (i-0, ...,Z andN o =N C)

N i :

Value ofN i in the maximum term (Eq. [8])

X A, XB, XC :

Atom fractions ofA, B, C

b :

Region composed of theC atoms and allA andB atoms which do not have a neighboringC atom (bulk solution)

c :

Region composed of allA andB atoms surroundingC atoms

A(b), B(b) :

Collection ofA, B atoms located in region b

N A(b), NB(b):

Total number ofA, B atoms in region b

N A(c), NB(c):

Total number ofA, B atoms in region c

C(A Z-i Bi):

Typical configuration in region c where aC atom is surrounded by (Z—i) atomsA andi atomsB (i = 0,...,Z)

q A, qB :

Partition functions of atomsA, B in the bulk solution (region b)

r i :

Partition function of atomC in the configurationC(A Z-i Bi)

q j :

Partition function of one of the atomsB, in the configurationC(A Z-i Bi) withj= 1,...,i(i=0)

q(i) :

Partition function of thei atomsB in the configurationC(A Z-i Bi)

Pk :

Partition function of one of theA atoms in the configurationC(A Z-i Bi) withk = 1,...,Z-i (i≠Z)

p(i) :

Partition function of the (Z-i) atomsA in the configurationC(A Z-i Bi)

Q t :

Total partition function

Q :

Partition function for a particular set of values ofN i

Q(cf) :

Partition function for the collection ofA andB atoms in the configurationsC(A Z-i Bi)

Q(b), Q′(b) :

Partition function of the atomsA andB distributed in the bulk solution (region b) in the case of substitutional, interstitial alloys

Q max :

Maximum value ofQ

s A :

Contribution from the interactions ofA atoms within the configurationC(A Z)

S A(i):

SB(i) Contributions from the interactions ofA andB atoms within the configurationC(A Z-iBi)

S(i) :

Term defined by Eq. [38]

S(0):

Value ofS(i) in pureA

S(Z) :

Value ofS(i) in pureB

γ m :

Activity coefficient of themth component (m = A, B, C)

γ C (A) :

Activity coefficient ofC defined so that it is unity at infinite dilution ofB andC in the solventA

γ B(A) :

Activity coefficient ofB at infinite dilution inA

γ C(B) :

Activity coefficient ofC dilute in pureB

γ C(A) :

Activity coefficient ofC dilute in pureA

γ (A) C(B) :

Activity coefficient ofC dilute in pureB with a standard state defined in pureA

μ m :

Chemical potential of the mth component (m =A, B, C

μ o m :

Chemical potential of purem (m =A, B, C)

μ /* m :

Standard chemical potential of the mth component (m =A, B, C)

n :

Number of interstitial sites for the occupation of theC atoms

α :

Proportionality constant (Eq. [17])

t :

Fraction of the totalZ′ interactions of anA orB atom influenced by aC atom

ΔA i :

“Specific bond free energy” for bonding the ithB atom to aC atom

ΔA o :

Term related to the difference between the properties of the soluteC in the pure solventsA andB (Eq. [39])

ΔA 01 :

Value of ΔA 1 at infinite dilution ofC andB

g i :

Total nonconfigurational free energy of thei atomsB and of the (Z-i) atomsA in theC(A Z-i Bi) configuration

Δg Ei :

Excess free energy of mixing of (Z-i) atomsA inC(A Z) withi atomsB inC(B Z) to form mixed species [C(A Z-i Bi)]

β j :

Terms related to interactions between the different atoms Eq. [10] (j = 1, ...,Z)

β0 :

Parameter defined by Eq. [28]

β:

Value of the β′ i s when all are equal (β = β1 = β2 = and so forth)

h lm :

Interaction parameter Eq. [43]

h h = :

h 11 h′ =h/2kT

λ:

Interaction coefficient for theA-B solvent

C i :

Fraction of theC atoms in the configurationC(A Z-i Bi)

LSA:

Limiting slope of the curve representing the variations of In γ C vs X A

LSB:

Limiting slope of the curve representing the variations of hi γ C vs X B

K BC, KAC :

Association constants for the formation of the speciesBC, AC from pureB andC and from pureA and C respectively.

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Authors and Affiliations

  1. Chemical Engineering Division, Argonne National Laboratory, 60439, Argonne, IL

    Milton Blander (Senior Chemist and Group Leader) & Marie-Louise Saboungi (Chemist)

  2. University of Provence, Marseille, France

    Pierre Cerisier (Maitre-Assistant)

Authors
  1. Milton Blander
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  2. Marie-Louise Saboungi
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  3. Pierre Cerisier
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Blander, M., Saboungi, ML. & Cerisier, P. A statistical mechanical theory for activity coefficients of a dilute solute in a binary solvent. Metall Trans B 10, 613–622 (1979). https://doi.org/10.1007/BF02662564

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  • DOI: https://doi.org/10.1007/BF02662564

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