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A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles

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Abstract

A general method of evaluation of configurational entropy of a liquid mixture is presented. It is based on a generalized lattice model with no restrictions due to particle shape being introduced. A general formula for the entropy is derived. Achieved results open a way for a rigorous analysis of particle shape effect on mixing process. As an example, a new formula for the entropy of mixing of hard spheres in continuous space is derived which may respect a physical bound for packing ratio. A systematic approach to improve the model accuracy is proposed. The resultant alternative models are discussed in details. A comparison with literature data and the Mansoori-Carnahan-Starling formula is presented. Very good agreement is shown.

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Abbreviations

A qj [O q]:

coefficients in formulae (47), (54) defined in eqs. (48)-(50);

A q :

coefficientA qj [j q]; for identical particles;

a o :

specific surface area of the oth particle;

B Dκi :

factor representing the effect of particles belonging toDκ (1st and further neighbors ofκ on probabilityακe Di r (eq. (28));

C D k ,C D k ,C D ke ,C Dk\i i :

the expected number of configurations of a single objectk, κ, κe, i, respectively, having a fixed (taken at random) configuration of objects belonging to the setD,Dk\r;

Dk\i ik :

the expected number of configurations of objecti (having a fixed configuration ofDk\i) in which i is adjacent toκ;

D :

set of objects arranged in lattice prior to that being placed;

Dk\i :

same asD, while objecti is removed andκ added to lattice;

eκ, ei :

index of an element of thekth,ith object, respectively;

e κ :

index of an element of previously formed partκ of objectk;

F q(·):

a general function representing effects of the shape and size of particles on coefficientsA qj [O q] (defined in eq. (89));

F ex :

Helmholtz free energy excess (see eq. (121));

G pj [O P]:

geometrical factors depending only on the shape and size of particles present in the mixture (defined in eq. (38));

G P :

constant factorG pj [j p] (for identical particles);

g eκ :

probability that a cell chosen for an elemente κ is eligible with respect to the shape of thekth object, provided that previously arranged elements ofk satisfy the same condition;

g eκ :

same asg eκ for the previously formed partκ of thekth object;κ B Boltzmann constant;

M :

the number of particles in the mixture;

M j :

the number of thejth component particles;

n :

the total number of lattice cells (volume of the mixture);

n Df :

the number of vacant cells left by objects belonging toD;

O q :

a sequence consisting of q indices, each of them representing geometrical parameters of a particle as the argument of functionsA qj [O q] orG qj [O q];

O q\j l) c :

cth combination ofl identical indicesj within the sequenceO q(j ε O q);

P Deκ :

probability that a cell chosen for thee κth element ofκe is vacant, providing that the objects ofD andκ are arranged in an admissible way (def. (eq. (11));

p, q :

superscripts used to indicate an order of products and of factors;

ℙ Dk\i i\f ,ℙ Dk\i i\ξ :

probability of non-conflict arrangement of the ith object with fixed configuration ofDk\i objects) provided that its first element is placed in any vacant cell, in ζ cell, respectively;

>ℙ Dk\i i\ξ :

same as ℙ Dk\i i\ξ but with the lack of conflict withκ assumed;

R k :

radius of a sphere representing the hard part of thekth object;

(R o)q :

qth order product ofR o1 , ... ,R o q;

r k :

current value for a radius of thekth particle being located;

δR :

thickness of a soft spherical layer in a particle;

Δr :

radius of a spherical region occupied by the vibrating center of a molecule in liquid state;

S M, So :

conffgurational entropy of a mixture and of its components in their standard states, respectively;

ΔS :

configurational entropy of mixing (eq. (1));

ΔS F :

approximation toΔS disregarding differences in the size of particles (def. eq. (118));

σ κSD :

component of the expression for configurational entropy contributed by inserting of thekth particle into the lattice (eq. (43));

S κ :

the number of elements in a surface layer of thekth object;

S κ :

the set of surface elements of thekth object;

v *j , vj :

volume fraction of thejth component and of the hard part of its particles, respectively, in the mixture;

x κ :

the number of hard part elements (volume) of thekth object;

x κ :

the set of elements of the hard part of thekth object;

x κ :

the number of elements (volume) of thekth object;

Z o :

compressibility (eq. (121));

α κe i D :

probability that an elemente i; of theith object, occupying the cellζ (chosen for thee κ,th element ofκe), is not in conflict withκ and with objects belonging to the setDk\i(def. (eq. (11));

α 0κei :

probability that an elemente i of the ith object occupying the cellζ is not in conflict withκ (def. eqs. (11), (16));

Γ mq[O q]:

mqth order function used to compute the value forA qj [];

κ :

a simply connected and properly shaped body, being a formed part of thekth particle, while itse κth element is being placed;

κe, κζ :

κ and the nexteth element ofk,κ andζ cell.

l, o, λ :

same asκ for theith,oth,l th object, respectively;

η oj ηM :

packing fraction ofthejth component particles in a standard state and in the mixture, respectively;

ζ :

eligible cell chosen for joining thee κth element toκ;

Φ κe :

the number of cells eligible for joining thee κth element toκ

φ :

constant factor taken in eqs. (101) and (115) to express the sequence {G p, p = 3,..., ∞};

Ω :

the number of configurations of objects in the mixture;

Ω D :

the number of configurations ofD-set objects in the lattice

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Authors and Affiliations

  1. Institute of Automatics, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059, Kraków, Poland

    Jan Duda

  2. Faculty of Energochemistry of Coal and Physicochemistry of Sorbents, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059, Kraków, Poland

    Janina Milewska-Duda

Authors
  1. Jan Duda
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  2. Janina Milewska-Duda
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Duda, J., Milewska-Duda, J. A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles. J Math Chem 17, 69–109 (1995). https://doi.org/10.1007/BF01165138

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  • DOI: https://doi.org/10.1007/BF01165138

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