On Mayer's Theory of Dilute Ionic Solutions

Haga, Eijirô

doi:10.1143/jpsj.8.714

Cited by 42 publications

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“…Knowing jUT(r), one then obtains the potentials of average force, the radial distribution functions, the osmotic pressure, and finally the osmotic coeffi cient. Comparison with the earlier calculations of Haga (5), based on Mayer's (6) theory of ionic solutions and presumably correct to a higher power of concentration than Moller's expression, indicates the limitations of Moller's theory (invalid at moderate concentration, small a, and large Ka). Judging by agreement with experimental data, the two theories appear comparable though the lack of data at low concentrations makes a comparison very difficult.…”

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confidence: 65%

Solutions of Electrolytes

Poirier¹

1959

Annu. Rev. Phys. Chem.

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Due to limitations of both space and time, the reviewer has restricted the scope of the following review, omitting certain subjects which might properly be included. Notable among these are: solutions of electrolytes in molten metals and salts, solutions of polyelectrolytes and soaps, solutions of chelate complexes, phase studies with water as one component, ion exchange, elec trode processes, and nuclear magnetic resonance. These topics are left to reviews, under other titles, appearing at least occasionally in the Annual Review of Physical Chemistry.The reviewer does not disguise his enthusiasm for theoretical work, especially that with a statistical mechanical flavor.The most recent discussion of The Faraday Society, Interaction in Ionic Solutions (1), has been published. In this one volume a number of very sig nificant and stimulating articles are contained or discussed. Most of the papers are mentioned individually below. EQUILIBRIUM PROPERTIESTheory.-During the past year, considerable progress has been made in this field. Falkenhagen & Kelbg (2) have briefly reviewed recent work in the statistical theory of ionic solutions and have presented a derivation of the relation between the potential of average force and the mean electrostatic potential employed in the Debye-Huckel theory.A new theory which, for I-I electrolytes, has been carried through to a comparison with osmotic coefficient data is due to Moller (3). The starting point is the Born-Green integral equation in the multicomponent form previ ously presented by Kelbg & Moller (4). The Kirkwood superposition approxi mation is contained in the initial integral equations; other limitations are introduced while solving the equations, a process which contains some novel features. The initial equations are integral equations for the potential of average force,

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confidence: 65%

Solutions of Electrolytes

Poirier¹

1959

Annu. Rev. Phys. Chem.

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“…There, the expansion of the second cluster integral was carried on up to the order n3 and the leading classical terms of order n512 coming from S3 and S, [17,3] were evaluated either analytically or with higher accuracy than before. Having extended the expansion of Ent, Eq.…”

Section: Density Expansion Of T H E Reduced Distribution Functionmentioning

confidence: 99%

Density Expansions of the Reduced Distribution Functions and the Excess Free Energy for Plasma with Coulomb and Short‐Range Interactions

Kahlbaum

1999

Contrib. Plasma Phys.

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A b s t r a c tThe equilibrium properties of a multi-component plasma of charged particles that interact through the Coulomb potential and general short-range potentials are investigated at low density within the framework of classical statistics. In particular, this article presents density expansions of (1) the reduced distribution functions for two and more particles up to second order as obtained from the screened Bogoliubov-BornGreen-Kirkwood-Yvon hierarchy and (2) the interaction contribution to the Helmholtz free energy including a new closed representation of the fourth duster integral derived with the help of a usual charging procedure. '1 IntroductionThe calculation of the equilibrium properties of many-particle systems for a given type of microscopic interaction is one of the most fundamental problems in statistical physics. In this article I consider a classical multi-component plasma at low density as a fluid mixture of N charged particles belonging to different species and interacting through the two-body Coulomb potential and additional short-range potentials between two and more particles.The systematic approach adopted here follows the traditional route of first solving the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced distribution functions by formal density expansion where Debye-Hiickel screening is consistently introduced, and then determining thermodynamic quantities as, e. g., the Helmholtz free energy from both the distribution functions and the interaction potentials by a charging process. In this way I calculate in Sec. 2 the first three coefficients in the expansion of the s-particle distribution function and in Sec. 3 the excess free energy up to the fourth cluster integral, thereby correcting and completing previous work of SCHMITZ [l, 21 from the late sixties on the same subject. All results are expressed in terms of the Mayer functions for two-, three-, and four-particle interaction potentials which makes it clear that the present method may be regarded as closely related to the diagrammatical derivation of cluster expansions from the ionic solution theory developed by MAYER and FRIEDMAN (for adetailed review see [3]). Density expansion of t h e reduced distribution functionThe reduced distribution function f51...a, for s particles, s < N , of species {a;} at positions {r;} (i = 1,. . . , s) from an N-particle system in the volume V at temperature T is given by where the configuration integral QN insures normalization of f5,...5, to Vd, and P = l/kwT.For a shorter notation I drop the dependence on r; always implying that, e. g., 351 = 3a, (rl).The central quantity in Eq. (1) is the total interaction potential, or direct potential, w , ,~. . .~~

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“…are described using the quantum statistics. [4] Hage [5] calculated the classical excess free energy and osmotic pressure for Coulomb potential. Path-Integral Monte Carlo (PIMC) methods are proven extremely useful and successful in predicting the thermodynamic properties of many condensed matter and more recently of finite atomic and molecular clusters.…”

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“…We used the results to calculate the equation of state; we used the thermal equilibrium plasma. [4] Hage [5] calculated the classical excess free energy and osmotic pressure for Coulomb potential. are described using the quantum statistics.…”

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The quantum equation of state until the fourth virial coefficient for many‐component plasmas

Eisa

2018

Contributions to Plasma Physics

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The aim of the present article is to calculate the quantum excess free energy until the fourth virial coefficient in terms of Slater sums for neutral many-component plasmas. We used the results to calculate the equation of state; we used the thermal equilibrium plasma. Bound states are not taken into account in this work. KEYWORDSfourth virial coefficient, quantum equation of state, quantum monte carlo, slater sum INTRODUCTIONThe properties and the behavior of systems composed of many elementary particles, atomic nuclei, atoms etc. are described using the quantum statistics. Quantum systems are studied using numerical simulations, and the most rigorous approaches at finite temperature are based on the path-integral treatment of quantum mechanics. Path-Integral Monte Carlo (PIMC) methods are proven extremely useful and successful in predicting the thermodynamic properties of many condensed matter and more recently of finite atomic and molecular clusters. PIMC results for dense hydrogen are published by Filinov et al. [1] and Bonitz et al. [2] . Besides simulations, the equilibrium behavior is determined either from the equations of state or from virial expansion. The word virial is derived from the Latin vis, which means force. The virial coefficients give the deviation ideality in terms of the forces between molecules. Mayer [3] calculated the osmotic pressure of the solution up to and including terms of order n ln(n) in the ionic concentration by considering also the terms corresponding to graphs of somewhat more complex topological types. The necessary tables, up to 16 terms of the infinite series for osmotic pressure of an ionic solution, were given by Poirier. [4] Hage [5] calculated the classical excess free energy and osmotic pressure for Coulomb potential. Meeron [6] proven that expressions for the logarithms of osmotic pressure series are equivalent to a series of certain well-known and accurately tabulated functions; he transformed Mayer's infinite series of integrals of a closed expression. Kelbg, [7] Ebeling, Kelbg, and Schmitz [8] derived the virial expansion of the excess free energy from the pair distribution function. Hirschfelder et al. [9] used the hard sphere potential, Ree and Hoover [10] used the Lennard-Jones potential to calculate the third virial coefficient of spherical gases. Stogryn [11] and Singh et al. [12] calculated the third virial coefficient of nonspherical molecules. Boushehri et al. [13] calculated the reduced third virial coefficients of several gases made of spherical molecules. At very low temperatures, a pure quantum mechanical method is referred. At high temperatures, the quantum effects in the equation state of gases are small and appear as corrections. It is well known that in the classical theory, which is for high temperatures, the virial coefficients are found explicitly in terms of the intermolecular forces. The quantum corrections to the equation state are calculated using the Wigner-Kirkwood (WK) method [14] when the potential is analytic and using Hermmer-Jancovici...

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On Mayer's Theory of Dilute Ionic Solutions

Cited by 42 publications

References 1 publication

Solutions of Electrolytes

Solutions of Electrolytes

Density Expansions of the Reduced Distribution Functions and the Excess Free Energy for Plasma with Coulomb and Short‐Range Interactions

The quantum equation of state until the fourth virial coefficient for many‐component plasmas

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