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Thus, the activity coefficient can be calculated if AGE is known; this is the difference between the work of the reversible transition of the real system at constant temperature and pressure from the standard to the actual state and the work in the same process in the ideal system: RT In Yi = ( ^) (1.3-3)
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As has already been mentioned, to carry out such a calculation is not a matter of thermodynamics, but requires adopting certain assumptions on the structure of the system and on interactions between particles. According to the statistical thermodynamics of solutions the logarithm of the activity coefficient for an electrolyte solution can be expanded in the series In y = <xx" + *i + yx\ + (1.3.4)
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where ny a> j3, . . . are constants (where a < 0, 0 < n < 1). These relationships have been verified experimentally. The term ax" describes the effect of long-range interactions in electrolyte solutions (i.e. ion-ion interactions in contrast to short-range dipole-dipole and ion-dipole interactions; see Section 1.2.3). At medium concentrations, the value of this term is comparable with that of the term fixly and at low concentrations up to 10~3 mol dm" 3 the term ax" predominates. The aim of the theory of electrolytes is to provide a theoretical interpretation of the coefficients in Eq. (1.3.4). At low concentrations, the Debye-Hiickel theory is valid; this theory neglects all types of interaction except for electrostatic, being satisfied with calculation of the term ax". The Debye-Hiickel theory forms the basis and is still the valid nucleus of all the other theories of strong electrolytes, which are more or less elaborations or modifications of the original concepts. 1.3.1 The Debye-Hiickel limiting law
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In infinitely dilute solutions (in the standard state) ions do not interact, their electric field corresponds to that of point charges located at very large distances and the solution behaves ideally. As the solution becomes more concentrated, the ions approach one another, whence their fields become deformed. This process is connected with electrical work depending on the interactions of the ions. Differentiation of this quantity with respect to nt permits calculation of the activity coefficient; this differentiation is identical with the differentiation 5GE/<9n/ and thus with the term RT In y,.
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The work connected with the mutual approach of the ions and deformation of their electric fields cannot be calculated directly. However, because this work depends on a change in the thermodynamic functions of state, its value depends only on the initial and final states, and the processes can be separated into several steps. According to P. Debye and E. Hiickel this separation is carried out so that the actual concentration change, i.e. the actual approach of the ions, occurs with uncharged species. The ions are supposed to be discharged in the standard state, i.e. they behave at the concentration 1 mol dm" 3 as if they were at infinite dilution. Work wl connected with this process can be readily calculated, as discharging of isolated ions is considered. The ion concentration is then changed from the standard state to concentration c. This transition is connected only with work corresponding to ideal behaviour, and not with electrical work. The action of forces other than electrostatic interionic forces in the electrolyte solution is neglected, as these are far more short-range interactions than electrostatic forces. For recharging of the ions at concentration c electrical work H>2 is expended. The calculation of work w2 is more complicated, as this work is connected with the formation of a space charge. Debye and Hiickel introduced the concept of an ionic atmosphere; i.e. an excess charge due to ions with the opposite charge is concentrated around each ion and the charge density is distributed in the ionic atmosphere according to the Maxwell-Boltzmann distribution law. This excess charge is equal to the charge of the ion considered (with opposite sign). The Debye-Huckel theory yields the coefficient yc, but the whole subsequent calculation is accompanied by numerous approximations, valid only at high dilutions, so that in the whole region where the theory is valid it may be assumed that yx~ ym~ ycFirst, electric work wx and w2 is calculated for a single ion species denoted as k. For wx the same procedure as for the quantity w2 in the Born treatment of solvation Gibbs energy (Eq. (1.2.5) will be used giving (e = Ds0))
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where r is the distance from the centre of the ion. At the final concentration c, the potential ipk at distance r is given not only by the potential of the ion, ty\, but also by the potentials of the surrounding ions. Debye and Hiickel assumed that a spherical ionic atmosphere of statistically prevailing ions with opposite charge forms around each ion, giving rise to the potential ipa. Thus y>k = ipok+ \pa. The potential of the space charge density p is given by the Poisson equation V2ii>= - (1.3.6)
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