Electrolyte solutes Gibbs-Duhem integration

In theory, once the activity of an electrolyte in solution is known, the activity of the solvent can be determined by the Gibbs-Duhem integration (see section 2.11). In practice, the calculation is prohibitive, because of the chemical complexity of most aqueous solutions of geochemical interest. Semiempirical approximations are therefore preferred, such as that proposed by Helgeson (1969), consisting of a simulation of the properties of the H20-NaCl system up to a solute... 

In the case of an electrolyte solute the Gibbs-Duhem integration is carried out in a slightly different way. Consider an electrolyte solute represented by the formula, where v+ and i represent the numbers of cations and anions in its formula. In the case of CaCb, = 1 and v = 2. The total number of ions in the formula is denoted by v, equal to - - v. The substance must be electrically neutral ... 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

An example drawn from Deitrick s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the thermodynamic goodness of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4, 5]. 

Determination of the activity coefficients of the non-volatile solute in a solution is difficult. If electrolytes (ions) are present, the activities can be obtained from experimental electromotive force (EMF) measurements. However, for non-electrolyte and non-volatile solutes an indirect method is applied to find initially the activity of the solvent over a range of solute concentrations, and then the Gibbs-Duhem equation is integrated to find the solute activity. If the solution is saturated, then it is easy to calculate the activity coefficient... 

As has been pointed out, the equations are all integrations of the Gibbs-Duhem relationship. They consequently cannot be applied to systems which when treated in the ordinary fashion apparently do not follow this basic relation, as in the case of dissociation of electrolytes in solution. 

For a nonvolatile substance we must find a way to determine its activity coefficient that does not depend on measuring its vapor pressure. We will discuss three different methods. The first is through integration of the Gibbs-Duhem equation. The second is through a theory due to Debye and Hiickel, which can be applied to electrolyte solutes. The third method for electrolyte solutes is an electrochemical method, which we will discuss in Chapter 8. Published data are available for common electrolytes, and some values are included in Table A. 11 in Appendix A. 

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