Real solutions, activity coefficient

In an ideal solution activities are equal to concentrations. In real solutions, activity coefficients are introduced to correct for the nonideal effects of the different solutes. 

In these equations addends J rdnic, and i T-lny. characterize deviation of the solutions from ideal and the work, which is necessary to expend in order to squeeze 1 mole of component i of the ideal solution into real solution. Activity coefficients can be greater or smaller than 1. When pressure of a gas solution or concentration of dissolved substances tends to 0, fugacity coefficients or activities coefficients approach 1. Even in diluted real solutions charged ions and dipole molecules experience electrostatic interaction, which shows up in a decrease of activities coefficient. Only in very diluted solutions this interaction becomes minuscule, and fugacity and activities values tend to values of partial pressure and concentration, respectively. Table 1.3 summarizes calculation formulae for activities values of groxmd water components under ideal and real conditions. 

Chemical Potentials of Real Solutions. Activity, a and Activity Coefficients, f... 

We will now have a look at the standard states for each of these cases. These states are very different from each other, and some are very hypothetical and seemingly very unrealistic. We will try to show how this arises, and that the standard states we use are actually quite reasonable. To do this with real numbers rather than just symbols, we need either experimental data or some equations that simulate or fit experimental data in a realistic way. Real data unfortunately have uncertainties, so we will borrow the concept of regular solutions from Chapter 10. For our purposes here, all we need to know is that in such solutions, activity coefficients follow the relationships... 

Flowever, in real systems, activity coefficients may play a small role in nonionic surface complexation reactions. In order for the cancellation to occur in Eqs (186)—(188) requires that the activity coefficient of the surface site and the nonionic complexed surface are the same. For the primitive interfacial model they are the same so the activity coefficient quotient reduces to unity. For nonionic—surface site complexes in real solid—solution interfacial systems the activity coefficients... 

A quite different approach was adopted by Robinson and Stokes [8], who emphasized, as above, that if the solute dissociated into ions, and a total of h molecules of water are required to solvate these ions, then the real concentration of the ions should be corrected to reflect only the bulk solvent. Robinson and Stokes derive, with these ideas, the following expression for the activity coefficient ... 

Standard potentials Ee are evaluated with full regard to activity effects and with all ions present in simple form they are really limiting or ideal values and are rarely observed in a potentiometric measurement. In practice, the solutions may be quite concentrated and frequently contain other electrolytes under these conditions the activities of the pertinent species are much smaller than the concentrations, and consequently the use of the latter may lead to unreliable conclusions. Also, the actual active species present (see example below) may differ from those to which the ideal standard potentials apply. For these reasons formal potentials have been proposed to supplement standard potentials. The formal potential is the potential observed experimentally in a solution containing one mole each of the oxidised and reduced substances together with other specified substances at specified concentrations. It is found that formal potentials vary appreciably, for example, with the nature and concentration of the acid that is present. The formal potential incorporates in one value the effects resulting from variation of activity coefficients with ionic strength, acid-base dissociation, complexation, liquid-junction potentials, etc., and thus has a real practical value. Formal potentials do not have the theoretical significance of standard potentials, but they are observed values in actual potentiometric measurements. In dilute solutions they usually obey the Nernst equation fairly closely in the form ... 

For ideal solutions, the activity coefficient will be unity, but for real solutions, 7r i will differ from unity, and, in fact, can be used as a measure of the nonideality of the solution. But we have seen earlier that real solutions approach ideal solution behavior in dilute solution. That is, the behavior of the solvent in a solution approaches Raoult s law as. vi — 1, and we can write for the solvent... 

The thermodynamic properties of real electrolyte solutions can be described by various parameters the solvent s activity Oq, the solute s activity the mean ion activities a+, as well as the corresponding activity coefficients. Two approaches exist for determining the activity of an electrolyte in solution (1) by measuring the solvent s activity and subsequently converting it to electrolyte activity via the thermodynamic Gibbs-Duhem equation, which for binary solutions can be written as... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

It is generally observed that as the temperature increases, real solutions tend to become more ideal and r can be interpreted as the temperature at which a regular solution becomes ideal. To give a physically meaningful representation of a system r should be a positive quantity and larger than the temperature of investigation. The activity coefficient of component A for various values of Q AB is shown as a function of temperature for t = 3000 K and xA = xB = 0.5 in Figure 9.3. The model approaches the ideal model as T - t. 

The values of activity a and concentration c are the same for very dilute solutions, so the ratio of a and c is one because the real and perceived concentrations are the same. If a = c, then Equation (7.25) shows how the activity coefficient y has a value of unity at low concentration. 

By contrast, the perceived concentration is usually less than the real concentration whenever the solution is more concentrated, so y < 1. To illustrate this point, Figure 7.9 shows the relationship between the activity coefficient y (as V) and concentration (as V) for a few simple solutes in water. The graph shows clearly how the value of y can drop quite dramatically as the concentration increases. 

The extent of ionic association depends on the ions we add to the solution. And the extent of association will effect the extent of screening, itself dictating how extreme the difference is between perceived and real concentration. For these reasons, the value of y (= a c) depends on the choice of solute as well as its concentration, so we ought to cite the solute whenever we cite an activity coefficient. 

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

Equation 1 implies that solubility is independent of solvent type, and is only a function of the equilibrium temperature and characteristic properties of the solid phase. In real systems the effect of non-ideality in the liquid phase can significantly impact the solubility. This effect can be correlated using an activity coefficient (y) to account for the non-ideal liquid phase interactions between the dissolved solute and solvent molecules. Eq. 1. then becomes [7,8] ... 

The case of a real solution, shown by equation 2.62, is handled by introducing a dimensionless correction, called the activity coefficient (y ). In the high dilution limit (mi -> 0), Xi = 1, and equation 2.62 reduces to 2.61. The activity of species i is simply a, = ypnjma. 

Thus, the ideal solution is a reference for the solvent in a real solution, and the activity coefficient of the solvent measures the deviation from ideality. 

The stability constants are defined here in terms of concentrations and hence have dimensions. True thermodynamic stability constants K° and (3° would be expressed in terms of activities (Section 2.2), and these constants can be obtained experimentally by extrapolation of the (real) measurements to (hypothetical) infinite dilution. Such data are of limited value, however, as we cannot restrict our work to extremely dilute solutions. At practical concentrations, the activities and concentrations of ions in solution differ significantly, that is, the activity coefficients are not close to unity worse still, there is no thermodynamically rigorous means of separating anion and cation properties for solutions of electrolytes. Thus, single-ion activity coefficients are not experimentally accessible, and hence, strictly speaking, one cannot convert equations such as 13.6 or 13.8 to thermodynamically exact versions. 

Generally, the expression f/fTe = y, xt = a, is referred to as the activity of the compound. That is, a, is a measure of how active a compound is in a given state (e.g., in aqueous solution) compared to its standard state (e.g., the pure organic liquid at the same T and p). Since yt relates a, , the apparent concentration of i, to the real concentration xt, it is only logical that one refers to yt as the activity coefficient. It must be emphasized here that the activity of a given compound in a given phase is a relative measure and is, therefore keyed to the reference state. The numerical value of Yi will therefore depend on the choice of reference state, since, as we have seen in Section 3.2, molecules of i in different reference states (i.e., liquid solutions) interact differently with their surroundings. 

Equation 6-24 and the equations that follow from it apply to molal activities. However, the concentration can be substituted for activity in very dilute solution where the behavior of the dissolved molecules approximates that of the hypothetical ideal solution for which the standard state is defined. For any real solution, the activity can be expressed as the product of an activity coefficient and the concentration (Eq. 6-26). 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

The third and fourth terms in (9.43) are usually ignored. This can be accepted for very dilute aqueous solutions where the activity coefficients tend to unity, their ratio converges even faster with dilution, and water is in a larger molar excess. Unfortunately, conditions in most real measuring situations are not such that these terms can be ignored with confidence. The variability of these two terms then introduces error into the optical determination of pH or other ions. 

These interactions exist in real systems for which the activity has the physical meaning of the effective concentration. Thus, only for dilute real solutions does A = a. In mixtures, the activity coefficient is usually, but not always, less than one and is affected by all the species in the multicomponent mixture. 

The three terms in these equations reading from left to right are related to 7, a , and to of Eq. 2.13, respectively. The activity coefficient and the osmotic coefficient measure the degree to which solute concentrations and the activity of water (aw) depart from ideal solutions, respectively. For ideal solutions, a = to and 7 = 1.0 (Eq. 2.13) or Gex = 0 (Eq. 2.32). Similarly, aw = 1.0 for an ideal solution. In the real world, solutions are rarely ideal, except in the infinitely dilute case we therefore need a model for calculating and (f>[= f(aw)]. An early model based on statistical mechanics was developed by Debye and Hiickel (1923). Their equations are... 

As real solutions may depart from ideality in a number of different ways, the activity coefficients in the above equations may take various expressions. We here discuss one... 

The basis of the ideal solution model is that the thermodynamic activities of the components are the same as their mole fractions. Implicit in this assumption is the idea that the activity coefficients are equal to unity. This is at best an approximation and has been found to be invalid in most cases. Solutions in which activity coefficients are taken into account are referred to as "real" solutions and are described by equation 3.4. 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

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