Dilute solution activity coefficients

The net retention volume and the specific retention volume, defined in Table 1.1, are important parameters for determining physicochemical constants from gas chromatographic data [9,10,32]. The free energy, enthalpy, and. entropy of nixing or solution, and the infinite dilution solute activity coefficients can be determined from retention measurements. Measurements are usually made at infinite dilution (Henry s law region) in which the value of the activity coefficient (also the gas-liquid partition coefficient) can be assumed to have a constant value. At infinite dilution the solute molecules are not sufficiently close to exert any mutual attractions, and the environment of each may be considered to consist entirely of solvent molecules. The activity... 

I. 46. The magnitude of the coefficient reflects the electric charge distribution of the ionic species. A 0.1 molal solution of Al2(S04)3 has an activity coefficient of only 0.035. It should also be noted that, in dilute solutions, activity coefficients of electrolytes decrease in magnitude with increasing concentration. A minimum is reached and the coefficient then increases with concentration. See Activity Debye-Huckel Law Biomineralization... 

It has been suggested that such a sensor can be used for the accurate measurement of Ca2+ activity in dilute sample solutions because, in dilute solutions, activity coefficients tend to unity. Is that statement correct or not ... 

For infinitely dilute solutions activity coefficients approach unity so the activity and the concentration of an ion will be equal. For calculations involving more concentrated solutions corrections must be made using activity coefficients, especially when relating the calculated concentration of species to an imposed mass (mole) balance constraint. The activity coefficients can be calculated from a number of ion activity theories and the relevant equations for some of the commonly used ones are shown below. 

A fundamental concept in all theories for determining activity coefficients is that ionic interactions are involved. These interactions cause a deviation in the free energy associated with the ions from what it would be if they did not occur. Consequently, at the limit of an infinitely dilute solution, activity coefficients go to 1 because there are no ionic interactions. This basic consideration also leads to the idea that as the concentration of ions increases, their extent of interaction must also increase. Ionic strength is a measure of the overall concentration of ions in a solution and the fact that more highly charged ions exert a greater influence on ionic interactions. It is calculated as ... 

Concentration and activity of a solute are only the same for very dilute solutions, i.e. yi approaches unity as the concentration of all solutes approaches zero. For non-dilute solutions, activity coefficients must be used in chemical expressions involving solute concentrations. Although freshwaters are sufficiently dilute to be potable (containing less than about 1000 mg total dissolved solids (TDS)), it cannot be assumed that activity coefficients are close to unity. 

Many nonionizable organic solutes in water are described thermodynamically on the mole fraction scale, although their solubilities may commonly be reported in practical units, for example, molality. [Refer to Schwarzenbach et al. (1993) and Klotz (1964) for detailed discussion of such aqueous solutions.] Here, the standard state is the pure liquid state of the organic solute, that is, Xj = 1. The reference state is Xi - 1, that is, a solution in which the organic solute molecules interact with one another entirely. Activity coefficients of solute molecules in dilute aqueous solutions are generally much greater than unity for this reference state choice, jc, 1. For example, with this reference state, aqueous benzene has an experimental infinitely dilute solution activity coefficient, T nzeno of 2400 for an infinite dilution reference state, jc, - 0, the activity coefficient would be approximately 1 (Tanford, 1991). 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

A unitless correction factor that relates the relative activity of a substance to the quantity of the substance in a mixture. Activity coefficients are frequently determined by emf (electromotive force) or freezing-point depression measurements. At infinite dilution, the activity coefficient equals 1.00. Activity coefficients for electrolytes can vary significantly depending upon the concentration of the electrolyte. Activity coefficients can exceed values of 1.00. For example, a 4.0 molal HCl solution has a coefficient of 1.76 and a 4.0 molal Li Cl has a value of... 

In addition to knowing the TP dependence of equilibrium constants (Eqs. 2.25 and 2.28), we must also know the T-P dependence of solute activity coefficients and the osmotic coefficient of the solution. A theoretical model, such as Pitzer s approach, is necessary for this purpose because activity coefficients and the osmotic coefficient must be defined at finite concentrations and not simply for the infinitely dilute state, which suffices for equilibrium constants (Eqs. 2.25 and 2.28). 

The standard state for solutes in the (HL) reference is therefore the hypothetical state of pure solute (x, = 1), but with solute molecules interacting only with solvent molecules (y, = 1). Practically, chemical potentials in the standard state are obtained by making measurements at very low concentrations and extrapolating them to X,- = 1, assuming that Henry s law continues to hold to this concentration. At nonzero concentration of solutes, activity coefficients in the (HL) reference measure deviations of the solution from ideally dilute behavior. 

As can be seen from these equations the specific nature of individual ions is not expressed any more in very diluted solutions and the activity coefficient is determined solely by the ionic strength and the valence type of the electrolyte. Assuming the solutions to be properly diluted, the activity coefficients of electrolytes of the same valence type are identical in all solutions of the same ionic strength. 

Data are available for equilibrium pressure-volume-temperature of pure polymer liquids, solvent activity coefficients at infinite dilution, solvent activity coefficients at finite concentrations, and liquid-liquid phase equilibria of binary and ternary polymer solutions. 

If the solution is sufficiently dilute, the activity coefficients are approximately unity, and so equation (94) reduces under these conditions to... 

If the ionic strengths in the two cells are kept equal, then provided the solutions are relatively dilute the activity coefficient actor will be virtually unity, and the second term on the right-hand side of equation (52) is zero hence under these conditions... 

If the solution is sufficiently dilute for activity coefficients to be taken as unity, then the equations for fci and fe, in place of Ki and K2, become [cf. equation (1)]... 

At infinite dilution the activity coefficients 7n and 7m are both unity, in accordance with equations (37.1) and (37.3), respectively, which define the standard states. In dilute solutions the values will also be approximately equal, but not necessarily unity because of failure to obey Henry s law. At appreciable concentrations, however, the molality of the solution is no longer proportional to the mole fraction of the solute, and even if the solution obeyed Henry s law 7m would not be unity. In such solutions 7n and 7m will be appreciably different. 

It may be noted that in an ideal solution where all ions are completely noninteracting (a situation realized only under conditions of high dilution), the activity coefficient term would equal unity, and then K would be equal to Kc-The ionic strength of a solution is defined as ... 

Also for dilute solutions activity is equal to concentration since activity coefficient is generally taken to be imity. 

Experimental measurements show that molecules in highly compressed gases or highly concentrated solutions, especially if electrically charged, abnormally affect each other. In such cases the true activity or effective concentration may be greater or less than the measured concentration. Hence when the molecules involved in equilibrium are relatively close together, the concentration should be multiplied by an activity coefficient (determined experimentally). At moderate pressures and dilutions, the activity coefficient for nonionic compounds is close to unity. In any event, the activity coefficient correction will not be made in the problems in this book. 

In the above equation the activity of the solid equals 1 and, since the solution is extremely dilute, the activity coefficients of dissolved ions also equals I. Substituting [S2 ] = 1.5[Bi3+] and solving for [Bi3+] gives [Bi3+] = 2.7 x lO-20 M. BiiSj has a solubility equal... 

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. 

For electrolyte solutions that are sufficiently dilute, the activity coefficient can be calculated from the limiting Debye-Huckel equation... 

If the dependence of the relative permittivity of the solvent on the electric field strength of the ions is also taken into account, then other thermodynamic parameters of electrolyte solutions (activity coefficient, heat of dilution, partial molar enthalpy content of the solute etc.) can likewise be calculated in better agreement with the experimental data. Although the introduction of the field-dependent relative permittivity into the ion-ion and ion-solvent interactions is accompanied by very great mathematical difficulties, the problem can be solved successfully by employing various approximations. 

The study of dilute fluid solutions has been essential to the foundation of solvation thermodynamics and the development of macroscopic modeling for the description of observed behavior and the correlation of experimental data. Typical studies of dilute solutions have dealt with the determination of limiting values for the solute activity coefficients and the corresponding slope of their composition dependence at infinite dilution (Jonah 1983, 1986), where these quantities played a crucial role in constraining the parameterization of excess Gibbs free-energy models (Van Ness and Abbott 1982 Lupis 1983 Wallas 1985 Chialvo 1990a). 

Another method using liquid-solid equilibria determines solute activity coefficients from temperature-dependent solubility data. The pure solute Y, is in equilibrium with the saturated solution. With reference to the state of the infinitely dilute solution [Eqs. (91a)-(91c)], the equilibrium condition is given by the relation... 

For a more concentrated solution (but still very dilute), the activity coefficient does not equal unity so ... 

Relationship between infinite dilution activity coefficient and interaction coefficient in binary mixtures At infinite dilution, the activity coefficient is solely a measure of solvent-solute interactions. The activity coefficient of component i at infinite dilution is shown by that is, 0. Derive the following expression for from the PR-EOS for a binary mixture and show that In is linear in... 

The prediction from the theoretical argument above, that a solute activity coefficient in a dilute solution is a linear function of the composition variable, is borne out experimentally as illustrated in Fig. 9.10 on page 264. This prediction applies only to a nonelectrolyte solute for an electrolyte, the slope of activity coefficient versus molality approaches —oo at low molality (page 290). 

---