Osmotic coefficient additivity

In addition to the activity and osmotic coefficients at room temperature, the first temperature derivatives and the related enthalpy of dilution data were considered for over 100 electrolytes (26, 29). The data for electrolytes at higher temperatures become progressively more sparse. Quite a few solutes have been measured up to about 50°C (and down to 0°C). Also, over this range, the equations using just first temperature derivatives have some validity for rough estimates in other cases. But the effects of the second derivative (or the heat capacity) on activity coefficients at higher temperatures is very substantial. 

In addition, the critical evaluation of enthalpies of dilution and solution, as well as evaluations of heat capacities have been initiated. These evaluations will allow calculations and correlations of activity and osmotic coefficients as a function of temperature and composition. 

In addition, the simple phenomenological relation (6.1.4), with a constant electro-osmotic coefficient lc, was replaced by a more elaborate one, accounting for the w dependence on the flow rate and the concentrations Ci, C2 via a stationary electro-osmotic calculation. This approach was further adopted by Meares and Page [7] [9] who undertook an accurate experimental study of the electro-osmotic oscillations at a Nuclepore filter with a well-defined pore structure. They compared their experimental findings with the numerically found predictions of a theoretical model essentially identical to that of [5], [6]. It was observed that the actual numerical magnitude of the inertial terms practically did not affect the observable features of the system concerned. 

Another function, the osmotic coefficient, has been used in place of the excess chemical potential or the activity coefficient. It is a multiplicative factor rather than additive, and is defined in terms of the chemical potential of the solvent. Two such functions are used, one based on molalities and the other on molarities. The first is defined, except for its absolute value, by... 

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). 

For hydrochloric acid, a strong electrolyte, calculate an experimental value of with Eq. (9) for each of the concentrations studied. In addition, use Eq. (13) to obtain a value of the osmotic coefficient based on the Debye-Hiickel theory for each concentration. Compare these experimental and theoretical values. 

Three systems were selected for examination, namely the solubilities of oxygen, carbon dioxide, and methane in water -1- sodium chloride. An accurate semiempirical equation [64] was used to express the composition dependence of the osmotic coefficient in water-r sodium chloride. The results of the calculations are presented in Fig. 1 and Table 1. One can see that Eq. (26) provides an accurate correlation for the gas solubility in solutions of strong electrolytes. In addition, the fluctuation theory allows one to use the experimental solubility data to examine the hydration in water (l)-gas (2)-cosolvent (3) mixtures. 

The lysozyme solubilities in aqueous solutions of sodium chloride are predicted for pH=4.5 and pH=6.5. In these predictions only the values of the preferential binding parameter were used and no additional (or adjustable) parameters were involved. The results are presented in Figs. 1 and 2 and the experimental preferential binding parameters used are listed in Table 2. The solubilities at pH=6.5 were predicted from the preferential binding parameter determined at pH=7.0 because the values for pH=6.5 were not available. The concentration dependence of the water activity in solutions of sodium chloride was obtained from Eq. (18) using an accurate semiempirical equation for the osmotic coefficient [37]. 

In order to obtain the number-average molar mass of a particular sample, osmotic coefficient (II/c) data, measured at various low concentrations, must be extrapolated to the zero concentration limit. In addition to the ideal gas contribution [Eq. (1.73)] that arises from individual polymers, the osmotic pressure also has a contribution from polymer-polymer interactions. The contribution to osmotic pressure from interaction... 

Those findings can be related to the measurements of the distribution functions presented in Sec. V, since the osmotic coefficient is a measure of the free counterions contributing to the osmotic pressure. The investigation of the distribution functions showed that the amount of condensed ions is always larger than the prediction from the PB theory. This entails a smaller osmotic coefficient. In addition, the stronger deviation at higher density, as well as the asymptotic correctness in the dilute limit, is in accord with the corresponding behavior of p. 

Since the experimentally determined osmotic coefficient appears to be smaller even than the molecular dynamics results, this indicates effects to be relevant that go beyond the model used for simulation. Most obvious candidates for this are the neglect of additional chemical interactions between the ions and the polyelectrolyte as well as solvation effects, i.e., interactions between the ions or the polyelectrolyte with the water molecules from the solution. It is for instance demonstrated in Ref. 46 that the osmotic coefficient also depends on whether one uses chlorine or iodine counterions. While one could certainly account for the different radii of these ions when computing the distance of closest approach entering the PB equation, the implications of the different hydration energies is much less obvious to incorporate and in principle requires very expensive all-atom simulations. 

Much deeper insights into the water sorption properties of polyelectrolytes can be provided by evaluating the (nNa)p term directly. By the use of the activity coefficient of Na+ ions in the PA A polyelectrolyte phase, (yNa)p, (aNa)p can be expressed as (aNa)p = (yNa)P [Na]p. Two concentration terms, i.e., (1) Na+ ions present in the polyelectrolyte phase to neutralize free car-boxylate groups and (2) Na+ ions imbibed in the polyelectrolyte phase in the form of NaCl, contribute the [Na]p term. Escape of Na+ ions from the polyelectrolyte phase due to their thermal motion, which produces a site vacancy of the polyion, should also be taken into consideration. The fraction of site vacancy of polyelectrolytes is available as a practical osmotic coefficient, c/>p-Na, which can simply be related to the linear charge separation of the PAA polyion. It has been revealed that PiNa is not affected by the change in the polyion concentration nor Cs, which is known as an additivity rule [16,17]. Thus the (aNa)p term can finally be expressed as... 

Most of the techniques mentioned above can be applied to electrolyte solutions in a straightforward manner. It may be necessary to use different apparatus materials due to the corrosive nature of some electrolyte systems, especially at high temperatures. In addition, special techniques exist to measure activity and osmotic coefficients in electrolyte solutions. These methods are discussed in more detail in reference [70]. 

The principal interests in this study are osmotic and activity coefficients of NaCl(ac ) and KCl(aq) solutions at temperatures to 350°C and up to saturation concentration. In the range 25-300 C and at 1 bar or saturation pressure, NaCl(aq) osmotic coefficients up to 4 m were taken from a comprehensive thermodynamic treatment of Pitzer et al. (9). Above 4 m, the values were taken from Liu and Lindsay (39). At temperatures above 300°C, osmotic coefficients were calculated from vapor pressure data of Wood et al. (4. Additional vapor pressure data are given in Refs. 41-47, but a critical evaluation of these data indicated that these are less precise measurements and were therefore given smaller weights in the regression. For KCl(aq), osmotic coefficients to 6 m at temperatures from 25-325 C at 1 bar or saturation pressure were taken from the ion interaction model of Pabalan and Pitzer (9). Additional values up to 350 C and saturation concentration were derived from Refs. 40,41, and 48. 

The references provide additional information on water activity, osmotic coefficient, and enthalpy of vaporization. 

For the transport properties, the conductivity is explained based on the pereolation eoneept where the percolation threshold occurs around X = 2 and is correlated with enough clusters being hydrated and connected by hydrated sulfonie acid sites to form a complete conductive pathway across the membrane [34,41]. With the addition of more water into the system, more elusters and channels form and the pathways become less tortuous the eonductivity increases. The electro-osmotic coefficient depends on the type... 

Figure 6 shows the potential of mean force (PMF) between a sodium ion and a chloride ion in water, at infinite dilution of the two ions, obtained from classical atomistic simulations [75]. The first minimum of the potential corresponds to the contact ion pair (CIP) distance, the second minimum corresponds to the solvent-shared ion pair (SIP) distance, and the third minimum to the solvent-separated ion pair (2SIP) distance. Figure 7a shows an example of a SIP in aqueous NaCl [75]. The infinite dilute potential of mean force in Fig. 6 can be used as an effective pah-potential in implicit solvent simulations. The osmotic coefficient (j) (ps) = nilpJc- T (with n the osmotic pressure and ps the salt number density) can be obtained through the virial route. For the case of a binary mixture of components i and j and pairwise additive, density-independent pair potentials, the virial equation can be expressed as... 

In addition to a sample peak, some anomalous peaks often appear in chromatograms. when multicomponent eluent is used [ref. 71, 72]. Such an induced peak can be observed when polyelectrolyte is chromatographed using a simple electrolyte solution as eluent the first peak corresponds to polyelectrolyte and the second to an induced peak which has the elution volume exactly the same as that of the eluent salt [ref. 33, 45-48, 73-76]. This effect of polyelectrolytes in exclusion chromatography has been explained in terms of Donnan salt exclusion established on the gel [ref. 33, 45-48, 73-76] a polyelectrolyte is barred from the gel interior and thus promote the diffusion of the penetrable eluent coion into the inner volume of the gel, the gel matrix acting as a semipermeable membrane. The eluent coion thus excluded from the polyelectrolyte zone produces an induced peak. It has been reported that the area of the induced peak allows to calculate the Donnan salt exclusion parameter [ref. 33, 74] or the osmotic coefficient [ref. 46-48, 76] of the polyelectrolyte. 

Some of the electrolytes to be tested precipitate in hydrated forms at 25 C. It is therefore necessary to calculate the water activities in addition to the electrolyte activity coefficients in order to model the solutions. The water activity may be calculated using an equation presented specifically for it s calculation, or from the results of an osmotic coefficient equation. 

The parameters were determined by fitting osmotic coefficients from solvent vapor pressure measurements with a least-squares method. The maximum concentration was about 6 mdlal and the standard deviation of fit was. 005. They were surprised to find a minimum in the degree of dissociation at about 2 molal and an increase in dissociation as the concentration increased. This behavior was attributed to the strong negative value of the HjPO -HjPO interaction. Additional parameters were presented based on a study of the KC1-KH P0 -H20 system ... 

This evaluation gives values for the osmotic coefficients and mean activity coefficients of seventy-nine uni-univalent electrolytes in aqueous solution at 25 C, with values expressed on the molality scale. The data from the literature were fitted, by statistical procedures, to equations which express the quantities as functions of electrolyte concentration. Literature references are given to fifty-one additional uni-univalent electrolytes. Also see item [159]. 

Parametrization of the thermodynamic properties of pure electrolytes has been obtained [18] with use of density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments it has been shown [19] that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature Adelman has formally shown [20] (Friedman extended his work subsequently [21]) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is experimental evidence [22] for being a function of concentration. There are two important thermodynamic quantities that are commonly used to assess departures from ideality of solutions the osmotic coefficient and activity coefficients. The first coefficient refers to the thermodynamic properties of the solvent while the second one refers to the solute, provided that the reference state is the infinitely dilute solution. These quantities are classic and the reader is referred to other books for their definition [1, 4],... 

The apparent osmotic coefficients (similariy as the apparent molar volumes, Eq. (2.54)) are usually assumed to be the sum of additive contributions coming from the unionized acid molecules HjCit and ions H+and H2Cit" (high-charged citrate anions are neglected) [89]... 

A comparison of the additivity rule with the limiting laws is instructive. The limiting law for the osmotic coefficient in the absence of added salt is given by Equation (77). If that result, with 2 i = l, is used in Equation (93), it is seen that the limiting law for (f) in the presence of added salt is identical to the additivity rule . Equation... 

Below the critical charge density, there should be no ion condensation but the activity coefficient or osmotic coefficient of the polyelectrolyte solution is explained on the assumption of the Debye-Huckel approximation. According to this theory, the additivity of osmotic pressure can be proved, though the additivity of counter-ion activity coefficient cannot be proved. 

The measurements shown in Figures 5 and 6 may be employed to test the Additivity Rule given in (A). This relation may be restated in terms of osmotic coefficients using the well-known equation relating the latter to the osmotic pressure. Hence,... 

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