Solvent osmotic coefficient

As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

The osmotic coefficient is often used as a measure of the activity of the solvent instead of a because a is nearly unity over the concentration range where 7 is changing, and many significant figures are required to show the effect of solute concentration on a. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a. ... 

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,... 

Thermodynamic methods also measure the activity coefficient of the solvent (it should be recalled that the activity coefficient of the solvent is directly related to the osmotic coefficient—Eq. 1.1.19). As the activities of the components of a solution are related by the Gibbs-Duhem equation, the measured activity coefficient of the solvent can readily be used to calculate the activity coefficient of a dissolved electrolyte. 

The ideality of the solvent in aqueous electrolyte solutions is commonly tabulated in terms of the osmotic coefficient 0 (e.g., Pitzer and Brewer, 1961, p. 321 Denbigh, 1971, p. 288), which assumes a value of unity in an ideal dilute solution under standard conditions. By analogy to a solution of a single salt, the water activity can be determined from the osmotic coefficient and the stoichiometric ionic strength Is according to,... 

Several authors have attempted to corrolate the degradation rate with such solvent parameters as osmotic coefficient [35], viscosity [36-38] and the Flory Huggins interaction parameter, % [39,40] - a low % value indicates a good solvent in which the polymer is expected to exhibit an open conformation (as opposed to coiled) and therefore is more susceptible to degradation (Fig. 5.15). 

To deal with this problem, Bjermm [4] suggested that the deviation of solvent behavior from Raoult s law be described by the osmotic coefficient g rather than by the activity coefficient 71. The osmotic coefficient is defined by the relationships... 

Once values of g as a function of solution composition have been obtained, the Gibbs-Duhem equation can be used to relate the osmotic coefficient of the solvent to the activity coefficient of the solute. For this purpose, the chemical potential of the solvent is expressed as in Equation (19.42), with the approximation given in Equation (19.53), so that... 

This expression is analogous to Eiq. (2.3), in that (1 — (p) expresses the contribution of the solvent and In y+ that of the electrolyte to the excess Gibbs energy of the solution. The calculation of the mean ionic activity coefficient of an electrolyte in solution is required for its activity and the effects of the latter in solvent extraction systems to be estimated. The osmotic coefficient or the activity of the water is also an important quantity related to the ability of the solution to dissolve other electrolytes and nonelectrolytes. 

The activity of water is obtained by inserting Eq. (6.12) into Eq. (6.11). It should be mentioned that in mixed electrolytes with several components at high concentrations, it is necessary to use Pitzer s equation to calculate the activity of water. On the other hand, uhjO is near constant (and = 1) in most experimental studies of equilibria in dilute aqueous solutions, where an ionic medium is used in large excess with respect to the reactants. The ionic medium electrolyte thus determines the osmotic coefficient of the solvent. 

The papers in the second section deal primarily with the liquid phase itself rather than with its equilibrium vapor. They cover effects of electrolytes on mixed solvents with respect to solubilities, solvation and liquid structure, distribution coefficients, chemical potentials, activity coefficients, work functions, heat capacities, heats of solution, volumes of transfer, free energies of transfer, electrical potentials, conductances, ionization constants, electrostatic theory, osmotic coefficients, acidity functions, viscosities, and related properties and behavior. 

Because of solvent loss, which occurred even with covered crucibles and which was of the same order as the weighing errors ( 5 X 10 4 g), the correction of weighing for the vacuum was not considered. Then the mean error in the molality determination was 0.3%. In each desiccator, 2-3 standard solutions (KCl or CaCk) were present. Their osmotic coefficient was taken from the literature (11). In the system considered equilibrated, the molal concentrations of the standard solutions differed by no more than 0.5%. 

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... 

If we choose to express the chemical potential of the solvent in terms of the osmotic coefficient, so that (Eq. (8.119))... 

In order to discuss thermodynamic properties in dilute aqueous solutions at temperatures other than 298.15 K, it is necessary to have the standard enthalpies of the species involved. Over narrow ranges of temperature, calculations can be based on the assumption that Af// values are independent of temperature, but more accurate calculations can be made when Cpm(i) values are known. It is also necessary to take into account the temperature dependencies of the numerical coefficients in equations 3.6-4 to 3.6-6. Clarke and Glew (1980) calculated the Debye-Hiickel slopes for water between 0 and 150°C. They were primarily concerned with electrostatic deviations from ideality of the solvent osmotic... 

Using the results of Example 1, calculate the osmotic coefficient for a chloroform-acetone solution with xchl = 0.424, treating acetone as the solvent. 

Instead of characterizing the deviation from ideality for the solvent 1 in terms of its activity coefficient y, we may introduce the osmotic coefficient defined by = (ln y J/lnjq as shown in Eq. 8.4 ... 

The non-ideality of the solvent in an aqueous 1 1 salt solution can be expressed in terms of a practical osmotic coefficient, 0 (17),... 

The osmotic coefficients do not appear to be very sensitive to solvent structural differences and, indeed, this also has been noted when results from salt solutions in water and deuterium oxide were compared178). 

If we consider a membrane having the same solute concentration on both sides, we have All 0 However, a hydrostatic pressure difference AP exists between the two sides, and we have a flow Jv that is a linear function of AP. The term Lp is called the mechanical filtration coefficient, which represents the velocity of the fluid per unit pressure difference between the two sides of the membrane. The cross-phenomenological coefficient Ldp is called the ultrafiltration coefficient, which is related to the coupled diffusion induced by a mechanical pressure of the solute with respect to the solvent. Osmotic pressure difference produces a diffusion flow characterized by the permeability coefficient, which indicates the movement of the solute with respect to the solvent due to the inequality of concentrations on both sides of the membrane. 

Similarly, for a real solution the excess solvent chemical potential y, may be shown as a function of the osmotic coefficient, ([) ... 

In order to evaluate (j), the total excess free energy expression given in equation 17 will be differentiated with respect to the solvent concentration. This involves no changes in the concentration ratios of any ionic specie m to the total concentration of all ionic species m. Differentiation of equation 17 with respect to m results in the working equation for the osmotic coefficient of a mixed electrolyte solution. Details of the differentiation are given elsewhere (15,16). 

The polyion domain volume can be computed by use of the acid-dissociation equilibria of weak-acid polyelectrolyte and the multivalent metal ion binding equilibria of strong-acid polyelectrolyte, both in the presence of an excess of Na salt. The volume computed is primarily related to the solvent uptake of tighdy cross-linked polyion gel. In contrast to the polyion gel systems, the boundary between the polyion domain and bulk solution is not directly accessible in the case of water-soluble linear polyelectrolyte systems. Electroneutrality is not achieved in the linear polyion systems. A fraction of the counterions trapped by the electrostatic potential formed in the vicinity of the polymer skeleton escapes at the interface due to thermal motion. The fraction of the counterion release to the bulk solution is equatable to the practical osmotic coefficient, and has been used to account for such loss in the evaluation of the Donnan phase volume in the case of linear polyion systems. 

This term describes the number of moles of osmotically active solute particles per unit mass of solvent (osmol kg ). For an ideal solute, the osmolality can be determined by multiplying the molality by n, the number of solute particles produced in solution (e.g. for NaCl, n = 2). However, for real solutes, a correction factor (the osmotic coefficient, (j>) is used ... 

Eq. (26) can be used to calculate the Henry constant H2t in a binary solvent water (1) -r electrolyte (3) using experimental data for the osmotic coefficient and considering e as an adjustable... 

Instead of characterizing deviations from ideality of the solvent by its activity coefficient y, it is often advantageous to introduce the osmotic coefficient ( ) of Bjerrum and Guggenheim, and to write the chemical potential in the form... 

The chemical potential of the solvent can also be expressed in terms of the osmotic coefficient An approximate form of (20.8) for dilute solutions is... 

Alternatively, introducing the osmotic coefficient of the solvent by means of (20.9 ),... 

This last equation enables the activity coefficient of the solute to be evaluated if the osmotic coefficient of the solvent is known for all solutions more dilute than the solution under consideration. 

---