Osmotic coefficient defined

For very dilute solutions it is sufficient to retain only the term in djT. Then, by employing the osmotic coefficient defined by (20.9) we find that (c/. 20.55)... 

For the pure single-electrolyte MX, the osmotic coefficient defined by Pitzer (2000) ... 

Osmotic coefficient (([)) - Defined by ( > = In aJ(MJLm-, where is the molar mass of substance A (normally the solvent), is its activity, and the are molalities of the solutes. [1] Osmotic pressure (II) - The excess pressure necessary to maintain osmotic equilibrium between a solution and the pure solvent separated by a membrane permeable only to the solvent. In an ideal dilute solution n = c RT, where is the amount-of-substance concentration of the solute, R is the molar gas constant, and T the temperature. [1,2]... 

This difficulty can be overcome by writing equivalent expressions whose variables do not go to infinity at the limit of. vi—>0. A function known as the practical osmotic coefficient is one that can be used in a graphical method to obtain In 7 4.2- The practical osmotic coefficient expressed in terms of mole fraction is defined as... 

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... 

Figure 3. Gurney parameters for + — pairs determined by fitting data for osmotic coefficient and conductivity A (38J. The line in the figure represents ideal correlation. The data are for alkali halides, except fluorides, in water. The parameter d+. is defined for a simpler MM-level model than Equation 9 in Ref. 38 it is reported that d+./k T = 0.75 + 3.6 A+, /k T. These correlations have been found by Justice and Justice...

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... 

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. 

In accordance with (6.3.7a) define the dimensionless electro-osmotic coefficient ui as... 

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). 

Counterion binding is not a well defined quantity, with various experimental techniques weighing the ion distribution slightly differently. Thermodynamic methods (e.g. ion activities or osmotic coefficients) monitor the free counterion concentration, transport methods (e.g. ion self diffusion or conductivity) the counterions diffusing with the micelle, and spectroscopic methods (e.g. NMR) the counterions in close contact with the micelle surface. Measurement of the effect of Na+ counterions on the symmetric S-O stretching modes would also be expected to be highly dependent on the distance of the counterion from the micelle surface (similar to the NMR method). 

To avoid singularities in the integrand when integrating Eq. (40), it is advantageous to define a quantity , called the osmotic coefficient, as... 

To circumvent the last term blowing up when integrating this equation from infinite dilution, we define the osmotic coefficient for an electrolyte as... 

The product zc equals Qc where Q is the fraction of ionizable monomers in the polyelectrolyte and c is the monomolecular concentration of the polyelectrolyte. To account for nonideality due to electrostatic interaction etc., the osmotic coefficient is introduced and defined as 0=1 l/rfa with... 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

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 ... 

Figure 23. Dependence of MacMillan-Mayer osmotic coefficient on concentration for lithium chloride aqueous solutions at 298 K dotted line indicates calculated and full line indicates experimental dependence on IM, the ionic strength defined in terms of MacMillan-Mayer variables (Friedman and Krishnan, 1973a).

The Osmotic Coefficient.—Instead of calculating activity coefficients from freezing-point and other so-called osmotic measurements, the data may be used directly to test the validity of the Debye-Hiickel treatment. If 6 is the depression of the freezing point of a solution of molality m of an electrolyte which dissociates into v ions, and X is the molal freezing-point depression, viz., 1.858° for water, a quantity , called the osmotic coefficient, may be defined by the expression... 

This expression is seen to resemble the first two terms of equation (39.24) with hf defined by (39.51), replacing j, defined by (39.20). Although h and j become identical at infinite dilution, as will be shown below, there is an important difference between these two functions and hence between the activity coefficients derived from them. Whereas j applies to the freezing point of the solution, h refers to the particular temperature, e.g., 25 C, at which the osmotic coefficient is determined, e.g., from vapor pressure measurements. 

At this point, it is convenient to introduce a quantity called the osmotic coefficient, (p. It is defined according to the equation... 

We believe that fundamental assessment of the factors important in defining counterion distribution in charged polymers (crosslinked and linear) has been resolved with our interpretation of osmotic coefficient data. By our analysis, ion-solvent Interactions are believed to contribute most importantly in the polystyrene sulfonate-based resins to their ion-exchange selectivity patterns. The agreement obtained between the prediction of these patterns and their observed distribution without need to resort to a single measurement for calibration of the activity coefficient terms in the polymer provides strong support for the validity of the interpretations made. 

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