Osmotic coefficient point depression

As we saw in Section 17.5, the activity coefficient of a nonelectrolyte solute can be calculated from the activity coefficient of the solvent, which, in turn, can be obtained from the measurement of colligative properties such as vapor pressure lowering, freezing point depression, or osmotic pressure. We used the Gibbs-Duhem equation in the form [Equation (17.33)]... 

It can be observed that g is the ratio between the observed osmotic pressure and the osmotic pressure that would be observed for a completely dissociated electrolyte that follows Henry s law [see Equation (15.47)], hence the name, osmotic coefficient. A similar result can be obtained for the boiling point elevation, the freezing point depression, and the vapor pressure lowering. 

Equilibria among water ice, liquid water, and water vapor are critical for model development because these relations are fundamental to any cold aqueous model, and they can be used as a base for model parameterization. For example, given a freezing point depression (fpd) measurement for a specific solution, one can calculate directly the activity of liquid water (or osmotic coefficient) that can then be used as data to parameterize the model (Clegg and Brimblecombe 1995). These phase relations also allow one to estimate in a model the properties of one phase (e.g., gas) based on the calculated properties of another phase (e.g., aqueous), or to control one phase (e.g., aqueous) based on the known properties of another phase (e.g., gas). 

Just as we discussed in Chapter 9, we can use measured activities of solvents (determined from vapor pressure, freezing-point depression, boiling-point elevation, or osmotic pressure) to determine activity coefficients of electrolytes in solution. For an ionic substance, the Gibbs-Duhem equation is... 

There are many measurement techniques for activity coefficients. These include measuring the colligative property (osmotic coefficients) relationship, the junction potentials, the freezing point depression, or deviations from ideal solution theory of only one electrolyte. The osmotic coefficient method presented here can be used to determine activity coefficients of a 1 1 electrolyte in water. A vapor pressure osmometer (i.e., dew point osmometer) measures vapor pressure depression. 

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... 

The osmotic coefficient, which is equal to the ratio of the actual osmotic pressure to the ideal value, is then equal to the ratio of the observed freezing point depression to the ideal (infinite dilution) value v m. 

Values of electrolyte activities, as measured by osmotic pressures, freezing point depression, and other experimental methods are in the literature (References 5 and 6, for example) or one can calculate activity coefficients based on models of molecular-level interactions between ions in electrolyte solutions. For illustrative purposes, mean molal activity coefficients for various salts at different aqueous molal (mj concentrations at 25°C are listed in Table 26.3 [7]. 

A substance in solution has a chemical potential, which is the partial molar free energy of the substance, which determines its reactivity. At constant pressure and temperature, reactivity is given by the thermodynamic activity of the substance for a so-called ideal system, this equals the mole fraction. Most food systems are nonideal, and then activity equals mole fraction times an activity coefficient, which may markedly deviate from unity. In many dilute solutions, the solute behaves as if the system were ideal. For such ideally dilute systems, simple relations exist for the solubility of substances, partitioning over phases, and the so-called colligative properties (lowering of vapor pressure, boiling point elevation, freezing point depression, osmotic pressure). 

In a similar fashion, solubility measurements (of a gas in a liquid, a liquid in a liquid, or a solid in a liquid) can be used to determine the activity coefficient of a solute in a solvent at saturation. Also, measurements of the solubility of a solid solute in two liquid phases can be used to relate the activity coefficient of the solute in one liquid to a known activity coefficient in another liquid, and freezing-point depression or boiling-point elevation measurements are frequently used to determine the activity of the solvent in a solute-solvent mixture. We have also showed that osmotic-pressure measurements can be used to determine solvent activity coefficients, or to determine the molecular weight of a large polymer or protein. 

The activity coefficients y are determined experimentally by a series of methods including vapor pressure, freezing-point depression, osmotic pressure, and solubility measurements (Denbigh 1981). 

In physical chemistry, we apply the term coUigative to those properties that depend upon number of molecules present. The principal coUigative properties are boiling point elevation, freezing point depression, vapour pressure lowering, and osmotic pressure. All such methods require extrapolation of experimental data back, to infinite dilution. This arises due to the fact that the physical properties of any solute at a reasonable concentration in a solvent are determined not by the mole fraction of solute, but by the so-called activity of the solute. This takes a value less than the actual mole fraction, and is related to it by the activity coefficient ... 

Using the freezing point depression 0, the osmotic coefficient can be calculated ... 

In Chapter 8 various equations were derived relating to the partifid pressures, freezing-point depression, osmotic pressure, etc., of an ideal solution. The corresponding expressions which are applicable to non-ideal solutions may be obtained simply by substituting in place of The coefficient has, in fact, been constructed to have this property. It foUows that an activity coefficient may be determined from experiment by application of these equations. As soon as a value of y hai been calculated from a measured property of the solution, for example a partial pressure, it may be used immediately to calculate the value of some other property of the same solution, e.g. its osmotic pressure, at the a ime temperature and pressure. 

As described in 9 6 the immediate result of measuring a fireezing-point depression or an osmotic pressure is the activity coefficient of the solvent. Provided that these results are available over a range of concentrations which extend up to very high dilution it is x>ossible to calculate the activity coefficient of the solvie by integration of the Gibbs-Duhem equation. 

In a solution of an electrolyte the activity coefi cientf of the eolverU can be determined by measiirement of its partial pressure, or from the freezing-point depression or the osmotic pressure. The relevant equations are the same as those developed in 9 6. Provided that values of the activity or osmotic coefficient have been determined over a wide range of concentrations, including some solutions which are very dilute, it is possible to calculate the activity coefficient of the solute in some particular solution by application of the Gibbs-Duhem equation. This procedure, as applied to solutions of nonelectrolytes, was described in 9 7 and 9 8. 

The general procedure described in this section for evaluating y requires knowledge of the osmotic coefficient (pm as a function of molality, (pm is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2). 

Table 2.4 Calculated freezing-point depressions of citric acid solutions, osmotic and activity coefficients of citric acid...

Freezing point depression Relative permittivity of pure water Permittivity of vacuum Osmotic coefficient Osmotic coefficient (for pressure)... 

Because polyelectrolytes are nonvolatile, the most important thermodynamic property for vapor + liquid phase equilibrium considerations is the vapor pressure of water above the aqueous solution. Instead of the vapor pressure, some directly related other properties are used, e.g., the activity of water a, the osmotic pressure 71, and the osmotic coefficient < . These properties are defined and discussed in Sect. 4. Membrane osmometry, vapor pressure osmometry, and isopiestic experiments are common methods for measuring the osmotic pressure and/or the osmotic coefficient. A few authors also reported experimental results for the activity coefficient y i of the counterions (usually determined using ion-selective electrodes) and for the freezing-point depression of water AT p. The activity coefficient is the ratio of activity to COTicentration ... 

The freezing point depression, refractive index, osmolality and specific conductance are all listed in Table 2.30. An et al. (1978) list activity and osmotic coefficients. The water absorbed by solid calcium chloride is listed in Table 2.41 and Fig. 2.81. 

Boiling point elevation and freezing point depression can be tied to the osmotic coefficient, (p, and are practical means for its measurement. We start with the Gibbs-Duhem equation at constant pressure and temperature ... 

This is the desired equation for the osmotic coefficient in terms of the boiling-point elevation, where we have inserted the van t Hoff equation for Hj. An analogous relation follows for the osmotic coefficient in terms of the freezing point depression ... 

The osmotic coefficient (j) is thermodynamically a well defined property. It can be evaluated from the freezing point depression according to the relation... 

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