Coefficient, activity freezing point

Elaborate procedures have been developed for obtaining activity coefficients from freezing-point and thermochemical data. However, to avoid duplication, the details will not be outlined here, because a completely general discussion, which is applicable to solutions of electrolytes as well as to nonelectrolytes, is presented in Chapter 21 of the Third Edition of this book [6]. 

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... 

Even in 1928, Harman (34) concluded from conductivity, transfer numbers, activity coefficients, hydrolysis, osmotic activity, freezing point data, phase relations, and diffusion experiments that there are only two simple silicates, NajSiOj and NaHSiOa, and that silicates in the SiO rNajO ratio range of 2 1 to 4 1 become increasingly colloidal. ... 

Similarly, concepts of solvation must be employed in the measurement of equilibrium quantities to explain some anomalies, primarily the salting-out effect. Addition of an electrolyte to an aqueous solution of a non-electrolyte results in transfer of part of the water to the hydration sheath of the ion, decreasing the amount of free solvent, and the solubility of the nonelectrolyte decreases. This effect depends, however, on the electrolyte selected. In addition, the activity coefficient values (obtained, for example, by measuring the freezing point) can indicate the magnitude of hydration numbers. Exchange of the open structure of pure water for the more compact structure of the hydration sheath is the cause of lower compressibility of the electrolyte solution compared to pure water and of lower apparent volumes of the ions in solution in comparison with their effective volumes in the crystals. Again, this method yields the overall hydration number. 

From a thermodynamic standpoint, freezing point measurements and isopiestic measurements are similar since both yield directly the activity of the solvent. When done carefully, freezing point data can generate activity coefficient values at concentrations down to 0.001 molal. During the first half of this century, much activity coefficient data was obtained from freezing point measurements. However, the popularity of this technique has decreased and is seldom used for aqueous solutions at the present time. 

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)]... 

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... 

ACTIVITY COEFFICIENT. A fractional number which when multiplied by the molar concentration of a substance in solution yields the chemical activity. This term provides an approximation of how much interaction exists between molecules at higher concentrations. Activity coefficients and activities are most commonly obtained from measurements of vapor-pressure lowering, freezing-point depression, boiling-point elevation, solubility, and electromotive force. In certain cases, activity coefficients can be estimated theoretically. As commonly used, activity is a relative quantity having unit value in some chosen standard state. Thus, the standard state of unit activity for water, dty, in aqueous solutions of potassium chloride is pure liquid water at one atmosphere pressure and the given temperature. The standard slate for the activity of a solute like potassium chloride is often so defined as to make the ratio of the activity to the concentration of solute approach unity as Ihe concentration decreases to zero. 

Activity coefficients of ions are determined using electromotive force, freezing point, and solubility measurements or are calculated using the theoretical equation of Debye and Htickel. 

In Equation 4.21, the activity of pure water (a) is unity and the activity of the water with the inhibitor (a ) is the product of the water concentration (xw) and the activity coefficient (xw). The water concentration is known and the activity coefficient is easily obtained from colligative properties for the inhibitor, such as the freezing point depression. For instance the activity of water in aqueous sodium chloride solutions may be obtained from Robinson and Stokes (1959, p. 476) or from any of several handbooks of chemistry and physics. 

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). 

Example 4. The freezing point of a 20% by weight aqueous solution of ethanol is 10.92°C. What is the activity coefficient of water in this solution ... 

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... 

Activity coefficient the ratio of the activity (of an electrolyte) as measured by some property, such as the depression of the freezing point of a solution, to the true concentration (molality). It is usually less than 1 and increases as the solution becomes more dilute, when the attractive forces between oppositely charged ions become negligible. 

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 activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

Values of Activity Coefficients.—Without entering into details, it is evident from the foregoing discussion that activities and activity coefficients are related to chemical potentials or free energies several methods, both direct and indirect, are available for determining the requisite differences of free energy so that activities, relative to the specified standard states, can be evaluated. In the study of the activity coefficients of electrolytes the procedures generally employed are based on measurements of either vapor pressure, freezing point, solubility or electromotive force. The results obtained by the various methods arc... 

The experimentally determined activity coefficients, based on vapor pressure, freezing-point and electromotive force measurements, for a number of typical electrolytes of different valence types in aqueous solution at 25 , are represented in Fig. 49, in which the values of log / are plotted against the square-root of the ionic strength in these cases the solutions contained no other electrolyte than the one under consideration. Since the Debye-Htickel constant A for water at 25 is seen from Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should be equal to —0.509 the results to be expected theoretically, calculated in this manner, are shown by the dotted lines. It is evident that the experimental results approach the values required by the Debye-Hiickel limiting law as infinite dilution is attained. The influence of valence on the dependence of the activity coefficient on concentration is evidently in agreement with theoretical expectation. Another verification of the valence factor in the Debye-Hiickel equation will be given later (p. 177). 

Individual Ion Activities.—The methods described in Chap. V for the determination of the activities or activity coefficients of electrolytes, e well as those depending on vapor pressure, freezing-point or other osmotic measurements, give the nean values for b >th ions into which the solute A convenient form of this equat ion for approximate purposes is... 

Nonelectrolytes in nonaqueous solvents Activity coefficients of dilute solutions of solutes can be studied experimentally by liquid-liquid chromatography as well as techniques such as solvent extraction, light scattering, vapor pressure, and freezing point depression. 

Determination of Activity Coefficients at Temperatures Other than the Freezing Point... 

The difficulty with Eq. (3.13.1) is that it yields only at the freezing point Tf of the solution. One generally wishes to know the activity coefficient at some... 

The integrations can be executed numerically after the freezing point depression has been determined empirically as a function of the molality m and vice versa. This then yields the activity coefficient of the solute at the freezing point of the solution. 

To find the activity coefficient at a standard temperature Tg other than the freezing point we set... 

In the case of a system where, although the phases are not ideal, they can be treated as regular solutions, the liquidus and solidus curves can be calculated from (23.1) and (23.2), using (21.52 ) for the activity coefficients. This has been done by Scatchard and Hamer J for a number of systems the results for the silver -h palladium and gold -h platinum systems are shown in figs. 23.2 and 23.3. The agreement between the calculated and observed freezing point curves is entirely satisfactory. 

The freezing points of three glycerol solutions in water are — 1.918 C for 1.0 molal, — 3.932 C for 2.0 molal and — 10.68 for 5.0 molal. Determine the activities and activity coefficients of the water in these solutions on the basis of the usual standard state, and consider the departure from Raoult s law. The vapor pressure of pure (supercooled) water at — 1.92 C is 3.980 mm. what would be the aqueous vapor pressure of the 1.0 molal glycerol solution at this temperature ... 

ACTIVITY COEFFICIENTS OF SODIUM CHLOBIDE FROM FREEZING POINT MEASUREMENTS... 

The amount of reliable data available for the purpose of correcting activity coefficients obtained from freezing point measurements is not large. The freezing point method has thus been mainly used for the study of dilute solutions. 

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. 

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