Mean molal property

It is sometimes convenient to consider an intensive property a mean molal property, defined by... 

We shall now derive a differential equation for the mean molal property g which is of considerable importance. We consider (7 as a function of r,p, i,. . . , and write its total differential in the form... 

The dependence of the generalized mean molal property g on the mole fraction of component 2, Xj, can be determined by experiment. A plot of g versus X2 yields a curve (Fig. 2-1). The equation of the tangent to this curve at the point X2 = a is... 

Fig. 2-1. Plot of the mean molal property g versus the mole fraction of component 2, X2.

As die molality of an ion (/w and mg.) cannot be altered independently, it is not possible to measure either G + or Ggi.. In order to overcome this difficulty, we introduce mean thermodynamics properties of two ions. 

Two additional observations should be made. First, the methods used here treat each of the ionic types as a separate species that influences the thermodynamic properties of solutions very strongly by virtue of its associated charge. Second, it is instructive to examine the dependence of the mean molal activity coefficient for several different electrolytes as a function of the molality. Representative examples are shown in Fig. 4.3.1. One sees at first a very steep drop in as m is increased, and then either a gradual or a very sharp... 

When the measured pH is used, the following conditions prevail (1) the individual-ion activities and individual-ion activity coefficients again depend on choice of scale (regardless of the inconsistency of the measured pH (2) the mean-activity coefficients are independent of the measured pH and, thus, independent of scale, and (3) the calculated mean activities, mean molalities, ion-activity products, saturation indices and CO2 partial pressure are not independent of scale. This dependency of thermodynamic properties on scale when the measured pH is used in calculations is particularly acute in the carbonate... 

On the basis of the above result, it is suggested that the osmotic pressure approach may be as useful for the estimate of thermodynamic properties of simple electrolyte mixtures in mixed solvents as in aqueous media. Additional research with alkali chlorides in alcohol-water mixtures is in progress to substantiate this conclusion. The mean molal coefficient data that are published for the alkali chlorides (12) are expected to facilitate a meaningful prognosis of the osmotic pressure model. 

From this expression comes the definition of the mean activity coefficient. Y+. in terms of the ionic activity coefficients Yc and Ya The mean activity coefficient is the property which is determined or calculated from experimental measurements. A similar expression results for the mean molality m, which is not generally used in reporting experimental measurements ... 

The same equation applies in a 3-component system containing diluent at constant mole fraction of diluent. Therefore, pressure drop can be related to conversion, if data on the partial molal properties of monomer and polymer and the compressibility of the mixture axe available. Since such data are unavailable, conversion must be obtained by sampling and gravimetric determinations of yield or by means of a previously established calibration of 3deld vs. pressure drop 34). 

If the validity of Eq. (1.3.31) is assumed for the mean activity coefficient of a given electrolyte even in a mixture of electrolytes, and quantity a is calculated for the same measured electrolyte in various mixtures, then different values are, in fact, obtained which differ for a single total solution molality depending on the relative representation and individual properties of the ionic components. 

Most of the properties change somewhat from one end to the other of industrial columns for effecting separations, so that the mass transfer coefficients likewise vary. Perhaps the property that has the most effect is the mass rate of flow which appears in the Reynolds number. Certainly it changes when there is a substantial transfer of material between the two phases in absorption or stripping and even under conditions of constant molal overflow in distillation processes, the mass rate of flow changes because of differences of the molecular weights of the substances being separated. As a practical expedient, however, mass transfer coefficients are evaluated at mean conditions in a column. 

In the previous chapter, we described the thermodynamic properties of nonelectrolyte solutions. In this chapter, we focus on electrolytes as solutes. Electrolytes behave quite differently in solution than do nonelectrolytes. In Chapter 11, we described the strong electrolyte standard state and summarized relationships between the activity of the solute ai, the mean activity coefficient 7 , and the molality m in Table 11.3. 

The inclusion of activity coefficients into the simple equations was briefly considered by Purlee (1959) but his discussion fails to draw attention to the distinction between the transfer effect and the activity coefficient (y) which expresses the non-ideal concentration-dependence of the activity of solute species (defined relative to a standard state having the properties of the infinitely dilute solution in a given solvent). This solvent isotope effect on activity coefficients y is a much less important problem than the transfer effect, at least for fairly dilute solutions. For example, we have already mentioned (Section IA) that the nearequality of the dielectric constants of H20 and D20 ensures that mean activity coefficients y of electrolytes are almost the same in the two solvents over the concentration range in which the Debye-Hiickel limiting law applies. For 0-05 m solutions of HC1 the difference is within 0-1% and thus entirely negligible in the present context. Of course, more sizeable differences appear if concentrations are based on the molality scale (Gary et ah, 1964a) (see Section IA). 

This is used to express the concentration of solute relative to the mass of solvent, i.e. molkg . Molality is a temperature-independent means of expressing solute concentration, rarely used except when the osmotic properties of a solution are of interest (p. 49). 

A single homogeneous phase such as an aqueous salt (say NaCl) solution has a large number of properties, such as temperature, density, NaCl molality, refractive index, heat capacity, absorption spectra, vapor pressure, conductivity, partial molar entropy of water, partial molar enthalpy of NaCl, ionization constant, osmotic coefficient, ionic strength, and so on. We know however that these properties are not all independent of one another. Most chemists know instinctively that a solution of NaCl in water will have all its properties fixed if temperature, pressure, and salt concentration are fixed. In other words, there are apparently three independent variables for this two-component system, or three variables which must be fixed before all variables are fixed. Furthermore, there seems to be no fundamental reason for singling out temperature, pressure, and salt concentration from the dozens of properties available, it s just more convenient any three would do. In saying this we have made the usual assumption that properties means intensive variables, or that the size of the system is irrelevant. If extensive variables are included, one extra variable is needed to fix all variables. This could be the system volume, or any other extensive parameter. 

At this point we have shown how the HKF model develops expressions for the standard state parameters V° and C° and hence S°, H°, and G° at high temperatures and pressures. The standard state universally used is the ideal one molal solution, which means that these parameters refer to the properties of ions or electrolytes in infinitely dilute solutions. You might suppose that therefore they would not be of much... 

This table contains standard state thermodynamic properties of positive and negative ions in aqueous solution. It includes en-thcdpy and Gibbs energy of formation, entropy, and heat capacity, and thus serves as a companion to the preceding table, Standard Thermodynamic Properties of Chemical Substances . The standard state is the hypothetical ideal solution with molality m = 1 mol/kg (mean ionic molality in the case of a species which is assumed to dissociate at infinite dilution). Further details on conventions may be found in Reference 1. 

Predict the mean activity coefficient of Na2S04 in a 1.5 molal aqueous solution at 298.15 K using the LIQUAC model. For the calculation the following properties should be used ... 

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