Activity of the solvent in a solution

The solid curve in Figure 16.1 shows the activity of the solvent in a solution as a function of the mole fraction of solvent. If the solution were ideal, Equations (14.6) and (16.1) would both be applicable over the whole range of mole fractions. Then, fli =Xi, which is a relationship indicated by the broken line in Figure 16.1. Also, because Equation (16.1) approaches Equation (14.6) in the limit as Xj 1 for the real solution, the solid curve approaches the ideal line asymptotically as Xj 1. 

It is important that the distinction between the similar equations (38.5) and (38.13) sliould be clearly understood. The former gives the variation with temperature of the activity of the solvent in a solution at its freezing point, which varies with the composition. The latter applies to the activity of the solvent in a solution of constant composition, t It is also the differential heat of dilution of the given solution (cf. 44b.)... 

According to equation (38.42) it should be possible to determine the activity of the solvent in a solution by means of osmotic pressure measurements. Since such measurements are not easily made, the procedure is not convenient. Nevertheless, the accuracy of equation (38.42) may be tested by using the known activity of the solvent to calculate the osmotic pressure of the solution, and comparing the result with the experimental value. Provided the vapor pressures are not high, ai, which is equal to fi/fi, may be replaced by pi/p°, and so the osmotic pressure can be derived from vapor pressure measurements. For aqueous solutions, a is of the order of 4 X 10 atm. " and may be neglected, except for solutions of high... 

Use the Flory-Huggins equation for Api to calculate an estimate of x for solutions of natural rubber (M = 2.5x10 ) in benzerje, given that vapor pressure measurements show that the activity of the solvent in a solution with ( 2 = 0.250 is 0.989. 

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. 

Equation 1.49 is applicable to any solution, ideal or non-ideal, provided only that the vapour behaves as an ideal gas comparison of this with Equation 1.49 shows that the activity of the solvent in a solution must be proportional to the vapour pressure of the solvent over a given solution. If a represents the activity of the solvent in the solution and p is its vapour pressure, then fl = kp, where k is a proportionality constant. The value of this constant can be determined by making use of the standard state postulated above, namely that a = 1 for the pure solvent, i.e. when the vapour pressure is p it follows, therefore, that k, which is equal to olp. is l/p°, and hence... 

The activity of the solvent in a solution ean thus be determined from measurements of the vapour pressure of the solution, p, and of the pure solvent, p° at a given temperature. It is obvious that for an ideal solution obeying Raoult s law p/p° will be equal to v. the mole fraction of solvent. The activity coefficient as given by Equation 1.56 will then be unity. It is with the object of obtaining this result that the particular standard state of pure solvent was chosen. For a non-ideal solution the activity coefficient of the solvent will, of course, differ from unity, and its value can be determined by dividing the activity as given by Equation 1.57 by the mole fraction of the solvent. 

For ideal solutions, the activity coefficient will be unity, but for real solutions, 7r i will differ from unity, and, in fact, can be used as a measure of the nonideality of the solution. But we have seen earlier that real solutions approach ideal solution behavior in dilute solution. That is, the behavior of the solvent in a solution approaches Raoult s law as. vi — 1, and we can write for the solvent... 

Although the phrase activity of a solution usually refers to the activity of the solute in the solution as in the preceding section, we also can refer to the activity of the solvent. Experimentally, solvent activity may be determined as the ratio of the vapour pressure Pi of the solvent in a solution to that of the pure solvent pf, that is... 

It is important to recognize that the experimentally determined interaction parameter is usually derived from the measurement of the activity of the solvent in a polymer solution, coupled with the application of the Flory-Huggins expression for the excess (denoted by superscript ) chemical potential... 

Assume that the activity of the solvent in a two-component solution obeys the formula... 

The discussion so far has been confined to systems in which the solute species are dilute, so that adsorption was not accompanied by any significant change in the activity of the solvent. In the case of adsorption from binary liquid mixtures, where the complete range of concentration, from pure liquid A to pure liquid B, is available, a more elaborate analysis is needed. The terms solute and solvent are no longer meaningful, but it is nonetheless convenient to cast the equations around one of the components, arbitrarily designated here as component 2. 

Osmotic pressure is one of four closely related properties of solutions that are collectively known as colligative properties. In all four, a difference in the behavior of the solution and the pure solvent is related to the thermodynamic activity of the solvent in the solution. In ideal solutions the activity equals the mole fraction, and the mole fractions of the solvent (subscript 1) and the solute (subscript 2) add up to unity in two-component systems. Therefore the colligative properties can easily be related to the mole fraction of the solute in an ideal solution. The following review of the other three colligative properties indicates the similarity which underlies the analysis of all the colligative properties ... 

The activity of any ion, a = 7m, where y is the activity coefficient and m is the molaHty (mol solute/kg solvent). Because it is not possible to measure individual ionic activities, a mean ionic activity coefficient, 7, is used to define the activities of all ions in a solution. The convention used in most of the Hterature to report the mean ionic activity coefficients for sulfuric acid is based on the assumption that the acid dissociates completely into hydrogen and sulfate ions. This assumption leads to the foUowing formula for the activity of sulfuric acid. 

In the osmotic pressure method, the activity of the solvent in the dilute solution is restored to that of the pure solvent (i.e., unity) by applying a pressure m on the solution. According to a well-known thermodynamic relationship, the change in activity with pressure is given by... 

The activity of the solvent often can be obtained by an experimental technique known as the isopiestic method [5]. With this method we compare solutions of two different nonvolatile solutes for one of which, the reference solution, the activity of the solvent has been determined previously with high precision. If both solutions are placed in an evacuated container, solvent will evaporate from the solution with higher vapor pressure and condense into the solution with lower vapor pressure until equilibrium is attained. The solute concentration for each solution then is determined by analysis. Once the molality of the reference solution is known, the activity of the solvent in the reference solution can be read from records of previous experiments with reference solutions. As the standard state of the solvent is the same for all solutes, the activity of the solvent is the same in both solutions at equUibrium. Once the activity of the solvent is known as a function of m2 for the new solution, the activity of the new solute can be calculated by the methods discussed previously in this section. 

Divalent or higher-valent cations and, in particular, transition metal cations, are likely to be covalently solvated by solvents that are strong electron pair donors (have large solvatochromic P values). This solvation often persists in crystals, so that the salt that is in equilibrium with the saturated solution in such solvents may not be the anhydrous salt (nor the salt hydrate). Equation (2.56) omits any consideration of the solvent of crystallization and pertains to the solventless (anhydrous) salt. For a salt hydrated by n water molecules in the crystal, the activity of water raised to the nth power must multiply the right-hand side of Eq. (2.56) for it to remain valid. A similar consideration applies for salts crystallizing with other kinds of solvent molecules, the activity of the solvent in the saturated solution replacing that of water. Such situations must be... 

The osmotic pressure determination of molecular weights is based on the thermodynamic interaction of solvent and solute to lower die activity of the solvent. Experimentally, the solution is separated from the solvent by a semipermeable membrane. The solvent tends to pass through the membrane to dilute the solution and bring the activity of the solvent in both phases to equilibrium. The quantitative measurement of this tendency is obtained by allowing tile liquid solution to rise ill a vertical capillary connected to the solution compartment. The equilibrium height it achieves or the rate at which it rises can be measured. 

Osmosis is the selective passage of particular components of a solution through a semipermeable membrane. Usually, it is the solvent that passes through the membrane, because the solute is blocked. However, some membranes also allow small solute molecules to pass through as well and only block the passage of macromolecular solute molecules. The osmotic pressure of a solution is the pressure difference produced at equilibrium across the membrane, with the solution on one side of the membrane and pure solvent on the other side. As shown in Fig. 4, the reduced activity of the solvent in solution is compensated for by an increase in the pressure of the solution ... 

The above procedure is used to predict activity coefficients of the solvents in a defined polymer solution mixture. The method yields fairly accurate predictions. Although Procedure D is a good predictive method, there is no substitute to reducing good experimental data to obtain activity coefficients. In general, higher accuracy can be obtained from empirical models when these models are used with binary interaction parameters obtained from experimental data. 

The activity coefficient of the solvent in a binary solution is given by... 

We haven t discussed activities, vapor pressure and other aspects of the thermodynamics of liquids (optimistically assuming you ve done all this in P. Chem). Nevertheless, we are sure that you recall that the vapor pressure of the solvent in a polymer solution relative to the vapor pressure of the pure solvent may, to a first approximation, be equated to the activity of the solvent, which in turn is related to the chemical potential by ... 

Colligative properties reflect the chemical potential of the solvent in solution. Alternatively, a colligative property is a measure of the depression of the activity of the solvent in solution, compared to the pure state. Colligative properties include vapor pressure lowering, boiling point elevation, freezing point depression, and membrane osmometry. The latter property is considered here, since it is the most important of the group as far as synthetic polymers are concerned. 

Although, in principle, equation (39.49) provides a method for determining activity coefficients, the details require consideration. The osmotic coefficients, in the first place, are determined from vapor pressure measurements. The activity ai of the solvent in a given solution is equal to/i// , by equation (31.5), or, approximately, to pi/p d. equation (38.1)], where pi is the vapor pressure of the solvent over the solution and is that of the pure solvent at the same temperature. Hence, by equation (39.46),... 

Determination of the activity coefficients of the non-volatile solute in a solution is difficult. If electrolytes (ions) are present, the activities can be obtained from experimental electromotive force (EMF) measurements. However, for non-electrolyte and non-volatile solutes an indirect method is applied to find initially the activity of the solvent over a range of solute concentrations, and then the Gibbs-Duhem equation is integrated to find the solute activity. If the solution is saturated, then it is easy to calculate the activity coefficient... 

The correlation between the availability of water and the rate constant of proton dissociation was measured in two systems. In one system, the ratio water methanol of a mixed solution modulated the availability of water [38]. In the other system, made of concentrated electrolyte solutions, the activity of the water was modulated by the salt [39]. The dependence of the measured rate of dissociation [60, 67, 68], either from photoacid or ground state acids, on the activity of the solvent yielded a straight log-log correlation function with respect to the activity of the water... 

An explanation of the phenomenon of osmosis is provided in most textbooks of physical chemistry. Suppose that a pure solvent and the solvent containing some solute are separated by a membrane that is permeable only for the solvent. In order to obtain pure solvent from the solution by filtering the solute molecules with the membrane, a pressure which is higher than the osmotic pressure of the solution must be applied to the solution side. If the external (total) pressures of the pure solvent and the solution were equal, however, the solvent would move into the solution through the membrane. This would occur because, due to the presence of the solute, the partial vapor pressure (rigorously activity) of the solvent in the solution would be lower than the vapor pressure of pure solvent. The osmotic pressure is the external pressure that must be applied to the solution side to prevent movement of the solvent through the membrane. 

For the solvent in a solution, i = y/X, whereX is the mole fraction. As the solution becomes more dilute, y approaches 1. The activity generally is assumed to be 1 for the dilute solutions of concern to us. 

FIG. 11 Combination of two factors regulating enzyme catalytic activity variation in the surfactant concentration and addition of water-miscible organic solvents, (a) Peroxidase in the system AOT-water/glycerol-octane at water/glycerol volume ratios (O) 100 0 ( ) 20 80. (b) a-Chymotrypsin in the system AOT-water/glycerol-octane at water/glycerol volume ratios (O) 100 0 ( ) 6 94. Dashed lines show the catal5dic activities of the enzymes in aqueous solution. (From Ref. 44.)... 

The correlation of the rates of decarboxylation of the difluoro diethyl lactone with solubdity parameter shows that the rate increases dramatically as the value of 8 increases. In fact, there was a factor of almost 500 difference in the rate constants depending on the solvent chosen. This is in agreement with the conclusion that the transition state for the reaction carried out in solution involves a substantial separation of charge. Furthermore, Figure 5.11 shows that the solubihty parameter can be a useful index for assessing the role of the solvent in a reaction of a greatly different type than those described earlier in this chapter. The entropy of activation for the reaction was reported... 

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