Activity from Vapor Pressure Measurements

IV. Vapor Pressure Method. —If the free weak acid or weak base is appreciably volatile, it is possible to determine its concentration or, more correctly, its activity, from vapor pressure measurements. In practice the actual vapor pressure is not measured, but the volatility of the substance in the hydrolyzed salt solution is compared with that in a series of solutions of known concentration. In the case of an alkali cyanide, for example, the free hydrogen cyanide produced by hydrolysis is appreciably volatile. A current of air is passed at a definite rate through the alkali cyanide solution and at exactly the same rate through a hydrogen cyanide solution the free acid vaporizing with the air in each case is then absorbed in a suitable reagent and the amounts are compared. The concentration of the hydrogen cyanide solution is altered until one is found that vaporizes at the same rate as does the alkali cyanide solution. It may be assumed that the concentrations, or really activities, of the free acid are the same in both solutions. The concentration of free acid cha in the solution of the hydrolyzed salt of the weak acid may be put equal to cx (cf. p. 374) and hence x and kh can be calculated. 

For nonideal liquid mixtures, more complex relations involving activity coefficients have been derived. It is always necessary, however, to obtain the vapor-pressure data from experimental measurements. 

Example 2.3. Activity and Activity Coefficient of Aqueous NaCl Water vapor pressures have been measured over NaCl solutions of varying molal concentrations. Results of such measurements allow calculations of relative vairor pressure lowering, 4>, and 7., for a range of NaCl concentrations in water. Robinson and Stokes (1959) provide such data for concentrations ranging fn)m 0.1 to 6.0 molal. Table 2.1 shows results for three different concentrations. The mole fraction of H2O, Xfy, is also included. From such data the activity of aqueous NaCl can be computed, (see also Figure 2.4.)... 

Finally, the activity coefficient can be calculated from the molar fraction using Eq. (9.4). pI J(T) denotes the vapor pressure of the measuring gas at the measuring temperature. Vapor pressures are available in the literature for many volatile compounds [8]. 

The activity coefficients of sulfuric acid have been deterrnined independentiy by measuring three types of physical phenomena cell potentials, vapor pressure, and freeting point. A consistent set of activity coefficients has been reported from 0.1 to 8 at 25°C (14), from 0.1 to 4 and 5 to 55°C (18), and from 0.001 to 0.02 m at 25°C (19). These values are all based on cell potential measurements. The activity coefficients based on vapor pressure measurements (20) agree with those from potential measurements when they are corrected to the same reference activity coefficient. 

The solvent s activity can be determined by measuring the saturation vapor pressure above the solution. Such measurements are rather tedious and their accuracy at concentrations below 0.1 to 0.5M is not high enough to produce reliable data therefore, this method is used only for concentrated solutions. The activity can also be determined from the freezing-point depression or boiling-point elevation of the solution. These temperature changes must be ascertained with an accuracy of about 0.0001 K, which is quite feasible. This method is used primarily for solutions with concentrations not higher than 1M. 

Rard (1992) reported the results of isopiestic vapor-pressure measurements for the aqueous solution of high-purity NiCl2 solution form 1.4382 to 5.7199 mol/kg at 298.1510.005 K. Based on these measurements he calculated the osmotic coefficient of aqueous NiCb solutions. He also evaluated other data from the literature and finally presented a set of smoothed osmotic coefficient and activity of water data (see Table IV in original reference). 

In the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

Because the chemical potentials of water distributed in two phases (i.e., solution and vapor) must be equal, the water activity of a food can be measured by bringing the food into equilibrium with the air above it. At equilibrium, under conditions of constant temperature and pressure, the aw values of the aqueous phase of a food (aw l) and of the air (aw v) are equal and can be estimated from the ratio of the partial vapor pressure of water above the food (pv) to the vapor pressure of pure water (p") at the same temperature (Walstra, 2003) ... 

In writing the Etudes de dynamique chimique (1884), van t Hoff drew on Helmholtz s 1882 paper but especially on the work of August Horstmann, a student of Bunsen, Clausius, and H. Landolt.59 As has often been discussed, van t Hoffs was an ambitious and original synthesis of disconnected ideas and theories about opposing forces, equilibrium, active masses, work and affinity, electromotive force, and osmotic pressure. He demonstrated that the heat of reaction is not a direct measure of affinity but that the so-called work of affinity may be calculated from vapor pressures (the affinity of a salt for its water of crystallization), osmotic pressure (affinity of a solute for a solution), or electrical work in a reversible galvanic cell (which he showed to be proportional to the electromotive force). 

Values for the parameters are determined by a least squares fit of experimental data using eq (5) for experiments such as galvanic cells measurements that measure solute activity and thus y/Yref values, and eq (6) for experiments such as vapor pressure measurements that measure solvent activity and thus (f) values. All the original data are used in a single fitting program to determine the best values for the parameters. A detailed description of the evaluation procedure has been illustrated for the system calcium chloride-water (Staples and Nuttall, 1977), and calculations deriving activity data from a variety of experimental technique measurements have also been described. 

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)]... 

The solvent activity in a solution of polybutadiene in benzene was determined by measuring the vapor pressure / , of benzene over solutions containing various concentrations of poly-mer.f A plot of ln(p,// ) — In 4>, — (1 — /ri) 2 versus — in which/ is the vapor pressure of the pure benzene —yields a straight line having an intercept of zero and a slope equal to 0.33. Evaluate the interaction parameter x from this result. Is the 0 temperature above or below the experimental temperature Explain. 

The influence of curvature on phase equilibria is most readily understood for liquids, for which the activity is measured by the vapor pressure of the liquid. Accordingly, suppose we consider the process of transferring molecules of a liquid from a bulk phase with a vast horizontal surface to a small spherical drop of radius Rs. 

Assuming again that y follows the Debye-Hiickel law, the total pressure P is measured as a function of the solute concentration, then the vapor phase y, the only unknown in Equation 4, can be calculated, and hence the activities a and a2 can also be calculated, provided the activities ai° and a of each solvent prior to the addition of the solute are known dG°/dZ can be obtained next from Equation 1. Finally, integration of dG°/<9Zi with respect to Z leads to the standard molar free energy of transfer AG°t between Z = 1 (if water is chosen as the reference solvent) and any value of Z. ... 

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. 

Water activity (aw) is the ratio of the partial vapor pressure of water above a solution to that of pure water at the same specific temperature. It plays an important role in evaluating the microbial, chemical, and physical stability of foods during storage and processing. The vapor pressure in the headspace of a food sample can be measured directly by a manometer. A manometer has one or two transparent tubes and two liquid surfaces where pressure applied to the surface of one tube causes an elevation of the liquid surface in the other tube. The amount of elevation is read from a scale that is usually calibrated to read directly in pressure units. Makower and Myers (1943) were the first to use this method to measure vapor pressure exerted by food. Later, the method was improved, in terms of design features of the apparatus, by various scientists (Taylor, 1961 Labuza et al., 1972 Lewicki, 1987). Trailer (1983), Lewicki (1989), and Zanoni et al. (1999) used a capacitance manometer instead of a U-tube manometer for the measurement of vapor pressure. Lewicki et al. (1978) showed that the precision and reproducibility of the method can be improved by the simultaneous measurement of the water vapor pressure and temperature of the food sample. The method is reviewed in detail by Rizvi (1995) and Rahman (1995). 

The water activity of food samples can be estimated by direct measurement of the partial vapor pressure of water using a manometer. A simple schematic diagram is shown in Figure A2.4.1. A sample of unknown water activity is placed in the sample flask and sealed onto the apparatus. The air space in the apparatus is evacuated with the sample flask excluded from the system. The sample flask is connected with the evacuated air space and the space in the sample flask is evacuated. The stopcock across the manometer is closed and temperatures are read. The equilibrium manometer reading is recorded (/, ). The stopcock over the sample is closed and the air space is connected with the desiccant flask. The manometer reading in the legs is read to give h2. The water activity of the sample is then calculated (Labuza et al., 1976) as ... 

Similarly, Thurmond (150) and Arthur (151) found that the interaction coefficients obtained from a fit of the experimental liquidus or vapor pressure in the arsenide and phosphide systems did not produce the same temperature dependence. Panish et al. (142, 154) pointed out that these discrepancies may be due to (1) errors resulting from the assumed values for AH/j and the approximation ACp[ij] = 0 in 0, (2) deviations from simple-solution behavior, or (3) uncertainties in the interpretation of the vapor pressure data, because some of the quantities necessary in the calculations are not accurately known (e.g., reference-state vapor pressures for pure liquid As and P). Knobloch et al. (184, 185) and Peuschel et al. (186, 187) have obtained excellent agreement between calculated and experimental activities and vapor pressures with the use of Krupkowski s asymmetrical formalism for activity coefficients, whereas Ilegems et al. (Ill) demonstrated that satisfactory agreement between liquidus and vapor pressure measurements exists when an accurate expression for the liquidus is used. 

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