Water, virial coefficient

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

Harvey A N 1999 Applications of first-principles calculations to the correlation of water s second virial coefficient Proc. 13th Int. Conf of the Properties of Water and Steam (Toronto, 12-16 September 1999)... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (J, 2 to obtain cross term second virial coefficients Bj2- At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy-pressure coefficient pressure yield the quantity 4>c = B - TdB/dT where B is the second virial coefficient. From values of obtain values of B without recourse to pVT measurements. 

The importance of the virial-coefficient equations is especially great for mixed electrolytes. Of the needed virial coefficients for a complex mixture such as sea water, most are determined by the pure electrolyte measurements and all the others of any significance are determined from data on simple mixtures such as NaCl-KCl, NaCl-MgC, NaCl-Na.SO, etc., which have been measured. The effect of the terms obtained from mixtures is very small in any case and these terms can be ignored for all but the most abundant species. 

A very severe test of these virial-coefficient equations for the sea-water-related Na-K-Mg-Ca-Cl-S0,-H 0 system has been made by Harvie and Weare (37) who calculated tne solubility relationships for most of the solids which can arise from this complex system. There are 13 invariant points with four solids present in the system Na-K-Mg-Cl-SO - O and the predicted solution compositions in all 13 cases agree with the experimental values of Braitsch (38) substantially within the estimated error of measurement. In particular, Harvie and Weare found that fourth virial coefficients were not required even in the most concentrated solutions. They did make a few small adjustments in third virial coefficients which had not previously been measured accurately, but otherwise they used the previously published parameters. 

It is shown that the properties of fully ionized aqueous electrolyte systems can be represented by relatively simple equations over wide ranges of composition. There are only a few systems for which data are available over the full range to fused salt. A simple equation commonly used for nonelectrolytes fits the measured vapor pressure of water reasonably well and further refinements are clearly possible. Over the somewhat more limited composition range up to saturation of typical salts such as NaCl, the equations representing thermodynamic properties with a Debye-Hiickel term plus second and third virial coefficients are very successful and these coefficients are known for nearly 300 electrolytes at room temperature. These same equations effectively predict the properties of mixed electrolytes. A stringent test is offered by the calculation of the solubility relationships of the system Na-K-Mg-Ca-Cl-SO - O and the calculated results of Harvie and Weare show excellent agreement with experiment. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

Here, m is the molal concentration of the /-component in mol/1000 g water ct is the weight concentration of the /-component in g/ml Vi and vz (with bars) are the specific volumes of water and biopolymer in ml/g and Mf is the molar weight of the biopolymer in g/mol (Da). In turn, the relationship between the second virial coefficients expressed in the different units (molal, At weight, ) is as follows (Wells, 1984) ... 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

Here, as previously, the solvent is taken as component 1, one of the biopolymers is the /-component, while another is the /-component m, mr and m,j are the concentrations (moles per kg of water) of the components An and Ajj are the second virial coefficients (m3/mol) characterizing the like pair interactions of the types biopolymer -biopolymer and biopolymer,-biopolymer, respectively and Ay is the cross second virial coefficient corresponding to the biopolymer -biopolymer pair interaction. 

The above procedure is now applied to two ethanol-water (8, 9) and five 1-propanol-water systems (9) which have been saturated with an inorganic salt and which show partial miscibility. The vapor pressures and molar volumes (10), and second virial coefficients of water (11), ethanol (12), and 1-propanol (IS) were obtained by interpolation of literature data. The vapor pressures of water saturated with salts over a temperature range are available for all salts (14) except lead nitrate. Such data are unavailable for both alcohols saturated with salt. Hence a correction to the saturation vapor pressure is made by multiplying by the ratio of the vapor pressure of alcohol saturated with salts to the vapor pressure... 

The calculations were carried out for various values of the parameters, the aspect ratio of segment p, and the number ratio of ionizable groups in the chain f. The other parameters were estimated for NIPA gel. All the values of parameters used are summarized in Table 3. The value of v0 was determined by taking the intermediate value between water and NIPA [20]. The parameters Ca, Cb and Cc for the hydrophobic interaction were determined from the values of isobutyl substituents of amino acids, determined by Nemethy and Scheraga [19]. Since there are no data for the 6 temperature and the virial coefficients of this system, we assumed Te to be 273.15 K, and estimated the virial coefficients... 

Hydrogen-bond formation is of importance also for various other properties of substances, such as the solubility of organic liquids in water and other solvents, melting points of substances under water,1 viscosity of liquids,14 second virial coefficient of gases,18 choice of crystal structure, cleavage and hardness of crystals, infrared absorption spectra, and proton magnetic resonance. Some of these are discussed in the following sections of this chapter. 

The experimentally fitted hydrate guest Kihara parameters in the cavity potential uj (r) of Equation 5.25 are not the same as those found from second virial coefficients or viscosity data for several reasons, two of which are listed here. First, the Kihara potential itself does not adequately fit pure water virials over a wide range of temperature and pressure, and thus will not be adequate for water-hydrocarbon mixtures. Second, with the spherical Lennard-Jones-Devonshire theory the point-wise potential of water molecules is smeared to yield an averaged spherical shell potential, which causes the water parameters to become indistinct. As a result, the Kihara parameters for the guest within the cavity are fitted to hydrate formation properties for each component. 

As already noted, we suggest that the behavior of the second virial coefficient of the apoferritin in acetate buffer is due to the adsorption of Na+ ions upon the negative sites of the protein surface, which depends on the concentration of the Na+ ions in the liquid in the vicinity of the surface. In what follows, the adsorption of acetate ions upon the positive sites or of neutral Na+—CH3COO " pairs on the neutral sites of the protein surface will be neglected and it will be assumed that only the dipoles of the ion pairs formed through the association ofNa4 to the acidic sites of the surfece polarize the neighboring water molecules. 

Simulations of the liquid water properties have been the subject of many papers, see Ref. (374) for a review. Recently a two-body potential for the water dimer was computed by SAPT(DFT)375. Its accuracy was checked375 by comparison with the experimental second virial coefficients at various temperatures. As shown on Figure 1-16, the agreement between the theory and experiment is excellent. Given an accurate pair potential, and three-body terms computed by SAPT376, simulations of the radial 0-0, 0-H, and H-H distribution functions could be... 

Figure 1-16. Theoretical (full line) and experimental (open triangles) second virial coefficient for gaseous water at various temperatures...

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