Virial coefficients For water

Fig. 1.11 Secoii(d virial coefficient for water and heavy water, based on data from Lemmon et al. (2007).

The above estimates of AX were made based on the second virial coefficient for water. Besides the uncertainty of extrapolating to low temperature, these values are relevant to the interaction between exactly two real water molecules. When we need to treat the water in the liquid state, we cannot rely on these estimates but use AX as an adjustable parameter. 

Rowlinson (1951) used a modified Stockmayer potential to calculate the second virial coefficient for water [see Ben-Naim (1974), page 247, Table 6.1.]... 

Fig. 14.23. Second thermal conductivity virial coefficient for water vapor as a function of temperature.

The SAPT method has recently been used to compute the complete six-dimensional water dimer potential. A global elaborate analytic potential has been fitted to about 1000 points. This potential has been used to compute the second virial coefficient for water. Results are presented in Figure 2. As one can see, the SAPT potential reproduces the experimental data very accurately. Compared to the popular empirical potentials as well as to the MCY ab initio potential the SAPT values are nearly an order of magnitude more accurate. Notice that the virial coefficients computed from the empirical potentials do not include quantum corrections but these are not important at this level of accuracy. The recent polarizable point charge (PPC) potential of Ref. 176, which gives the best virial coefficient of all empirical potentials, has to a lesser extent the effective character typical of bulk empirical potentials as it models explicitly the nonadditive induction energy. Thus, the pairwise additive component is less biased by efforts to mimic nonadditive forces. 

Figure 2 The second virial coefficient for water from ab initio and empirical potentials. SAPT Wig.c denotes classical coefficient while SAPT Wig.c -t- s-s.q includes quantum correction. Other potentials are from Refs. 173-176...

A potential for the water dimer with flexible monomers has recently been developed [206]. The resulting potential energy surface is 12-dimensional, which is at the limits of complexity that one can deal with. About half a million points have been computed (compared to about 2500 in [30] using rigid monomers). Fitting these data required a very substantial effort. The potential will be used in calculations of the spectra of water dimer and of the second virial coefficient for water. 

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. 

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... 

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

This potential was devised by Rowlinson (1951) for the computation of the virial coefficients of water. 

C. H. J. Johnson and T. H. Spurling, Aust. J. Chem., 24, 1567 (1971). Pairwise Additive Third Virial Coefficients for Multipolar Molecules Application to Water Vapour. 

Pratt, L.R.,and Chandler, D., 1980, Hydrophobic interactions and osmotic second virial coefficient for methanol in water, J. Solution Chem., 9 1. 

Second osmotic virial coefficient for interactions between solutes S in water... 

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. 

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. 

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. 

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