Virial coefficients of water

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 Temperature dependence of the second virial coefficient of water as predicted by six models in comparison with experimental data...

A much more careful calculation was carried out by Rowlinson (1951) using first a three-charge and then a four-charge model whose parameters were deduced from the interaction potential of water molecules as expressed by the second virial coefficient of water vapour. The four-charge model so deduced had charges of + 0 32e on each of the protons and two similar negative charges... 

Rowlinson, J. S. (1951). The lattice energy of ice and the second virial coefficient of water vapour. Trans. Faraday Soc. 47,120-9. [10,42,45] Samoilov, O. Ya. (1965). Structure of Aqueous Mectrolyte Solutions. 

Problem 2.15 a) Calculate the second virial coefficient of water at 200 °C using only data from the steam tables. 

The virial coefficients of water are defined as the coefficients in the density expansion of the pressure ... 

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

The next development was the calculation of the second virial coefficient of water based on a variant of what is now known as the Stockmayer potential. The energy of molecules separated by a distance r is taken to be... 

An effective potential that includes the dipole-dipole, dipole-quad-rupole, and quadrupole-quadrupole multipole interactions was used to estimate the third virial coefficient of water. Again, it was noted that the potential energy of interaction between water molecules is not additive, we can only conclude that the nonadditivity is successfully mimicked by our additive potential function. ... 

J. S. Rowlinson, Trans. Faraday Soc., 47, 120 (1951). The Lattice Energy of Ice and the Second Virial Coefficient of Water Vapor. 

Rowlinson IS (1951) The lattice energy of ice and the 2nd virial coefficient of water vapour. Trans Faraday Soc 47 120-129... 

Using the Steam Tables, determine the second and third virial coefficients of water at 300°C by using ... 

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. 

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. 

In the above equations denotes the vapor mole fraction of component i, Pf is the vapor pressure of the pure component i, Bu is the second virial coefficient of component i, dn = 2Bn — Bn — B22 and B 2 is the crossed second virial coefficient of the binary mixture. The vapor pressures, the virial coefficients of the pure components and the crossed second virial coefficients of the binary mixtures were taken from [32], The Wilson [38], NRTL [39] and the Van Ness-Abbott [40] equations were used for the activity coefficients in Eq. (17). The expressions for the activity coefficients provided by these three methods were differentiated analytically and the obtained derivatives were used to calculate D = 1 -I- Xj(9 In Yil 2Ci)pj. There is good agreement between the values of D obtained with the three expressions for the systems V,V-dimethylformamide-methanol and methanol-water. For the system V,V-dimethylformamide-water, the D values calculated with the Van Ness-Abbott equation [40] were found in good agreement with those obtained with the NRTL equation, but the agreement with the Wilson expression was less satisfactory. 

These macroscopic viscosity measurements have been confirmed at the molecular level. For example, dynamic light-scattering methods show an average hydrodynamic diameter (D J of about 370 A for a 50/50 copolymer of NVP/SPE in low salt (<2%) and a of about 390 A in high-salt (20% NaCl) concentrations (18). Moreover, in solutions of water or low concentrations of salt, solvent quality (as measured by second virial coefficient, A2) decreases with increasing levels of SPE. LALLS measurements for NVP/SPE 80/20 and 10/90 copolymers in 2% NaCl solution yielded molecular weights of 1.1 X 10 and 1.4 x 10 g mol respectively. The same compositions yielded second virial coefficients of 9.0 X 10 and 0.6 X 10 , respectively. In this case, a higher virial coefficient means better solution quality. 

Akin-Ojo, 0., and Szalewicz, K. (2005). Potential energy surface and second virial coefficient of methane-water from ab initio calculations, / Chem. Phys. 123, p. 134311, doi 10.1063/1.2033667. 

The solubility of gases is generally described by the Ostwald coefficient L = Vs/Tw, where subscripts s and w denote the solute and the solvent (water), and Vs denotes the volume of pure gas at a given temperature and pressure sorbed by (dissolved in) a volume Vw of pure water. For the standard pressure of =0.1 MPa 1 atm (the Ostwald coefficient is not appreciably dependent on the pressure at moderate pressures) the standard state molar volume of the gas is Vs° =RT/P° + Bss, the second term is the second virial coefficient of the gas (describing interactions of the gas molecules with each other) and generally Bss RT/P° and can be ignored. The molar concentration (in Vw = 1 dm of water) of the gas is then LP°IRT (with... 

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. 

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