Coefficients of Water

At low densities, the pressure can be expanded in a power series in the density q, 

Note that (6.23) differs from the expression given in Section 1.8. The latter is obtained from the former if the total potential energy C/3(Xi, X2, X3) is pairwise additive. Equations (6.22) and (6.23) are special cases of a more general scheme which provides relations between virial coefficients and integrals involving interactions among a set of a small number of particles. This is known as the Mayer cluster theory [see, for example, Mayer and Mayer (1940), Hill (1956), and Munster (1969)]. 

The computation of the third virial coefficient poses far more serious difficulties. First, the integral in (6.23) is diflBcult to carry out numerically since it involves a twelvefold integration. Second, we cannot, at present, estimate the contribution from nonadditivity of the total potential to the integral. 

It has recently been established that three-body forces have an important effect on the value of even for nonpolar molecules [see Mason 

For all the above reasons, it is safe to conclude that computations of the third virial coefficient and comparison with experimental values are, at present, impractical for a test of the intermolecular potential chosen for water. In fact, all of the attempts made so far in this direction have failed to reproduce even qualitatively the temperature dependence of For 

---

The film pressure of a myristic acid film at 20°C is 10 dyn/cm at an area of 23 A per molecule the limiting area at high pressures can be taken as 20 A per molecule. Calculate what the film pressure should be, using Eq. IV-36 with / = 1, and what the activity coefficient of water in the interfacial solution is in terms of that model. 

Diffusion of the molecular gases can be compHcated by reactions with the glass network, especially at the sites of stmctural defects. The diffusion coefficient of water, for example, shows a distinct break around 550°C (110). Above 550°C, the activation energy is approximately 80 kj /mol (19 kcal/mol), but below 550°C, it is only 40 kJ/mol (9.5 kcal/mol). Proposed explanations for the difference cite the fact that the reaction between water and the sihca network to form hydroxyls is not in equiUbrium at the lower temperatures. 

The activity coefficient of water is related to the osmotic coefficient by the formula ... 

The values for the lipid molecules compare well (althoughJgiey are still somewhat larger) with the experimental value of 1.5x10 cm /s as measured with the use of a nitroxide spin label. We note that the discrepancy of one order of magnitude, as found in the previous simulation with simplified head groups, is no longer observed. Hence we may safely conclude that the diffusion coefficient of the lipid molecules is determined by hydrodynamic interactions of the head groups with the aqueous layer rather than by the interactions within the lipid layer. The diffusion coefficient of water is about three times smaller than the value of the pure model water thus the water in the bilayer diffuses about three times slower than in the bulk. 

Figure 41-6. Permeability coefficients of water, some ions, and other small molecules in lipid bilayer membranes. Molecules that move rapidly through a given membrane are said to have a high permeability coefficient. (Slightly modified and reproduced, with permission, from Stryer L Biochemistry, 2nd ed. Freeman, 1981.)...

N + 4H + 302N + 2H+30 Focus first on the elements that appear in only one reactant and one product, nitrogen and hydrogen in this case. Nitrogen is already balanced. To balance the H atoms, change the stoichiometric coefficient of water from 1 to 2 ... 

Fig. 2.7.5 Two-dimensional D—T2 map for Berea sandstone saturated with a mixture of water and mineral oil. Figures on the top and the right-hand side show the projections of f(D, T2) along the diffusion and relaxation dimensions, respectively. In these projections, the contributions from oil and water are marked. The sum is shown as a black line. In the 2D map, the white dashed line indicates the molecular diffusion coefficient of water,...

Farng and Nelson [1] applied the capillary method to the determination of the diffusion coefficient of salicylic acid in the presence of polyelectrolytes. The reported variability in terms of the coefficient of variation ranged from 0.89% to 8.3%. Stout et al. [2] showed the tube method to be useful for determining diffusion coefficients of water-insoluble pharmaceuticals such as sulfonamides and steroids. The coefficient of variation associated with the diffusion coefficient for sulfisoxazole is 5.5%. 

DWi = diffusion coefficient of water in air Xw = mole fraction of water vapor in air at a point x... 

The fitting to the data in Figs. 5.7 and 5.8 gives the values of oqlbs to about 5% precision. The value found for the bare sphere is consistent with the values found at other pressures and for overcoupled modes. Using (5.14) and the known bulk absorption coefficient of water (at these wavelengths, water has ab 800 m l)14 we estimate <5 6.7 pm from Fig. 5.7. This corresponds to water coverage on the... 

Figure 31. Diffusion coefficient of water in a DPPC bilayer as a function of position. Results of MD simulations of Marrink and Berendsen [129]. Redrawn by permission of the American Chemical Society...

The ammonia partial pressures given in Tables 1 and 2 are based on the concentration of ammonia found in the vapor stream times the total pressure. The actual pressures applied at each run condition are summarized in Table 3 where the pressures varied from 15 psia at 80°C to 90 psia at 120°C. Because nitrogen was used as a pressurizing fluid, the partial pressure of water and the total pressure excluding nitrogen have been computed in Tables 1 and 2 based on Raoult s law for water as noted at the bottom of Table 1. Raoult s law applies for the partial pressure of water because the activity coefficient of water is virtually unity at the low levels of ammonia used in the liquid phase. Minor effects due to vapor non- ideality have not been applied. 

Equations 8 and 9 can be used with the Gibbs-Duhem equation to calculate pf,., the activity coefficient of water, for each of the binary systems. The Gibbs-Duhem equation for a binary aqueous electrolyte solution is written ... 

The activity coefficient of water in the NaCl-HpO system can be well described by substitution of only the first term of Equation 9 into Equation 15. The resulting expression for the activity coefficient of water is ... 

The second term of Equation 9 is only important for the calculation of the activity coefficient of NaCl at low concentrations, and makes little contribution to the integral in Equation 15. Equation 17 predicts the activity coefficients of water within 1% of the experimental values for molalities above 0.2. 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

The activity coefficient of water in the ternary solution f> 3)> ls estimated by ... 

Figure 6 compares experimental and calculated activity coefficients of water in the ternary system at 25°C and a total molality of 3.0. Equation 18 was used to express the experimental activity coefficients. Agreement between experimental and calculated values is surprisingly good considering that Equation 19 contains no ternary parameters. The activity coefficient of water in the HC1-NaCl -H O system is not a strong function of composition, and Equation T9 provides an adequate description of the activity coefficients. 

Figure 6. Activity coefficients of water in the HCl-NaCl-H20 system at 25°C and total molality of 3.0 ((0) experimental)...

---