Hard sphere diameters, calculation

Calculations of the capacitance of the mercury/aqueous electrolyte interface near the point of zero charge were performed103 with all hard-sphere diameters taken as 3 A. The results, for various electrolyte concentrations, agreed well with measured capacitances as shown in Table 3. They are a great improvement over what one gets104 when the metal is represented as ideal, i.e., a perfectly conducting hard wall. The temperature dependence of the compact-layer capacitance was also reproduced by these calculations. 

For poly(methylene), an exclusion distance (hard sphere diameter) of 2.00 A was used to prevent overlap of methylene residues. The calculation reproduced the accepted theoretical and experimental characteristic ratios (mean square unperturbed end-to-end distance relative to that for a freely jointed gaussian chain with the same number of segments) of 5.9. This wps for zero angular bias and a trans/gauche energy separation of 2.09 kJ mol". ... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

Figure 3.20 The effective hard sphere diameter, r0, calculated from Equation (3.65) for 100 nm radius particles with ( = 50 mV...

The hard sphere diameters were then used to calculate the theoretical Enskog coefficients at each density and temperature. The results are shown in Figure 3 as plots of the ratio of the experimental to calculated coefficients vs. the packing fraction, along with the molecular dynamics results (24) for comparison. The agreement between the calculated ratios and the molecular dynamics results is excellent at the intermediate densities, especially for those ratios calculated with diameters determined from PVT data. Discrepancies at the intermediate densities can be easily accounted for by errors in measured diffusion coefficients and calculated diameters. The corrected Enskog theory of hard spheres gives an accurate description of the self-diffusion in dense supercritical ethylene. 

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

Figure 3. The ratio D/D as a function of packing fraction for supercritical ethylene. A indicates ratios calculated using hard sphere diameters determined from diffusion data. 0 indicates ratios calculated using hard sphere diameters determined from compressibility data. The solid lines are the molecular dynamics results, extrapolated to infinite systems, of Alder, Gass and Wainwright (Ref. 24).

Figure 10 (a) Ti (p, T) vs. density for the solvent ethane at 34°C and the best fit theoretically calculated curve. The theory was scaled to match the data at the critical density, 6.88 mol/L. The best agreement was found for u> = 150 cm-1, (b) Ti (p, T) vs. density for the solvent ethane at 50°C and the theoretically calculated curve. The scaling factor, frequency >, and the hard sphere diameters are the same as those used in the fit of the 34°C data. Given that there are no free parameters, the agreement is very good. 

Figure 13 shows a comparison of Ti(p, T) data with calculated curves for the CO2 solvent at two temperatures, 33°C (2 K above Tc) and 50°C. The calculated curves are scaled to the data at 33°C and 2 mol/L. Again, ( is set to 150 cm 1. The solvent hard sphere diameter is the literature value, 3.60 A (104). The agreement between theory and experiment, while not poor, is clearly not as good as that displayed for ethane and fluoroform. Adjusting o> does not improve the agreement between theory and data, nor does a further variation of the solvent diameter. The T1 (p, T) data appear to differ from the data in the other two solvents. After the initial rapid decrease in the lifetime at low densities, the data curve with CO2 as the solvent is much flatter than the data obtained in the other two solvents. The theory does well up to 6 mol/L but then overestimates the decrease in lifetime at higher densities. 

F.q. (16-23). Subsequently, it became clear that a theoretical form for S (q)S(q) given by Percus and Yevick (1958) and Pcrcus (1962) was more convenient, and probably as accurate as the experiment for the resistivity calculation. This approach was used by Ashcroft and Lckner (1966) for an extensive study of the resistivity of all the simple liquid metals. The form due to Pcrcus and Yevick depends only upon two parameters, a hard-sphere diameter and a packing fraction these lead to a simple form in terms of elementary functions Ashcroft and Lckner discuss the choice of parameters. This form is presumably just as appropriate for other elemental liquids. 

Figure 8.13 The function G (A) for water at 300 K at the liquid saturation conditions (Ashbaugh and Pratt, 2004). The points are obtained by direct Monte Carlo calculation, and the solid line by matching an empirical thermodynamic model for the large solute case. The dashed lines are the classic scaled-particle model (Pierotti, 1976) predictions for several solvent hard-sphere diameter parameters between cr = 2.6 A and 3.0 A in 0.1 A increments. Notice that the parameter value that provides the best fit of the classic scaled-particle model for small radii is not the same as that for the large radii results.

Experimental results for the static structure factor of polyethylene are given in Figure 19, which shows results from PRISM calculations for compari-The calculations used hard sphere diameters of 3.70 and 3.90 A. The... 

To examine the potential of this new approach, we analyze the experimental data for the osmotic pressure of bovine serum albumin (BSA) in 0.15 mol dm-3 sodium chloride [112] and human serum albumin (HSA) solution in 0.1 molx dm-3 phosphate buffer [111]. According to a previous experimental and theoretical study [111] the two solutions differ substantially in the degree of protein association. The theoretically determined osmotic coefficient can be fitted to the experimental results to obtain the fraction of dimers in the solution. The results of our analysis are presented in Figs. 11 and 12. The protein molecular weights used in these calculations were 69,000 g/mol for BSA and 66,700 g/mol for HSA. The hard-sphere diameter of spherical proteins was assumed to be 6.0 nm. For the case of the multicomponent model, the ions of the low-molecular weight +1 — 1 electrolyte were modelled as charged hard spheres with diameter 0.4 nm. 

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

For (17 — V)/RT no hard-sphere property calculations are made and the a0 term of the quadratic fit along the compressibility factor isochores can be equated to ZHs(pd3). This is then solved for the diameter used in the pseudo parameter computations. 

The theoretical curve defined by the reciprocal of Eq. (4-14) is calculated as follows. is obtained from a plot of In/oo versus 1/T the high-pressure rate constant 4 0 is measured directly or obtained from extrapolation of the plot of l//exp versus 1/P the effective number of oscillators n/2 is obtained by locating the pressure at which begins to fall off it is assumed that at this pressure the rate of activation is equal to the first-order rate of reaction, that is, ac W = > exp 9 relation which will yield a value of n/2 after insertion of the experimental value of and a reasonable value for the elastic-hard-sphere diameter d. 

Intermolecular repulsion forces will define a hard-sphere diameter for the molecules allowing the calculation and/or measurement of the density fluctuation as one moves away from the surface. As shown in Figure 4.86, the result is a damped density curve (density distribution versus distance in molecular diameters from the surface). For a liquid between two surfaces at short separation distances, the two effects will overlap producing an interference pattern as illustrated in Figure 4.9. When the distance of separation is some integral multiple of the hard sphere diameter, nAhs, reinforcement occurs producing a local free energy minimum. For separations that correspond to fractional... 

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