Hard sphere diameters, calculation coefficients

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. 

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. 

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... 

Calculations of departures from ideality in ionic solutions using the MSA have been published in the past by a number of authors. Effective ionic radii have been determined for the calculation of osmotic coefficients for concentrated salts [13], in solutions up to 1 mol/L [14] and for the computation of activity coefficients in ionic mixtures [15]. In these studies, for a given salt, a unique hard sphere diameter was determined for the whole concentration range. Also, thermodynamic data were fitted with the use of one linearly density-dependent parameter (a hard core size o C)., or dielectric parameter e C)), up to 2 mol/L, by least-squares refinement [16]-[18], or quite recently with a non-linearly varying cation size [19] in very concentrated electrolytes. 

Table A. 19 in the appendix gives some experimental values for self-diffusion coefficients, obtained from tracer diffusion experiments. When these data are used to calculate effective hard-sphere diameters the calculated hard-sphere diameters depend on temperature, with smaller diameters corresponding to higher temperatures. This is explained by the fact that the actual intermolecular repulsive potential is not infinitely steep like the hard-sphere potential. When two molecules strike together more strongly, as they more often do at higher temperature, the distance of closest approach is smaller and the effective hard-sphere diameter is smaller.

Calculate the viscosity coefficient of O2 gas at 292 K from its hard-sphere diameter. Compare your result with the value in Table A. 18 in the appendix. 

Calculate the hard-sphere diameter of water molecules at 100°C from the coefficient of viscosity in Table A.18. Compare your result with the hard-sphere diameter in Table A.15. 

Calculate the effective hard-sphere diameter of argon atoms from the value of its self-diffusion coefficient at 273 K in Table A. 19. Compare it with the value from Table A.15. 

Various diffusion and thermal diffusion processes were used in World War II to separate gaseous molecules from uPg molecules. Calculate the mutual diffusion coefficient of these substances at 60°C and 1.000 atm. Calculate the self-diffusion coefficient of at this temperature and pressure. Make a reasonable estimate of the effective hard-sphere diameter of UFs and see Problem 10.26 for the necessary equation. 

The calculation of TCF in multicomponent systems has been done only for spherical cavities with the formalism developed by Lebowitz et al. Methods IV and V can in principle be extended for TCF calculation for nonspherical cavities in multicomponent systems. An artificial binary system (benzene-water) was selected here to illustrate the computational methodology. In practice, these two solvents mix very little, and their mixture can be of little interest, but they are quite different in their chemical nature and this makes such a system interesting. The method is by no means limited to certain mixtures and is universally applicable to any mixture if the molecular and physical parameters of the pure components are known (hard sphere diameter, number density, thermal expansion coefficient, dielectric constant). Figure 9 displays TCF calculated as a function of solvent mole fractions for a spherical cavity of cyclohexane size created in a hypothetical water-benzene mixture. Gc (Figure 9) increases with the increase of water mole fraction, but there is little difference between pure benzene and a mixture containing around 50% water as far as solvation of cyclohexane is concerned. 

For hard spheres, the coefficients are independent of temperature because the Mayer/-fiinctions, in tenns of which they can be expressed, are temperature independent. The calculation of the leading temiy fy) is simple, but the detennination of the remaining tenns increases in complexify for larger n. Recalling that the Mayer /-fiinction for hard spheres of diameter a is -1 when r < a, and zero otherwise, it follows thaty/r, 7) is zero for r > 2a. For r < 2a, it is just the overlap volume of two spheres of radii 2a and a sunple calculation shows tliat... 

In this experiment the mutual diffusion coefficients for the Ar—CO2 and He—CO2 systems are to be measured using a modified Loschmidt apparatus. These transport coefficients are then compared with theoretical values calculated with hard-sphere collision diameters. 

The HOMO and the LUMO are both 7r-type orbitals for which an HMO calculation has been made. " Figure 3.8 shows a diagram commonly used to show such orbitals. The diameters of the spheres represent the coefficients of each atomic orbital in the MO. We readily see that carbon atom 3 is the reactive site for reaction with soft electrophiles. We can assume that the oxygen atoms will be more negative than any of the carbon atoms, and will react with hard electrophiles. 

Figure 19-8. Measured second virial coefficients ofSTA (soUd squares) in dffferent background salt concentrations compared with data on a number of proteins (Lysotyme, BPTI open circles) in different buffer solutions. The second virial coefficients are nondimensionalized with the hard sphere value and plotted against the solubility (volume fraction 0at) of the respective species. The solid lines are calculations of the attractive Yukawa potential with two different ranges of attractions (2ak) of 7 and 15. The values of 7 and 15 indicate that attractions between the particles are short ranged. The experimental datafor STA (at high salt concentrations) and proteins collapse within the narrow range of attractions which are only a fraction of the particle diameter. The collapse also indicates thatproteins and STA are thermodynamically similar, iftwo suspensions have the same B2 then they have the same solubility. This plot also provides an opportunity to extract interaction potential parameters for a given experimental system in a model independent manner. For detailed discussions, please refer to (Ramakrishnan, 2000).

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