Virial charges

Virial charges have been criticized because (a) they show little similarity to Mulliken charges, (b) they are often very large and (c) they lead to bond polarities in contradiction to established chemical thinking. For example, virial charges suggest a —H bond polar-... 

Before the data of Table 11 are shortly discussed, one has to stress that the partial charges derived from adiabatic infrared intensities are not related to Mulliken charges, virial charges or most other atomic charges used in ab initio theory. The partial charges p are effective charges which in addition to the atomic monopole contribution, cover the atomic dipole contribution as well. They are... 

We now consider more interesting properties that can be extracted in our approach which cannot be extracted in a standard X-ray charge analysis. For a system at equilibrium, the virial theorem gives the total energy as... 

If the gas of charges (plasma) is sufficiently dilute, we could hope a priori that its equation of state would be described by the virial expansion ... 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

The Huggins coefficient kn is of order unity for neutral chains and for polyelectrolyte chains at high salt concentrations. In low salt concentrations, the value of kn is expected to be an order of magnitude larger, due to the strong Coulomb repulsion between two polyelectrolyte chains, as seen in the case of colloidal solutions of charged spheres. While it is in principle possible to calculate the leading virial coefficients in Eq. (332) for different salt concentrations, the essential feature of the concentration dependence of t can be approximated by... 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

When space is partitioned with discrete boundaries, as in Eq. (6.7) and in the Bader virial partitioning method, the moments can be derived directly from the structure factors by a modified Fourier summation, as described for the net charge in chapter 6. 

Figure 6.9 Effect of CITREM concentration on the molecular and thermodynamic parameters of complex protein-surfactant nanoparticles in aqueous medium (phosphate buffer, pH = 7.2, ionic strength = 0.05 M 20 °C) (a) extent of protein association, k = Mwcomplex/Mwprotem (b) structure-sensitive parameter, p (c) second virial coefficient, A2 (rnolal scale) (d) effective charge, ZE (net number n of moles of negative charges per mole of original sodium caseinate nanoparticles existing at pH = 7.2 (Mw = 4xl06 Da)). The indicated cmc value refers to the pure CITREM solution. Reproduced from Semenova et al. (2007) with permission.

We have now looked at two models for the second virial coefficient of uncharged colloidal solutes. In Section 3.5b we see that B depends on the magnitude of the particle charge for polyelectrolyte solutes. 

Table 3.3 also includes an approximation for the case in which the concentration of the salt exceeds that of the colloid, but not to the swamping extent, mM > mP. Comparison of that case with the result given in Equation (34) suggests that the contribution of charge to the second virial coefficient of the solution is given by... 

We have already seen that the second virial coefficient may be determined experimentally from a plot of the reduced osmotic pressure versus concentration. Since all other quantities in Equation (99) are measurable, the charge of a macroion may be determined from the second virial coefficient of a solution with a known amount of salt. As an illustration of the use of Equation (99), we consider the data of Figure 3.6 in Example 3.5. 

Since the limiting value of Tr/mPshown in Figure 3.6 applies to these data also, it is possible to evaluate the second virial coefficient at these pH levels. Evaluate B and z, the effective protein charge, at the pH values shown. (Note that a slight variation in NaCl concentrations at these different pH levels should be taken into account for a more accurate determination of z.)... 

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