Manning theory

The Manning theory(16,17) predicts a critical value c above which counterions are condensated on polyion. In the case of monovalent ions, =1 and for > c, the chain expansion... 

According to theoretical considerations, fa increases if the charge distance on the polyelectrolyte chain becomes larger. This could be shown for DADMAC/AAM copolymers for charge distances in the range of 0.5 to 1.9 nm (Table 10). For b < 1B the fa values are within the limits of the Manning theory and below the values calculated by Gueron. However, for b > 1B the experimental values are remarkably below the theoretical values (Table 10). 

Assuming infinitely long polyelectrolyte chains, the Manning theory neglects the influence of molar mass on the equivalent conductivity. It could be shown in... 

Counterion condensation theory, however, does not provide a detailed picture of the distribution of the condensed Ions. Recent research using the Poisson-Boltzmann approach has shown that for cylindrical macroions exceeding the critical linear charge density the fraction of the counterions described by Manning theory to be condensed remain within a finite radius of the macroion even at infinite polyion dilution, whereas the remaining counterions will be infinitely dispersed in the same limit. This approach also shows that the concentration of counterions near the surface of the macroion is remarkably high, one molar or more, even at infinite dilution of the macromolecule. In this concentrated ionic milieu specific chemical effects related to the chemical identities of the counterions and the charged sites of the macroion may occur. 

As typically observed for polyelectrolytes, a large proportion of small co-ions and counterions is held within the domain of the macro-ion. Activity measurements made with ion-specific electrodes have shown that 66% of the Na" " ions of sodium heparinate are bound to the polyanionic chain. Binding of Na" " by heparin was also studied by Na-n.m.r. spectroscopy. Ion-activity data have been interpreted in the framework of the Manning theory, providing intercharge distances ranging from 0.24 to 0.47 nm, depending on the heparin preparation and the experimental conditions (see Ref. 7, and references cited therein). 

Equations are derived from the Manning theory of polyelectrolyte solutions that permit the calculation of the charge fraction of ionic polysaccharides using experimental counterion and coion diffusion coefficients. Results are reported for the sodium salts of heparin, dextran sulfate, alginate and the calcium salt of heparin. 

It will now be shown how counterion and coion diffusion results can be used to estimate the degree of dissociation of a polyelectrolyte if the experimental diffusion coefficients approximate those predicted from the Manning theory. The Manning thecry... 

Figure 2. Chloride ion diffusion ratio in aqueous sodium heparin solutions containing sodium chloride. The solid line is predicted from the Manning theory (8).

Table I. A Comparison of the Experimental Charge Fraction with Those Predicted from the Manning Theory for Several Ionic Polysaccharides...

Counterion binding to polyelectrolytes appear to be adequately accounted for in the operationally defined condensation term in the Manning theory. However, there are some experimental results that indicate that counterion binding may be present for polyelectrolytes where which is contrary to the Man-... 

Money is often called the root of all evil, and in safety, money paid to be injury-free fits that description. I have seen employees hide injuries, lie about injuries, and do anything in their power to cover up injuries, all for the money. Paying employees a safety bonus is tantamount to encouraging them to be dishonest about their injuries and to purposefully hide them so that they can collect the monthly so-called safety bonus. This does not make employees work safer It makes them more clever at hiding injuries and deceiving the system. The reasonable man theory (i.e., how the average person would behave) says that most workers would do the same. 

There are multiple theories for determining the amount of compensation. Three major ones are the whole-man theory, the lost wages theory, and loss of earning capacity. One or more of the theories may apply under a particular law. 

Whole-Man Theory The whole-man theory considers only the functional effect of the loss—its impact on normal functions and abilities. The theory defines disability as a percentage of a whole, fiiUy-fiinctional person. A formula... 

On the contrary, the Q-branch of the anisotropic Raman spectrum broadens monotonously until it merges with the 0-and S-branches (11). The major restriction of these theories is that they are essentially gas phase theories however, this restriction is less stringent for the theory by Burshtein et al. Moreover, the rotation-vibration coupling effects due to intermolecular forces are neglected here again. This is done in the Sobel man theory by neglecting the v-dependence of the S matrix. 

This theory allows the PE to bent due to entropy, including at low ionic strength, and allows PE of a Unite length, in contrast of Mannings theory. For an alternative definition of F(, and its derivation, see Ref [59]. 

The values of the counterion condensation degree predicted from the limiting Manning theory according to the equation ... 

Results are represented in Figure 15. The probability of the distribution can be computed from (23) the function has a minimum at r = Tq in Figure 16, a typical result is presented. The fraction of counterions contained between ro and R is a= /2X for all values of 2/2q. This is also found in the Manning theory when 2> 1. 

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