Adding Ionic Terms

For the H2 molecule there are two possible ionic states (H+H ) and (H H+). The wave functions for these two ionic states are 

Adding the contributions from these ionic wave functions to the ground state of the H2 molecule gives 

The incorporation of ionic terms in molecular wave functions plays a major role in the description of solvent (and protein) effects on chemical processes. This point will be emphasized repeatedly throughout this book. 

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The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

Usnally, only very dilute solutions can be considered ideal. In most aqueous solutions, ions are stabilized because they are solvated by water molecules. As the ionic strength is increased, ions interact with each other. Thus, when calculating the chemical potential of species i, a term that takes into account the deviation from ideal conditions is added. This term is called an excess term and can be either positive or negative. The term usually is written as 7 riny., where y. is the activity coefficient of component i. The complete expression for the chemical potential of species i then becomes... 

Therefore, the principal role of the inclusion of the ionic term in the wave function is the reduction of the kinetic energy from the value in the purely covalent wave function. Thus, this is the delocalization effect alluded to above. We saw in the last section that the bonding in H2 could be attributed principally to the much larger size of the exchange integral compared to the Coulomb integral. Since the electrical effects are contained in the covalent function, they may be considered a first order effect. The smaller added stabilization due to the delocalization when ionic terms are included is of higher order in VB wave functions. 

At first glance the use of solid nitrile rubber in place of liquid plasticizers would appear to improve the volume resistivity of plasticized polyvinyl chloride somewhat but when the lower plasticizing efficiency of the nitrile rubber is considered, only little improvement remains at equal tensile modulus or hardness. This is difficult to explain in terms of the flow of ions through a liquid plasticizer medium. As we can see, the volume resistivity of nitrile rubber alone is much lower than that of polyvinyl chloride, and the volume resistivity of these blends is simply the resultant of the two components. Actually the same reasoning might well apply to conventional blends of good quality polyvinyl chlorides with good quality liquid plasticizers, in the absence of any added ionic soluble impurities, as we can see from our earlier data. 

We have shown that the values of oxyacids of the type HOX, where X is [M(OH) 0 ], are well described by equation 23 with q representing the charge on the X group. This result is in accord with equation 41. We conclude that, in general, ionic substituents can be described by simply adding the term a q to the equation used to estimate parameter values for uncharged groups. Thus ... 

This increase in y values with ionic strength can be modeled by adding positive terms to some form of the extended Debye-Hiickel expression for log y. Simple models using this approach have been proposed by Hiickel (see Harned and Owen 1958), and more recently by others, including... 

It should also be noted that the full complexity of the PB equation is not always necessary. In many cases mobile ion densities are small enough that the exponential factors in the mobile ion distribution can be approximated by keeping only terms linear in the potential, leading to the linearized Poisson— Boltzmann (LPB) equation. In this case the ionic terms can be added to the diagonal of the coefficient matrix A, and extremization along any component direction is analytic because the surface is parabolic. Another advantage of LPB is the additivity of terms in the electrostatic potential, which enables contributions made by any part of a molecule to be determined. 

On the basis of this mechanism, a great number of ionophore-based ISEs were developed. It should also he noted here that polyvinyl chloride (PVC) has been used since the early 1970s to support liquid membranes, and the term plasticized liquid membrane" has been coined to describe these systems. In those days, the addition of ionic sites was not explicitly realized, but instead, impure ionic sites in the PVC (anionic sites) played virtually the same role. However, the membrane including such an impurity does not allow quantitative control of the optimized amounts of ionic sites, and therefore, it was supplemented later by deliberately added ionic sites. 

Quasi-coupKng between the ionic and electronic currents is sometimes introduced by adding a term LieVAe to Eq. (1) and a term LeiVAi to Eq. (2). However, it was shown that these corrections are not really necessary (except perhaps in metals [33]) if the current density of all mobile species, as well as interactions between defects are all taken into consideration [36]. 

For example, Marshall and Slusher (1966) made a detailed evaluation of the solubility of ealeium sulphate in aqueous sodium chloride solution, and suggested that variations in the ion solubility product could be described, for ionic strengths up to around 2 M at temperatures from 0 to 100 °C, by adding another term in an extended Debye Hiickel expression. Above 2 M and below 25 °C, however, further correction factors had to be applied, the abnormal behaviour being attributed to an increase in the complexity of the structure of water under these circumstances. Enthalpies and entropies of solution and specific heat capacity were also reported as functions of ionic strength and temperature. 

The VB wavefunction suffers from the opposite problem because it does not allow for any ionic character to be present in the hydrogen bond. This situation can be improved by adding the ionic terms found in the MO wavefunction, but multiplied by a weighting factor A. With this modification, the spatial part of the VB wavefunction becomes ... 

Soon, Knox thought to use the hydrophobic effect to form ion-pairs in aqueous phases. He developed the first use of reversed-phase ion-pair chromatography in 1976 [6]. He termed it soap chromatography. He added ionic surfactants to the polar hydro-organic mobile phase. They adsorb on the alkyl-bonded stationary phase. They also associate with the hydrophobic and ionic analytes. 

We turn to the more complicated but important problem of ionic surfactant adsorption, and start with the salt-free case where strong electrostatic interactions are present. In Fig. 3 we have reproduced experimental results published by Bonfillon-Colin et al. for SDS solutions with (open circles) and without (full circles) added salt [13]. The salt-free ionic case exhibits a much longer process with a peculiar intermediate plateau. Similar results were presented by Hua and Rosen for DESS solutions [21]. A few theoretical models were suggested for the problem of ionic surfactant adsorption [22-24], yet none of them could produce such dynamic surface tension curves. It is also rather clear that a theoretical scheme such as the one discussed in the previous section cannot fit these experimental results. On the other hand, addition of salt to the solution screens the electrostatic interactions and leads to a behavior very similar to the non-ionic one. We shall return to this issue in Section 4. We thus infer that strong electrostatic interactions affect drastically the adsorption kinetics. Let us now study this effect in more detail. We follow the same lines presented in the previous section while adding appropriate terms to account for the additional interactions. 

A bonding factor corresponds to a single covalent bond, so the bond order for the wave function in Example 20.7 is equal to 2, as in the LCAOMO description of Example 20.6. This wave function does not include ionic terms, but ionic terms can be added to each bonding factor in the wave function to make an improved valence-bond function. 

The contribution of the elastic term in lightly cross-linked networks can be described by the Gaussian theory of rubber elasticity (2,3). In fully neutralized polyelectrolytes, in the presence of added salt, the ionic term is not expected to play an exphcit role. Ionic interactions, however, may modify the mixing free energy contribution. In neutral polymer solutions the Flory-Huggins theory (I), based on the lattice model of solutions, expresses the mixing pressure as... 

Were we to simply add the ionization energy of sodium (496 kJ/mol) and the electron affin ity of chlorine (—349 kJ/mol) we would conclude that the overall process is endothermic with AH° = +147 kJ/mol The energy liberated by adding an electron to chlorine is msuf ficient to override the energy required to remove an electron from sodium This analysis however fails to consider the force of attraction between the oppositely charged ions Na" and Cl which exceeds 500 kJ/mol and is more than sufficient to make the overall process exothermic Attractive forces between oppositely charged particles are termed electrostatic, or coulombic, attractions and are what we mean by an ionic bond between two atoms... 

Water. Latices should be made with deionized water or condensate water. The resistivity of the water should be at least lO Q. Long-term storage of water should be avoided to prevent bacteria growth. If the ionic nature of the water is poor, problems of poor latex stabiUty and failed redox systems can occur. Antifreeze additives are added to the water when polymerization below 0°C is required (37). Low temperature polymerization is used to limit polymer branching, thereby increasing crystallinity. 

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