Low-concentration behavior

FIGURE 5.18 Tungsten CMP removal rates R versus abrasive loading for five different abrasive sizes. Symbols are for data from Bielmann et al. [74] (a) plotted versus weight percent and (b) the same data plotted versus particle concentration, including inset for low-concentration behavior. 

The most important application of low concentration behavior of solutes is for solid solutes, especially electrolytes. Electrolyte solutions are examined in detail in chapter 3. Some general thermodynamic methods for describing the properties of very dilute solutions are considered in the following section. 

Various anomalies in low-concentration behavior have been reported. In precipitation, a trace-level radionuclide that does not exceed its solubility product may coprecipitate with a second substance. In electrodeposition, despite adjustment for concentration (or more accurately, thermodynamic activity) by the Nernst equation, the predicted voltage may not be suitable because of the factors discussed in Section 3.7 pertaining to overvoltage and an electrode surface not fully covered by the element of the radioisotope (Haissinsky 1964). In volatilization or sublimation of a radionuclide, distinctive effects have been attributed to the situation that atoms at trace-level are not bonded to each other but to the other atoms in the medium or on a surface. A radioactive cation sorbed on a negatively charged solid particle that it does not fully neutralize will behave like a negative particle. 

The type of behavior shown by the ethanol-water system reaches an extreme in the case of higher-molecular-weight solutes of the polar-nonpolar type, such as, soaps and detergents [91]. As illustrated in Fig. Ul-9e, the decrease in surface tension now takes place at very low concentrations sometimes showing a point of abrupt change in slope in a y/C plot [92]. The surface tension becomes essentially constant beyond a certain concentration identified with micelle formation (see Section XIII-5). The lines in Fig. III-9e are fits to Eq. III-57. The authors combined this analysis with the Gibbs equation (Section III-SB) to obtain the surface excess of surfactant and an alcohol cosurfactant. 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Fiend s Constant. Henry s law for dilute concentrations of contaminants ia water is often appropriate for modeling vapor—Hquid equiHbrium (VLE) behavior (47). At very low concentrations, a chemical s Henry s constant is equal to the product of its activity coefficient and vapor pressure (3,10,48). Activity coefficient models can provide estimated values of infinite dilution activity coefficients for calculating Henry s constants as a function of temperature (35—39,49). 

This study describes the direct photochemical degradation of the methomyl presents at low concentration in different ogranic solvents. Also the kinetic behavior of photolytic reaction of methomyl with solvents has been studied. 

Typically, elimination by the El mechanism is observed only for tertiary and some secondary alkyl halides, and then only when the base is weak or in low concentration. Unlike eliminations that follow an E2 pathway and exhibit second-order kinetic behavior ... 

Analysis of the physicochemical properties of fluoride and oxyfluoride melts reveals that the complex ions are characterized by coordination numbers that do not exceed seven. Fluoride melts consist of the complex ions MeF72 and MeFe. Molten chloride-fluoride systems initiate the formation of heteroligand complexes of the form MeFgCl2 . Oxyfluoride and oxyfluoride-chloride melts can contain oxyfluoride complexes MeOF63 at relatively low concentrations. The behavior of the more concentrated melts can be attributed to the formation of oxyfluorometalate polyanions. 

How does yield stress depend on a filler concentration It is shown in Fig. 9 that appreciable values of Y appear beginning from a certain critical concentration cp and then increase rather sharply. Though the existence of cp seems to be quite obvious from the view point of the possibility of contacts of the filler, i.e. the beginning of a netformation in the system, practically the problem turns on the accuracy of measuring small stresses in high-viscosity media. It is quite possible to represent the Y(cp) dependence by exponential law, as follows from Fig. 10, for example, leaving aside the problem of the behavior of this function at very low concentrations of the filler, all the more the small values of are measured with a significant part of uncertainty. 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Typical adsorption isotherms are shown in Figs. 16 and 17. Despite the large experimental scatter, a steep increase in adsorption can be seen at low concentrations, followed by a plateau at concentrations exceeding the CMC. Similar behavior has been observed before with model surfactants [49-54] and has also been predicted by modem theories of adsorption [54]. According to Fig. 16, adsorption increases modestly with salinity provided that the calcium ion concentration remains low. The calcium influence, shown in Fig. 17, cannot be explained by ionic strength effects alone but may be due to calcium-kaolinite interactions. 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

---