The dilute range

Tracer Diffusivity Tracer diffusivity, denoted by D g is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selr-diffusivities at the pure component limit. That is, at the limit of dilute A in B, D g D°g and... 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Now interpret phase X as pure solute then Cs and co become the equilibrium solubilities of the solute in solvents S and 0, respectively, and we can apply Eq. (8-58). Again the concentrations should be in the dilute range, but nonideality is not a great problem for nonelectrolytes. For volatile solutes vapor pressure measurements are suitable for this type of determination, and for electrolytes electrode potentials can be used. 

For solutions which do not follow Beer s Law, it is best to prepare a calibration curve using a series of standards of known concentration. Instrumental readings are plotted as ordinates against concentrations in, say, mg per lOOmL or lOOOmL as abscissae. For the most precise work each calibration curve should cover the dilution range likely to be met with in the actual comparison. 

A single extrapolation of the ratio D/s measured at a series of very low concentrations would then suffice. The graphical method advocated above for handling osmotic and turbidity data could be applied here also for the purpose of obtaining (D/s)o = RT/M, Again, if a poor solvent is used, so that T2 is very small, D/s should be nearly independent of c over the dilute range. 

Fig. 111.—Experimental values of the interaction parameter %i plotted against the volume fraction of polymer. Data for polydi-methylsiloxane M =3850) in benzene, A (New-ingi6). polystyrene in methyl ethyl ketone, (Bawn et aV ) and polystyrene in toluene, O (Bawn et alP) are based on vapor pressure measurements. Those for rubber in benzene, T (Gee and Orr ) were obtained using vapor pressure measurements at higher concentrations and isothermal distillation equilibration with solutions of known activities in the dilute range.

Although these examples demonstrate the feasibility of using calculated values as estimates, several constraints and assumptions must be kept in mind. First, the diffusant molecules are assumed to be in the dilute range where Henry s law applies. Thus, the diffusant molecules are presumed to be in the unassociated form. Furthermore, it is assumed that other materials, such as surfactants, are not present. Self-association or interaction with other molecules will tend to lower the diffusion coefficient. There may be differences in the diffusion coefficient for molecules in the neutral or charged state, which these equations do not account for. Finally, these equations only relate diffusion to the bulk viscosity. Therefore, they do not apply to polymer solutions where microenvironmental viscosity plays a role in diffusion. 

Debye-Huckel effects are significant in the dilute range and are not considered, and (2) the usual composition scale for the solute standard state is molality rather than mole fraction. Both of these problems have been overcome, and the more complex relationships are being presented elsewhere (17). However, for most purposes, the virial coefficient equations for electrolytes are more convenient and have been widely used. Hence our primary presentation will be in those terms. 

Prepare a standard graph by using the standard solution in the dilution range from 0.2 to 2 p,mol. 

The calibration graph should cover the dilution range likely to be used in the determination of the concentration of the unknown solution. If the solution of unknown concentration has an absorbance value outwlth those used in the calibration graph, another calibration graph should be prepared using more appropriate concentrations of the standard solutions. 

Pipet 100 pi of a suitable dilution of a secondary antibody-HRP conjugate into each well (the dilutions range from 1 500 to 1 100 000, dependent on the quality of the conjugate, and the amount of bound primary antibody the dilution has to be checked empirically). Incubate on a shaker at RT for 30 min. Wash with Soln. B at least three times to remove traces of unbound conjugate. 

Absorption isotherms were measured in the dilute range of the a phase from n = 0 to n = 0.012 and from 418° to 586°K. From four isotherms in the range... 

The fluidity of ammonia increases about 1.5% per degree, and in the dilute range the temperature coefficient of metal solutions is of this order of magnitude but from a concentration of approximately 0.9AT onward the temperature coefficient of sodium and potassium begins to increase, reaching a maximum of about 3.6% for sodium and 4.6% for potassium. The conductance increase owing to temperature increase can only be caused by an increased dissociation of sodium spinide. It follows that the conductance increase with increasing concentration of sodium solutions is to be expected and conforms with the assumptions of a micro Wien effect. 

Dilution of Samples. Samples are generally plated in dilutions of 1-100 or 1-1000. The dilution used will depend upon the number of colonies present on the plates after a 48-hour incubation period, or what previous experience with the product has shown. Dilutions should be adjusted so that the plate counts fall between 30 and 300 colonies per plate. If the count falls outside this range, higher or lower dilutions are necessary. Upon initial plating, when the dilution range is unknown, samples should be plated in dilutions of 1-100, 1-1000, 1-10,000 and made in duplicate. If spoilage is suspected the lower dilutions are not necessary and even higher dilutions made. 

Figure 2. Concentration dependence of V, scattering from Latex 42BRD47 at pH 3.64. The dilute range, expanded in the lower graph corresponds to residual molecular scattering in excess of the solvent.

However, in contrast to the ammonia system, metal solutions in amines, ethers, and related solvents are rich in information about distinguishable species existing in dilute solutions (53, 55). Sections II and III,B will similarly outline certain recent developments in these solvent systems, which have led to a fairly detailed picture of localized-electron states in the dilute range. 

In the weak-interaction model (85) developed in the previous section to explain ion-pairing in metal-ammonia solutions, aggregation interactions involving Ms+ and es are relatively weak, and leave the isolated solvated electron properties virtually intact. However, a major difficulty (29,54,134) arises with the type of model when one considers the precise nature of the corresponding electron spin-pairing interaction in ammonia solutions. It is worth expanding on this issue because it probably remains one of the fundamental dilemmas of metal-ammonia solutions in the dilute range (54). 

Finally, there is the matter of ion activity coefficients. The Davies equation given in Chapter 1 will be used, because all of the solutions are in the dilute range. In addition, ion pairing corrections will be made for the CaHCC>3+ and CaC03° ion pairs. This step requires iteration in calculation of the ion activity coefficients. The sequence demanded by the problems is that the concentrations must be initially calculated using ion activity coefficients from the previous case. These new concentrations are then used to calculate new ion activity coefficients, and the process is repeated until the desired degree of precision is reached. Neutral species will be assumed to have an ion activity coefficient of 1, and the ion activity coefficients of H+, OH-, and CaHCC>3+ will be assumed equal. 

Hydrogen bonding of associative groups is often characterized by a measurable shift of an absorption band. The measure of the intensity of these bands affords the equihbrium constant [90,107,185,199]. However, the use of FTIR spectroscopy is less versatile than NMR spectroscopy because solvent absorption often limits the dilution range accessible. 

Concentration-Dependent Da in Powders. It has been assumed in the powder techniques discussed so far that Da does not depend upon intracrystalline concentration. For the dilute range of sorptions—e.g., where Henry s law is valid—this is correct. However, as already considered (Table III), over extended ranges in concentration Tiselius s results for water in a single crystal of heulandite showed strong concentration dependence. For powders, 2 methods may be mentioned. 

The calculations have been carried out for those systems for which the solubility calculations have been performed. The dilute region of sodium chloride (c < 0.3) was selected to ensure that the condition F /x = s = constant is satisfied. The partial molar volume was estimated using literature data [67-69]. According to the latter data, depends weakly on C3 and this dependence is linear in the dilute range [68,69]. For sodium chloride and potassium chloride, decreases by at most 1 cm /mol when Cj is changed from 0 to 2mol/l. In our calculations, the above decrease was taken 1 cm /mol. On this basis the composition dependence of was evaluated in the composition range 0 < C3 < 0.3. The partial molar volumes Vi and V3 of water and sodium chloride in the binary mixture water (1) + sodium chloride (3) were obtained from data available in the literature [70,71], and the composition dependence of the isothermal compressibility of the mixed solvent (water (1) + sodium chloride (3)) was taken from reference [71]. 

Ann and Anjj are compared in Figure 4. One can see that the contribution of Awjj is important in the dilute range of the... 

Water (1)—Lysozyme (2)—NaCl (3). Data regarding the preferential binding parameter for these mixtures are not available in the dilute range. For this reason, the experimental OSVC were correlated using eq 10 with two adjustable parameters, J21 and 722- The partial molar volumes of the components of the protein-free mixed solvent (Vi and V3) were... 

---