Equivalent Conductivity Varies with Concentration

At first sight, the title of this section may appear surprising. The equivalent conductivity has been defined by normalizing the geometry of the system and the 

It would be awkward to have to refer to the concentration every time one wished to state the value of the equivalent conductivity of an electrolyte. One should be able to define some reference value for the equivalent conductivity. Here, the facts of the experimental variation of equivalent conductivity with concentration come to one s aid as the electrolytic solution is made more dilute, the equivalent conductivity approaches a limiting value (Fig. 4.57). This limiting value should form an excellent basis for comparing the conducting powers of different electrolytes, for it is the only value in which the effects of ionic concentration are removed. The limiting value will be called the equivalent conductivity at infinite dilution, designated by the symbol /1° (Table 4.12). 

Equivalent Conductivities of True Electroiytes in Dilute Aqueous Solutions at 296 K 

It may be argued that if at infinite dilution there are no ions of the solute, how can the solution conduct The procedure for determining the equivalent conductivity of an electrolyte at infinite dilution will clarify this problan. One takes solutions of a substance of various concentrations, determines the /r, and then normalizes each to the equivalent conductivity of particular solutions. If these values of A are then plotted against the logarithms of the concentration and this A versus log c curve is extrapo- 

Equivalent Conductivities at Infinite Dilution of Eiectrolytes In Aqueous Soiution at 298 K 

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The above argument brings out an important point about the limitations of the Nernst-Einstein equation. It does not matter whether the diffusion coefficient and the equivalent conductivity vary with concentration to introduce deviations into the Nernst-Einstein equation, D and A must have different concentration dependencies. The concentration dependence of the diffusion coefficient has been shown to be due to nonideality (f 1), i.e., due to ion-ion interactions, and it will be shown later that the concentration dependence of the equivalent conductivity is also due to ion-ion interactions. It is not the existence of interactions perse that underlies deviations from the Nernst-Einstein equation otherwise, molten salts and ionic crystals, in which there are strong interionic forces, would show far more than the observed few percent deviation of experimental data from values calculated by the Nernst-Einstein equation. The essential point is that the interactions must affect the diffusion coefficient and the equivalent conductivity by different mechanism and thus to different extents. How this comes about for diffusion and conduction in solution will be seen later. 

For practical systems, as will be seen in the part on transport with MSA (mean spherical approximation), the equivalent conductivity varies with concentration. 

Again, as expected, the equivalent conductance changes with concentration. However, it was noted by early investigators that for dilute (less than about 0.1 normal) solutions, A varied with the square root of the concentration, and the y-intercept of the straight line of A versus /n was a value of A that was characteristic of the ionic solute. This characteristic, infinitely diluted value is given the symbol Aq. Various values of Ag are listed in Table 8.4. Mathematically, the relationship between the equivalent conductance versus concentration can be expressed as... 

Fig. 6.1-4. Equivalent conductance versus concentration. Conductance varies with concentration, especially at high dilution. For strong electrolytes like KCI and CaCh, these variations are chemically interesting but practically unimportant. For weak electrolytes like acetic acid, the variation is larger (see Section 6.2).

Critical Micelle Concentration. The rate at which the properties of surfactant solutions vary with concentration changes at the concentration where micelle formation starts. Surface and interfacial tension, equivalent conductance (50), dye solubilization (51), iodine solubilization (52), and refractive index (53) are properties commonly used as the basis for methods of CMC determination. 

What does Eq. (4.163) reveal It shows that the equivalent conductivity will be a constant independent of concentration only if the electrical mobility does not vary with concentration. It will be seen, however, that ion-ion interactions (which have been shown in Section 3.3.8 to depend on concentration) prevent the electrical mobility from being a constant. Hence, the equivalent conductivity must be a function of concentration. 

For the polyion equivalent conductivity, conditions are different. Here an appreciable concentration dependence is expected even in very dilute solutions. This is partly due to the direct dependence of Ap on a, a quantity that may vary with concentration, and partly due to the concentration dependence of the friction coefficient/p. As in the case of polymeric solutes in general, the friction coefficient depends on the polyion chain conformation, which for flexible polyelectrolytes is strongly concentration dependent. Furthermore, the polyion friction coefficient also includes contributions from the fraction (1 — a) of the counterions, which form a kinetic unit with the polyion. The friction coefficient can therefore be written in the form... 

Chemical diffusion is sometimes referred to as diffusion under a chemical potential or concentration gradient. However, we have seen that self-diffusion properly describes this, and that all diffusion phenomena we consider here are merely minor perturbations of thermally induced self-diffusion. Why then do we need to be concerned with chemical diffusivity We use it whenever we want to or have to relate our flux to the concentration gradient. Under steady state processes, our flux vs force equivalents provides sufficient description, and can be integrated to yield steady state fluxes through membranes, as we have shown Thus, for this purpose, the self-diffusion coefficients and conductivities are useftil since they behave in a simple manner vs defect concentrations, and since we in general know the chemical and/or electrical potential gradents applied. However, if we know the concentration gradients and how the chemical diffusivity varies with concentration, we can use chemical diffusivity. 

It is not usual to talk about the resistance of electrolytes, but rather about their conductance. The specific conductance (K) of an electrolyte is defined as the reciprocal of the resistance of a part of the electrolyte, 1 cm in length and 1 cm2 in cross-sectional area. It depends only on the ions present and, therefore it varies with their concentration. To take the effect of concentration into account, a function called the equivalent conductance, A, is defined. This is more commonly (and conveniently) used than the specific conductance to compare quantitatively the conductivities of electrolytes. The equivalent conductance A is the conductance of that volume of the electrolyte which contains one gram equivalent of the ions taking part in the electrolysis and which is held between parallel electrodes 1 cm apart (units ohm-1 cm4). If V cubic centimeters is the volume of the solution containing one gram equivalent, then the value of L will be 1 cm and the value of A will be V square centimeters, so that... 

For a weakly ionized substance, A varies much more markedly with concentration because the degree of ionization a varies strongly with concentration. The equivalent conductance, however, must approach a constant finite value at infinite dilution, Ag, which again corresponds to the sum of the limiting ionic conductances. It is usually impractical... 

In the phenomenological treatment of conduction (Section 4.2.12), it was stated that the equivalent conductivity A varies with the concentration c of the electrolyte according to the empirical law of Kohlrausch Piq. (4.139)]... 

In solvents of moderate dielectric constant (e.g. THF, D 7.6 at room temperature) the polymerization systems show measurable conductance. It is clear that the concentration of free ions cannot be large, since conventional inorganic salts are not highly dissociated in such solvents. Figure 3 shows the variation of equivalent conductance with concentration for polystyrylsodium (two ended, Na CH(Ph)CH2. (CHPhCH2 ) q-CH2CH(Ph)"Na" ) in THF at 20°C. A varies as [C] - as expected if the degree of dissociation to free ions is low. A dissociation constant of... 

Theoretical interpretation of the concentration dependence of equivalent conductivity for simple binary mixtures was first presented by Markov and Shumina (1956). It should be emphasized that this theory, even when considering simple structural aspects, represents rather a method of interpretation of the experimental data than a genuine picture of the structure of the melt. In molten salts generally only ions and not molecules are present, hence the conception of Markov and Shumina (1956) is to be considered also from this aspect. Their theory is based on the assumption that the electrical conductivity of a mixture of molten salts varies with temperature like pure components. In this respect, general character of the electrical conductivity dependence on composition, indicating the interaction of components in an ideal solution, could be expected. 

Ionic conduction may dominate the electrical behavior of materials with small electronic conductivity, and its study is useful in the investigation of lattice defects and decomposition mechanisms. In order to establish that conduction takes place by the motion of ions and not of electrons or holes, one can compare the transport of charge with the transport of mass plated out on electrodes in contact with the sample. In practice, this approach is not always feasible because of the very low conductivities associated with ionic motion. When ionic conductivity is suspected one usually attempts to vary the concentration of defects by introducing impurities. For example, for cation conduction in monovalent ionic compounds, addition of divalent cations should enhance the conductivity, since the vacancies produced (in order to ensure charge compensation) lead to enhanced diffusion of the monovalent cation. (The diffusion of a vacancy in one direction is equivalent to the diffusion of an ion in the opposite direction). 

The equivalent conductance A can be extremely accurately measured, often to accuracies of 0.01%. It is known to vary slightly with concentration, as shown in Fig. 6.1 -4. This variation follows the equation... 

After the adjustments made in late August 1996, the gasoline-equivalent hydrocarbon recovery rate increased from about 0.55 gal/hr to between 0.65 and 0.70 gal/hr. AcuVac conducted mini-pilot tests in the SVE wells on September 27-28, 1996. These tests showed that the flow from individual wells varied from less than 1 to 16 cfm at a vacuum of approximately 90 inches of water. Hydrocarbon vapor contents ranged from 40 to 2,040 ppmv. Based on the observed vapor flows and hydrocarbon concentrations, and with the objective of maintaining hydraulic control by applying suction to raise water levels, 10 wells were selected for continued operation. These... 

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