Donnan equations

Building on initial work [47], the main focus of SECM in the study of ET at ITIES has been to identify and understand the potential-dependence of ET rates. In these studies, the potential drop across an ITIES has been controlled by varying the concentration of potential-determining ions in the two phases. The potential drop across an ITIES follows the Nernst-Donnan equation [74,75],... 

When the expressions (6.2.5) are substituted into the Henderson equation (2.5.34) A0l is obtained. Both contributions A0D are calculated from the Donnan equation. From Eq. (6.2.3) we obtain, for the membrane potential,... 

To describe macromolecular solutions, the Donnan theory is used in form of the Donnan equation... 

Fig. 21. (a) Rate constant for the hydrolysis on 0.1 N HCl of a cholesterol formate monolayer (50a). The incorporation into the film of a little long-chain sulfate (C22H46-SOi ) greatly accelerates the reaction. The calculated increases in reaction rate according to the Gouy (33) and Donnan equations are shown (b) For hydrolysis on 0.66 N HCl incorporation of CisH37N(CH3)+ into the film retards reaction because hydrogen ions are repelled from the surface (50a). 

Fig. 22. Equilibrium between bulk and surface phases for a film of insoluble acid HA (e.g., octadecanoic acid) and its anion A , in the presence of a small amount of HOI. The hydrogen and chloride ions can penetrate into the surface phase, remaining in equilibrium with the bulk. The final equilibrium is given by the Donnan equations (xxv) and (xxvi), the former referring to the equilibrium of HCl between the surface and bulk phases, and the latter to the electrical neutrality of the surface phase.

This method of finding the concentration of ions near the surface was applied by Davies (49,21) to the hydrolysis of ionized films of the ester monocetyl succinate. Table IX shows that the rate constant for this hydrolysis, which increased 300% if calculated using bulk concentrations of the catalytic hydroxyl ion, varied by not more than 36% when evaluated using the surface concentrations deduced from (xxv) and (xxvi). Figure 19 shows a similar effect for the addition of neutral salt, the marked catalysis by which is thus demonstrated to be due entirely to electrostatic effects. The acceleration in the rate of hydrolysis of a film cholesterol formate if the surface bears a negative charge can be predicted on the basis of the Donnan equations (xxv) and (xxvi). Values of 5 of 6 A. and 8 A. have been used, the results being compared with experiment in Fig. 21a. The calculated retardation is shown in Fig. 21b. 

If swelling effects are ignored, the usual simplified Donnan equation for co-ion uptake from dilute solutions is obtained namely ... 

The above equations can be compared with the GC equations (11), (12) and (19) for ipd and (7d respectively. In the Donnan equations the salt concentration appears directly, whereas in the GC equations k or c appears, this indicates that the salt effects are relatively large for the Donnan model. 

This is the fundamental Donnan equation and may be written for the above example in the form ... 

The TPM is generally designated for the chemical stimulation and can provide local chemical as well as mechanical unknowns. In most cases, the gel phase only is investigated by prescribing the concentrations at the boundary of the gel (at the gel-solution interface) by using the Donnan Equation (7) together with the electroneutrality condition of (6). 

Two further limiting cases are encountered experimentally, particularly in electrochemical approaches to measurement of charge transfer kinetics (Sections III and IV). The first is the case of an interface with a single potential determining ion. This corresponds to the addition of salts AX and CX to the system (i.e., a common ion (X) with hydrophilic and hydrophobic counterions), such that the distribution of X is controlled and can in turn be used to control the Galvani potential difference via the Nemst-Donnan equation ... 

According to the equation log m is a linear function of log m and the constant z can be found from the slope. The constant z is a measure of the heterogeneity of the membranes, becoming zero for a perfectly uniform membrane, as demanded by the approximate form of the Donnan equation for a 1 1 electrolyte. The z of various commercial ion exchange membranes has been measured using salt and acid solutions.111112 The complete distribution function is given by... 

In order to relate the ionic concentrations at the inner and outer edges of the membrane to that in the bulk solutions, the Donnan equation is utilized [Eq. (134)]. 

This equation, called the Donnan equation, is named after the British chemist Frederick George Donnan who published his important paper about the theory of membrane equilibrium in 1911. 

Chemical potentials inside the membrane are determined by the Flory-Rehner-Donnan equation of state for one-dimensional swelling (5), since the clamped membrane s area. A, is constant. The polymer-solvent interaction parameter, f, is a linear function of the polymer volume fraction, i.e. J + Xi as proposed by Hirotsu (/ 7). Transport of water into and out of the hydrogel membrane causes its thickness, I, to change at a rate that is proportional to the difference in chemical potential of water between the membrane and Cell I and Cell II. 

The ion activity can be replaced by concentration to a first approximation. The co-ion concentration can be calculated with the Donnan equation based on the capacity of an ion-exchange membrane. It is more difficult to determine the ions mobility ratio. The counterion mobility can be calculated based on isoconductivity point for the appropriate ionic form. The mobility of co-ions can be determined based on a seh-diffusion coefficient measurement. 

When a column system involving gel particles carrying a small number of fixed negative charges, which are distributed uniformly throughout the gel interior, is saturated with an aqueous solution of a 2 1 salt, K A , the relation between the distribution coefficient of anion A", K, and the salt concentration in the external liquid phase, C, can be derived from Donnan equation [ref. 49]... 

Thus, the diffusible ions are distributed as follows K C = Ki C i, according to the Donnan equation. The potential that develops at the surface of the cell as a result of the uneven distribution of the cations can be calculated with the aid of the Nernst equation ... 

The Galvani potential difference across the oil/water interface A< q (f e potential of the water (W) phase referred to that of the oil (O) phase) when both the O and W phases contain a common ion M which is transferable across the interface is given by the Nernst-Donnan equation [31, 39-42],... 

The Donnan equation involves nine factors concentration of dye in fibre, concentration of dye applied, liquor ratio, distribution of ions between solution and fibre, the ionic charge on the dyestuff molecule, the internal volume of the fibre, the affinity of the dye, the gas constant and the dyeing temperature. All together, these terms describe the dyebath conditions. 

The Additivity Rule has been assumed to describe the Donnan equilibrium between polyelectrolyte gels and aqueous solutions of low molecular weight electrolytes. [10] The mean molal activity coefficient, for the salt which invades the gel may be computed from experimental measurements of the equilibrium partition of electrolyte with the thermodynamic Donnan equation, and its value compared with that given by the Rule. According to the Rule, y+ should be constant and independent of salt concentration in fact, y was observed to increase with dilution. [11]... 

---