Calculation of the Diffusion Potential

While Equation 5.4 can be used to calculate the diffusion potential, it is not common practice due to lack of information for such calculations. Equation 5.5 can be used instead, which requires a simple computer code to avoid mistakes using the equation with a number of summations. 

In some cases, Equation 5.5 can be simplified when one of the ions in the solutions (I) and (II) are the same and the molar concentration of both electrolytes is 

Equation 5.7 can now be used for calculating the diffusion potentials in Equation 5.3. If we assume that concentrations of HCl(aq, I) and HCl(aq, II) are, respectively, 0.1 and 0.5 mol kg, using the transport numbers for H+(aq) and Cl (aq) from Table 3.2, and the mean activity coefficients from [Chapter 10, Table 10.17], the value can be calculated as -26.9 mV. This is a pronounced value comparing with the first term of Equation 5.3, which is -40.1 mV. 

Equation 5.7 also shows that using a salt bridge with an electrolyte that has similar cation and anion conductivities is beneficial for minimizing the diffusion potential. This is why KCl(aq) is suggested to be used as the salt bridge electrolyte, in which the limiting ionic conductivities of K+(aq) and Cl (aq) are very similar and, respectively, 73.48 and 76.31 cm S mol- [Chapter 10, Table 10.12]. 

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Consider a system in which the analyte contains both determinand J and interferent K, and where a diffusion potential is formed in the membrane as a result of their different mobilities. A simplification that provides the basic characteristics of the membrane potential employs the Henderson equation for calculation of the diffusion potential in the membrane. According to (2.1.9) the membrane potential is separated into three parts, two potential differences between the membrane and the solutions A 0 and Aq with which it is ip contact, and the diffusion potential inside the membrane... 

Calculation of the diffusion potential requires the integration of Eq. (31.71) over the diffusion region ... 

Dumont and Bougeard (68, 69) reported MD calculations of the diffusion of n-alkanes up to propane as well as ethene and ethyne in silicalite. Thirteen independent sets of 4 molecules per unit cell were considered, to bolster the statistics of the results. The framework was held rigid, but the hydrocarbon molecules were flexible. The internal coordinates that were allowed to vary were as follows bond stretching, planar angular deformation, linear bending (ethyne), out-of-plane bending (ethene), and bond torsion. The potential parameters governing intermolecular interactions were optimized to reproduce infrared spectra (68). 

By analyzing the interaction between a chain center and the centers from adjacent chains, a relation for the potential interaction energy in both the normal and activated state was obtained (43). This formula was eventually used to develop a relationship for the calculation of the diffusion activation energy Ed. To use this relation... 

Quantitative measurements of electrokinetic phenomena permit the calculation of the zeta potential by use of the appropriate equations. However, in the deduction of the equations approximations are made this is because in the interfacial region physical properties such as concentration, viscosity, conductivity, and dielectric constant differ from their values in bulk solution, which is not taken into account. Corrections to compensate for these approximations have been introduced, as well as consideration of non-spherical particles and particles of dimensions comparable to the diffuse layer thickness. This should be consulted in the specialized literature. 

As already mentioned, the magnitude of the diffusion potential can be approximately calculated only in some special instances in general, this problem lias not yet been solved by electrochemistry. One of the above mentioned special cases is the combination of two different uni-univalent electrolytes, having a common ion and an identical concentration, such as. 

A second instance, where the value of the diffusion potential can be under certain conditions ascertained by calculation, is the system in which two solutions of the same electrolyte, but of different concentration, are in contact. The method of such calculation will be illustrated by an example of a concentration cell with transference composed of two hydrogen electrodes, dipped into hydrochloric acid solutions with different concentrations which are in direct contact ... 

On this basis, three models will be discussed, which enable a calculation of the electrical potential, namely the constant-capacitance, the diffuse-double-layer, and the triple-layer model. 

Besides the bormdary potentials at the two interfaces of the membrane, the membrane potential ( m) is also dependent of the diffusion potential ( ) (Equation 1). When there is a difference in ion activity between both sides of the membrane, ions start to diffuse from the high to the low activity side. A diffusion potential (Sd) is then created, caused by differences in mobility of cationic and anionic species in the membrane. This diffusion potential can be calculated with the use of the... 

The Z-axis forms an angle of 17° with the Zn-Ow bond shown in Figure 1. The position of Zn in the plane below [i.e., (0.0, 0.0, 0.0)] is indicated by dotted lines. The molecular wavefunctions were calculated with an ab-initio LCAO-SCF method using minimal STO-4G atom optimized basis sets. The basis set for Zn was augmented by additional diffused 3d and 4p functions obtained from an STO-4G expansion of Slater orbitals with exponents of 1.6575 and 1.45, respectively. The method used for the calculation of the electrostatic potentials is described in Ref. 65. Units are Kcal/mol. 

Fig. 31.2 shows profiles of electrode potentials across the three-layer boundary which reveal the ingress of ions into the intact adhesive/metal interface as a function of time. The potential profile can be assumed to reflect the ion concentration at the interface. This was proven by XPS small-spot analysis of the adhesive/metal interface after the removal of the adhesive [35]. Based on this assumption the profile can be evaluated and used as a basis for the calculation of the diffusion coefficient according to one-dimensional molecular diffusion in a medium with a constant diffusion coefficient. 

Flexible macromolecules. Calculations of the attractive potential energy according to equation (15.11) show that for spherical colloidal particles immersed in a dilute solution of rigid spheres, the attraction rarely exceeds k T. For articulated macromolecules, the configurational entropy of the chains is decreased in the neighbourhood of the interfaces and this provides a source of non-zero values for w(x,d). Asakura and Oosawa (1954) approximated this entropy decrement by analysing the problem in terms of the classical theory (Carslaw, 1921) for diffusion in a vessel with walls that absorb diffusing particles. The end result for parallel flat plates of area A is... 

In practice, in place of model calculations and corresponding corrections, the elimination of the diffusion potential is conventionally applied. This is achieved by introducing the so-called salt bridges filled with concentrated solutions of salts, which contain anions and cations of close transport numbers. A widely known example is saturated KCl (4.2 M) in aqueous solutions, potassium and ammonium nitrates are also suitable. However, the requirement of equal transport numbers is less important as compared with that of high concentration of electrolyte solution, which fills the bridge [33, 34]. A suitable version of the salt bridge can be chosen for any type of cells, when taking into account the kind of studies and the features of chosen electrodes. 

Fig. 1 shows that the curves obtained with 0.5 and 5 mM K in the internal volume were displaced by only about 0.35 pH-units, or approx. 21 mV from one another. This is due to the effect of the membrane electrical capacitance on the distribution of at equilibrium [4,7]. We used the method of Apell and Bersch [ ] to calculate equilibrium values of the -diffusion potential after addition of proteoliposomes with a known internal -concentration to a medium with 50 mM K, in the presence of valinomycin. Fig. 2 shows the dependence of these diffusion potentials on the internal diameter of the proteoliposomes. The dashed line in Fig. 2 shows that with proteoliposomes of 27 nm internal diameter, the -diffusion potential obtained with an initial internal K -concentration of 0.5 mM is only 21 mV higher than the one obtained with an initial internal -concentration of 5 mM. The diffusion potential obtained in the latter case is 52 mV. These diffusion potentials correspond with ApH-values of 0.36 and 0.88 units, respectively. This is in good agreement with the results shown in Fig. 1, and the required internal diameter of 27 nm is in good agreement with electron-microscopic and other evidence on the size of the proteoliposomes [2]. Furthermore, Fig. 2 shows that vesicles of this diameter generate a K -diffusion potential of only 77 mV even if the initial internal -concentration is zero. Since ATP-synthesis was observed only above a threshold Apjj+ of 90 mV (Fig. 1), this explains why... 

Experimental [3] and Calculated (Using Equation 5.6) Values of the Diffusion Potential... 

The gradient of the diffusion potential, dil//dx)j p, may be calculated from the condition that, in the steady state, no electric current may pass through the solution, i.e.,... 

For an electrode with high interfacial rate constants, for example, relation (28) can be plotted, which yields the flatband potential. It allows determination of the constant C, from which the sensitivity factor S can be calculated when the diffusion constant D, the absorption coefficient a, the diffusion length L, and the incident photon density I0 (corrected for reflection) are known ... 

The ionic concentration gradients in the transition layer constitute the reason for development of the diffusion component E of electric field strength (the component arising from the difference in diffusion or mobihties between the individual ions). The diffusion potential between the solutions, 9 = - / can be calculated... 

The diffusion-potential reduction thus attained is entirely satisfactory for many measurements not demanding high accuracy. However, this approach is not feasible for the determination of the accurate corrected OCV values of cells with transference that are required for thermodynamic calculations. 

We shall write (p) and (q) for the membrane surface layers adjacent to solutions (a) and (p), respectively. Using the equations reported in Section 5.3, we can calculate the ionic concentrations in these layers as well as the potential differences and between the phases. According to Eq. (5.1), the expression for the total membrane potential additionally contains the diffusion potential within the membrane itself, where equilibrium is lacking. Its value can be found with the equations of Section 5.2 when the values of and have first been calculated. 

The equation obtained can be used when the electrode potential can be varied independent of solution composition (i.e., when the electrode is ideally polarizable). For practical calculations we must change from the Galvani potentials, which cannot be determined experimentally, to the values of electrode potential that can be measured E = ( q + const (where the constant depends on the reference electrode chosen and on the diffusion potential between the working solution and the solution of the reference electrode). When a constant reference electrode is used and the working solutions are sufficiently dilute so that the diffusion potential will remain practically constant when their concentration is varied, dE (i(po and... 

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