Henderson diffusion potential

Taking for the diffusion potential the Henderson diffusion potential and adding the two Donnan potentials one gets the well-known formula of K. H. Meyer (97, 98), J. F. Sievers (97, 98) and T. Teorell (168). 

The use of the Henderson diffusion potential involves that the mobilities u+ and m are supposed to be constant, and that ideal solutions are assumed. Water transport is neglected. 

The diffusion potential, >diff, is calculated from Eq. (2.40) on the assumption that ions are linearly distributed across the membrane (an assumption made by the Henderson diffusion potential),... 

When both solutions are binary and identical in nature and differ only by their concentration and the component E of the held strength is given by Eq. (4.18), the diffusion potential 9 can be expressed by Eq. (4.19). An equation of this type was derived by Walther Nemst in 1888. Like other equations resting on Eick s law (4.1), this equation, is approximate and becomes less exact with increasing concentration. For the more general case of multicomponent solutions, the Henderson equation (1907),... 

Here, x is the coordinate normal to the diaphragm, so that d — q—p. The liquid junction potential A0L is the diffusion potential difference between solutions 2 and 1. The liquid junction potential can be calculated for more complex systems than that leading to Eq. (2.5.31) by several methods. A general calculation of the integral in Eq. (2.5.30) is not possible and thus assumptions must be made for the dependence of the ion concentration on x in the liquid junction. The approximate calculation of L. J. Henderson is... 

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... 

The hydrated layer has finite thickness, therefore the exchanging ions can diffuse inside this layer, although their mobility is quite low compared to that in water (n 10-11cm2s-1 V-1). As we have seen in the liquid junction, diffusion of ions with different velocities results in charge separation and formation of the potential. In this case, the potential is called the diffusion potential and it is synonymous with the junction potential discussed earlier. It can be described by the equation developed for the linear diffusion gradient, that is, by the Henderson equation (6.24). Because we are dealing with uni-univalent electrolytes, the multiplier cancels out and this diffusion potential can be written as... 

Due to the different mobilities, concentration gradients and thus potential gradients will be established. In actual measurements these potentials will be added to the electrode potentials. A calculation of liquid junction potential is possible with the -> Henderson equation. As liquid junction potential is an undesired addition in most cases, methods to suppress liquid junction potential like -> salt bridge are employed. (See also -> diffusion potentials, -> electrolyte junction, -> flowing junctions, and -> Maclnnes.)... 

The Integration of the Differential Equation for Diffusion Potentials The Planck-Henderson Equation... 

The diffusion potential can be estimated on the basis of the Henderson equation (9.7.26). Since only one ion is moving in the membrane, the sums in this equation have only one term and the diffusion potential is... 

The diffusion potential is approximately given by the Henderson equation... 

This is the usual Planck-Henderson expression for the diffusion potential. In the present situation it is containing the contribution of the protons and only. Eliminating AVfrom Eqs. (1) and (4) we have... 

One problem is the existence of the diffusion potential ). Depending on the chemical potential differences between the solution in the reference half-cell and the calibrating or sample solution, the unknown value of the diffusion potential contributes to A . It gives uncertainty in direct potentiometry. In order to decrease this, a high concentration of KCl is used as an electrolyte in the current bridges, if it is possible (see the Henderson form). 

However, in this case the diffusion potential on the aqueous solution side of the interface due to the difference in and D may become significant [32], When the aqueous phase contains a salt MX, where and X do not partition in the ILSB, the Henderson equation [33] for the diffusion potential takes the form... 

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

This is known as the Planck-Henderson equation for diffusion or liquid-junction potentials. 

In real cells, multiple transmembrane pumps and channels maintain and regulate the transmembrane potential. Furthermore, those processes are at best only in a quasi-steady state, not truly at equilibrium. Thus, electrophoresis of an ionic solute across a membrane may be a passive equilibrative diffusion process in itself, but is effectively an active and concentra-tive process when the cell is considered as a whole. Other factors that influence transport across membranes include pH gradients, differences in binding, and coupled reactions that convert the transported substrate into another chemical form. In each case, transport is governed by the concentration of free and permeable substrate available in each compartment. The effect of pH on transport will depend on whether the permeant species is the protonated form (e.g., acids) or the unprotonated form (e.g., bases), on the pfQ of the compound, and on the pH in each compartment. The effects can be predicted with reference to the Henderson-Hasselbach equation (Equation 14.2), which states that the ratio of acid and base forms changes by a factor of 10 for each unit change in either pH or pfCt ... 

The derivation presented here gives a greatly oversimplified picture of the actual situation in a membrane with fixed ion exchange sites. Use of the Henderson equation implies that the diffusion process in the membrane has reached a steady state with a linear concentration distribution. This is certainly not the case for solid membranes such as glass which are thick with respect to the diffusion length of the ion. Moreover, it is probably not valid to assume that the ionic mobility is independent of position in the membrane. More complex models for the membrane potential have been developed but they lead to essentially the same result. More details can be found in monographs devoted to this subject [9]. 

At a diaphragm two solutions are separated from each other. When these two solutions consist of different electrolytes or electrolytes in different concentrations, a diffusion of the constituent of the solutions occurs. This leads to a potential difference, which is called a liquid-junction potential, the magnitude of which depends strongly on the composition of the solutions. Ideally, this liquid-junction potential should be very small and constant to minimise errors. To estimate liquid-junction potentials, the Henderson equation is applicable [23] ... 

A liquid junction potential arises at the interface because in the case that r+ r, the different rates of diffusion of the two ions from high to low concentration generates a charge separation at the interface and hence a potential difference known as a liquid junction potential, ljp- According to the classical theory of liquid junction potentials [P. Henderson, Z. PhysiL Chem. 59 (1907) 118], for a salt and X , this has a limiting magnitude of... 

Both internal and external reference electrodes possess an interface between the internal solution and the external environment. This interfaee is eommonly established within a porous junction and is designed to permit electrolytic communication while preventing flow. In any event, the junction gives rise to the isothermal liquid junetion potential (ILJP), Ed(T2), which develops, because some ions diffuse faster than others, thereby generating an eleetrie field that opposes the proeess. Integration of the electric field across the junetion yields the isothermal liquid junction potential. Bard and Faulkner provide a detailed discussion of the thermodynamics of the isothermal liquid junction. For dilute solutions, the potential ean be ealeulated from Henderson s equation. In the ease of Thermoeell I, the isothermal liquid junetion potential is expressed by ... 

The ion transport number quantifies the amount of charge that is transported through the ion exchange membrane by that specific ion. Several methods are available to determine the ion transport number. A static method derived from Henderson s equation determines the transport numbers under diffusion [204]. In this method, a two-compartment cell is separated by a membrane and each of the compartments is filled with a KOH solution, both of different concentration. The resulting potential over the membrane is measured and can be used to calculate the transport number using Henderson s equation [204] ... 

In the derivation of the formula for calculating the liquid junction potential, the electric work done in separating the charges is set equal to the work of diffusion that is, the change in chemical potential arising from the diffusion of the ions. Only after making certain approximations can one arrive at the so-called Henderson solution [56] of the Nernst-Planck equation [57] ... 

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