Planck-Henderson equation

Equation (3) suggests that the membrane potential in the presence of sufficient electrolytes in Wl, W2, and LM is primarily determined by the potential differences at two interfaces which depend on charge transfer reactions at the interfaces, though the potential differences at interfaces are not apparently taken into account in theoretical equations such as Nernst-Planck, Henderson, and Goldman-Hodgkin-Katz equations which have often been adopted in the discussion of the membrane potential. 

Finding an equation for jjp is surprisingly difficult, but it is finally shown that [cf. the deduction of the Planck-Henderson equation, Eq. (4.291)]... 

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

Fig. 4.86. In the derivation of the Planck-Henderson equation, a linear variation of concentration is assumed in the interphase region, which commences atx=0 and ends at x=L...

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

Einstein-Schmolukowski, 378, 405 Gibbs-Duhem, 262 LaPlace, 392 Leonard-Jones, 45 Nernst-Einstein, 456 Nernst Planck, 476 Onsager, 494 Planck-Henderson, 500 Poisson, 235, 344 Poisson-Boltzmann, 239 Sackur-Tetrode equation, 128 Setchenow s, 172 Tafel, 2... 

From the century s beginning, through its midpoint, the electrochemistry of electrodes was based upon the treatment given by Nemst (Section 7.2.36). This had been derived first, for an interface between a metal and its ions in solution, but the treatment had spread (Planck and Henderson, 1890-1907) to the potential difference between two liquids containing different concentrations of electrolytes. The first of these two treatments yields an equation (Nemst equation) identical in form to the... 

The solution is given for the case of a smeared-out boundary and linear spatial distributions of concentrations. Generally, Henderson and Planck equations yield similar results however, for junctions with a pronounced difference in ion mobilities (like HCl-LiCl), the deviation can reach about 10 mV. A specific feature of the Planck equation is the existence of two solutions, the firstbeing close to that of Henderson, and the second one being independent of the solution concentration and of no physical meaning [iv]. Two particular types of liquid junctions are (a) two solutions of the same electrolyte at different concentrations and (b) two solutions at the same concentration with different electrolytes having an ion in common. For type (b) junctions, the simplification of the Henderson model results in the Lewis-Sargent equation ... 

III the two special cases considered above, first, two solutions of the same electrolyte at different concentrations, and second, two electrolytes with a common ion at the same concentration, the Planck equation reduces to the same form as does the Henderson equation, viz., equations (43) and (44), respectively. It appears, therefore, that in these particular instances the value of the liquid junction potential does not depend on the type of boundary connecting the two solutions. 

When different electrolytes are present on either side of the boundary, the electrolyte distribution is time dependent. This means that an exact thermodynamic solution to the problem is not possible. The solution to the problem given here is a steady-state solution, that is, the solution appropriate to a system in which mass transfer is occurring but under conditions that the liquid junction potential is independent of time. The porous diaphragms described earlier are examples of junctions which meet this condition. There are two well-known solutions to equation (9.7.17), one by Planck [6] and the other by P. Henderson [7]. The latter solution is more often used in practice and therefore is presented here. 

The Integration of the Differential Equation for Liquid Junction Potential. To account for the results of such measurements as have been described in the previous section there have been a number of integrations of the fundamental differential equation (3). Of these the following will deal only with the integrations by Henderson and by Planck. In addition a graphical integration method devised by Maclnnes and Longsworth will be discussed. 

This equation contains only measurable quantities. However in order to integrate it, information concerning the point to point variation of the concentrations in the boundary is necessary, since the values of the transference numbers and the activity coefficients depend both upon the total concentrations of the solutions I and II and upon the proportions in which these solutions are mixed. The distribution of electrolytes in the boundary assumed by Planck and by Henderson have already been discussed. These were chosen, it is well to repeat, not because of their inherent probability, but because with them analytical integrations could be carried out. 

Equation or Method 26b Author Henderson o Planck Cell Potential millivolts E Liquid Junction Potential, millivolts Eh 26.85... 

Concerns with liquid junctions—that is, electrolytes with different ionic concentrations or different ionic species meeting at a junction, such as a membrane or simply a small hole in a Luggin capillary, go back at least to the works of Nernst [4, 5], Planck [6] in the 1880s and 1890, and that of Henderson [7] in 1907. It is Henderson who is credited with the derivation of the equation named after him, for the potential difference across such a junction, see below, although we find essentially the same equation in the 1890 work of Planck [6]. These works were concerned with steady state solutions. Helfferich (in 1958) [8] and Cohen and Cooley [9] computed, by finite differences, time-dependent behaviour at liquid junctions. Many subsequent works were of course published since then, including the recent work of Strutwolf et al. [10, 11], Dickinson et al. [12] and Britz and Strutwolf [13],... 

If one considers a sample solution and bridge electrolyte of any desired composition, one must make assumptions about the concentration gradient within the liquid junction in order to arrive at simplified expressions. While it is possible to numerically simulate these profiles on the basis of the Nemst-Planck equation,for practical purposes it is perfectly acceptable to use the Henderson equation as approximation. It assumes linear concentration gradients across the Jimction and is written as ... 

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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