Henderson boundary

When specialized for the Henderson boundary, Eq. (13-105) can easily be integrated if the ionic components are assumed to be ideal. This implies that the chemical potential of ion i is given by the expression... 

The path of integration of the line integral in Eq. (13-114) is specified by the assumption of a Henderson boundary. The Henderson boundary requires the concentration of component i in the boundary to be given as a function of x by the expression... 

Fig. 13-4. Concentration of component i versus position in the Henderson boundary.

One reason is that firms who want to employ the new technology cannot merely invest in basic research. Cockburn and Henderson (1998) emphasize that it is also important for the firm to be actively connected to the wider scientific community (p. 158). They developed the concept of connectedness, as measured by the extent of collaboration in writing scientific papers across institutional boundaries, and conclude that firms wishing to public sector research must do more than simply invest in in-house basic research they must also actively collaborate with their public sector colleagues. The extent of this collaboration... is positively related to private sector research productivity (p. 180). The process by which firms acquire this new technology is not simple or direct, nor obtained without cost. 

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 ... 

P. Henderson, An Equation for the Calculation of Potential Difference at any Liquid Junction Boundary, Z. Phys. Chetn. (Leipzig) 59 118 (1907). 

I. The Continuous Mixture Boundary.—This type of boundary, which is the one postulated by Henderson, consists of a continuous scries of mixtures of the two solutions, free from the effects of diffusion. If the two solutions are represented by the suffixes 1 and 2, and 1 — x is the frac-... 

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. 

An interesting equation, derived by Henderson J, relates g(z) to the solvent structure-originated part of the disjoining pressure /7soi(h) for the case of spherical molecules and hard wall boundaries ... 

McConnell J. C., Henderson G. S., Barrie L., Bottenheim J., Niki H., Langford C. H., and Templeton E. M. J. (1992) Photochemical bromine production impheated in Arctic boundary-layer ozone depletion. Nature 355, 150-152. 

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. 

This equation is due to Henderson.14 It is important to note that this derivation requires no particular spatial arrangement of the layers of solution in the boundary, the only requirement being that the solutions composing the boundary should be a series of mixtures of the end solutions I and II. 

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. 

Chuang, S. H. F., and Henderson, M. K. Using Subgraph Isomorphisms to Recognize and Decompose Boundary Representation Features, Journal of Mechanical Design, vol. 116 (1994) pp. 793-800. 

Second, several Cp exp T) dependences in Lyon et al. (1978), Charlu et al. (1970), Spedding and Henderson (1971), Spedding et al. (1974), and Lyapunov et al. (2000a,b) were decreasing in the temperature region adjacent to the lower boundary of measurements, which resulted in the appearance of a minimum in the corresponding curves. 

It should be stressed that this stationary situation is still a nonequilibrium one. The quantitative result depends on parameters mentioned above and on the geometry of the boundary region. The latter, in its turn, depends on the degree of spreading of the initial sharp boundary during the period until the steady state is established. Typically, the steady-state A jg values are calculated in the framework of the models of Planck [26, 27] and Henderson [26, 28, 29], as the most conventional. 

Henderson equation, which has gained wider acceptance, corresponds to the so-called free diffusion boundary, i.e., to a degraded diffusion layer with approximately linear concentration distribution along the normal to the boundary. The traditional form of Henderson equation (for concentrations c having the units of normality) is as follows ... 

Equation (3.9) is also known as the Lewis-Sargent formula [31]. This formula operates with equivalent conductivities of solutions A, not of the ions. The paper [31] published 1 year after Henderson s papers played a historical role. The cells of the same geometry as were used in [31] are still utilized for emf measurements. These cells with free diffusion boundary are sometimes called Lewis-Sargent cells. Another important note in ref. [31] concerns the search of conductivity values. 

To judge what boundary is sharp (Planck s model) or smoothed (Henderson s model), one should consider the thickness of the real diffusion layer under steady-state conditions. Typically it is of the order of pm, i.e., much larger than the ionic size, if the shape of the boundary is not regulated by some mechanical means. The specific size effect appears if the boundary is formed in microchannel configuration and when the diffusion length is comparable with the total distance between the channel walls. The problem is considered quantitatively in refs. [32, 33], in relation to devices for microanalysis. 

Henderson, D., D. Gillespie, T. Nagy, and D. Boda. 2005. Monte Carlo simulation of the electric double layer Dielectric boundaries and the effects of induced charge. Molecular Physics 103 (21-23 SPEC. ISS.) 2851-2861. 

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