Henderson equation

When a number of assumptions are made, an analytical expression, which is well known as the Henderson equation, can be obtained  

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

The assumptions in deriving Equation 5.5 are as follows (1) linear dependence of concentrations with distance between electrolytes (I) and (II), (2) nonideality is ignored in estimating the transport numbers, and (3) ionic mobility is independent of concentration. 

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The measurement of pK for bases as weak as thiazoles can be undertaken in two ways by potentiometric titration and by absorption spectrophotometry. In the cases of thiazoles, the second method has been used (140, 148-150). A certain number of anomalies in the results obtained by potentiometry in aqueous medium using Henderson s classical equation directly have led to the development of an indirect method of treatment of the experimental results, while keeping the Henderson equation (144). 

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),... 

For binary solutions of symmetric z z electrolytes having a common ion and the same concentration c a = Cma general Henderson equation changes to... 

In titrations we normally have to deal mainly with weak to fairly strong acids (or bases), so that for acids we can use the equation Ka = [H+ ] [A- ]/[HA] hence [H+] = KB [HA]/[A ]. When only a part X of the acids has been titrated, we find [H+ ] = Ka (1 - A)// this equation is approximately valid, because the salt formed is fully dissociated, whereas the dissociation of the remaining acid has been almost completely driven back. Hence for the pH curve we obtain the Henderson equation for acid titration ... 

On the basis of the Henderson equation for titration of acid or base one can prove mathematically that the half-neutralization point represents a true inflection point and that as the titration end-point dpH/dA is maximal or minimal, respectively (the latter is only strictly true for titration of a weak acid with a weak base and vice versa). 

Final remarks on end-point detection. In addition to our remarks above on the types of titration curves and the Henderson equation or more extended relationships, we can state that in Gran s method activity coefficients are taken into account however, these were assumed to be constant, which is incorrect, and therefore the addition of an ISA (ion strength adjuster) must be recommended (for errors of the Gran method see ref.66). 

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,... 

Another less precise but frequently used method employs a liquid bridge between the analysed solution and the reference electrode solution. This bridge is usually filled with a saturated or 3.5 m KCl solution. If the reference electrode is a saturated calomel electrode, no further liquid bridge is necessary. Use of this bridge is based on the fact that the mobilities of potassium and chloride ions are about the same so that, as follows from the Henderson equation, the liquid-junction potential with a dilute solution on the other side has a very low value. Only when the saturated KCl solution is in contact with a very concentrated electrolyte solution with very different cation and anion mobilities does the liquid junction potential attain larger values [2] for the liquid junction 3.5 M KCl II1 M NaOH, A0z, = 10.5 mV. 

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 rate equation to be used for kinetic analysis of enzyme depletion is that for simple noncompetitive inhibition. If the Henderson equation or similar types are not employed, keep in mind that the inhibitor concentration [I] is the free inhibitor concentration. Determination of Ki may not be feasible if the rate assay is insensitive and requires an enzyme concentration much greater than K[. Alternatively, Ki may be obtained by measuring the on-off rate constants of the E l complex, provided the rate constants for any conformation change steps involved are also known. 

Recently, Fuchs etal. [15], using the streaming mercury electrode and applying the Henderson equation, have determined the pzc value in the solutions of tetraethy-lammonium perchlorate in DMSO as —0.515 0.001 V (versus Ag/0.01 M Ag+ (DMSO) reference electrode). This value was corrected for the liquid junction potential and was independent of tetraethyl ammonium perchlorate (TEAR) concentration within the range 0.02 to 0.75 M. Using the same methodology, KiSova et al. 

In aqueous solutions, the method of measuring electrode potentials has been well established. The standard hydrogen electrode (SHE) is the primary reference electrode and its potential is defined as zero at all temperatures. Practical measurements employ reference electrodes that are easy to use, the most popular ones being a silver-silver chloride electrode and a saturated calomel electrode (Table 5.4). The magnitude of the liquid junction potential (LJP) between two aqueous electrolyte solutions can be estimated by the Henderson equation. However, it is usual to keep the LJP small either by adding the same indifferent electrolyte in the two solutions or by inserting an appropriate salt bridge between the two solutions. 

If two electrolyte solutions that are of different concentrations but in the same solvent contact each other at a junction, ion transfers occur across the junction (Fig. 6.3). If the rate of transfer of the cation differs from that of the anion, a charge separation occurs at the junction and a potential difference is generated. The potential difference tends to retard the ion of higher rate and accelerate the ion of lower rate. Eventually, the rates of both ions are balanced and the potential difference reaches a constant value. This potential difference is called the liquid junction potential (LJP) [10]. As for the LJP between aqueous solutions, the LJP between non-aqueous solutions can be estimated using the Henderson equation. Generally the LJP, Lj-, at the junction Ci MX(s) c2 NY(s) can be expressed by Eq. (6.1) ... 

Tab. 6.5 Ionic molar conductivities (A00) in some organic solvents and LJPs between solutions in the same solvent (Ej) calculated by the Henderson equation...

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... 

To maintain the condition Eref+Ed = const, numerical corrections are possible, e.g., using the Henderson equation for liquid-junction potential. 

Here it is not very correct to assume that the concentration gradients vary linearly through the junction, especially because the concentration profiles depend on the technique of junction formation. Assuming that activities are equal to concentrations and that there is, in fact, a linear transition, we obtain the Henderson equation... 

There are interferences to this simple functioning of the ISE according to (13.9). These are due to the fact that membranes are not perfectly selective and respond to some extent to species other than the desired ion. If we consider a linear concentration gradient within the membrane then the Henderson equation (equation (2.60)) can be applied, writing it in the form... 

The alkaline error , often found in pH electrodes, arises because in very alkaline solution [Na+] or [K+] is normally very high, making a significant extra contribution to the potential as expressed through the Henderson equation (13.10). Minimization of this error is done by using glass of special composition and with very low selectivity coefficients for Na+ and K+. 

Electrolyte junction — A liquid junction is the region of contact of two different -> electrolyte solutions kept apart by a porous -> diaphragm, such as sintered glass or ceramic. At the contact a -> Galvani potential difference appears, which is called -> liquid junction potential (Ej). In the case of two solutions of the same electrolyte, but with different concentrations (c(a) and c(/S)), the potential Ej is defined by the equation Ej = (t+-t-) ln ry, where t+ and t are - transport numbers of the cation and anion, respectively. If the concentration of one of the ions is the same in both solutions, but the other ion differs (e.g., NaCl and KC1), the potential Ej is given by the Henderson equation, which is reduced to the Lewis-Sargent relation for a 1 1 electrolyte Ej = ln, where A (/3) and A (a) are molar conductivities of the electrolytes in the com-... 

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 Henderson equation which has gained wider acceptance, can be written as follows for concentrations c having the units of normality ... 

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. 

Two special cases of the Henderson equation are of interest. If the two solutions contain the same uni-univalent electrolyte at different concentrations, then... 

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