Henderson approximation

In contrast to the Planck solution, the Henderson approximation enjoys considerable use [ 10,11 ]. Henderson s liquid-junction model is based on the assumption that the concentrations of the ions in the liquid junction change linearly withx between values corresponding to the edges of the liquid junction. This assumption is equivalent to the concept of a mixture of electrolytes changing unifonnly between the two edges of the liquid junction. Then... 

Henderson approximation [27], which assumes a linear concentration profile of the ions in the membrane (Equation 6) ... 

The product a0 oq has a maximum at pH = pKa, where a0 = ax = Vi, so that the product has the value A. For a single monoprotic buffer mixture (i.e., the mixture of a monoprotic acid and its conjugated base), the pH in the region of maximal buffer action, pH = pKa, can often be estimated from the Henderson approximation... 

It is difficult to give a reference for the HNC closure because so many different authors developed this approximation at about the same time. A full set of references can be found in the review of Barker and Henderson [4,5]. 

FIG. 1 Total local density p(z) for bulk density p = 0.821 and e /k T = 4.25. The solid line is for PYl theory, the dashed line is for HNCl approximation and the points denote the Monte Carlo simulation results. (Reprinted from S. Sokolowski, D. Henderson, A. Trokhymchuk, O. Pizio. Density profiles of associating fluid near a hard wall PY/EMSA and HNC/EMSA singlet theory, Physica A, 220, 22-32. (1995), with permission from Elsevier Science.)... 

FIG. 6 A comparison of the Monte Carlo (points), HHNCl, HNCl, and PYl density profiles. The results are for bulk density 0.403 (lower group of curves) and 0.741 (upper group of curves). The curves at z — 0, are, from the bottom, from HHNCl and HNCl approximations. (Reprinted from A. Trokhymchuk, D. Henderson, S. Sokolowski. Local density of overlapping spheres near a hard wall A density functional approach. Physics Letters A 209, 317-320. 1995, with permission from Elsevier Science.)... 

Carnie and Chan and Blum and Henderson have calculated the capacitance for an idealized model of an electrified interface using the mean spherical approximation (MSA). The interface is considered to consist of a solution of charged hard spheres in a solvent of hard spheres with embedded point dipoles, while the electrode is considered to be a uniformly charged hard wall whose dielectric constant is equal to that of the electrolyte (so that image forces need not be considered). 

The values of [HA] and [A ] in this expression are the equilibrium concentrations of acid and base in the solution, not the concentrations added initially. However, a weak acid HA typically loses only a tiny fraction of its protons, and so [HA] is negligibly different from the concentration of the acid used to prepare the buffer, [HA]initia. Likewise, only a tiny fraction of the weakly basic anions A- accept protons, and so [A-] is negligibly different from the initial concentration of the base used to prepare the buffer. With the approximations A ] [base]initia and [HA] [acid]initia, we obtain the Henderson-Hasselbalch equation ... 

A note on good practice Keep in mind the approximations required for the use of the Henderson-Hasselbalch equation (that the concentrations of both the weak acid and its conjugate base are much greater than the hydronium ion concentration). Because the equation uses molar concentration instead of activities, it also ignores the interactions between ions. 

Heisenberg uncertainty principle If the location of a particle is known to within an uncertainty Ax, then the linear momentum parallel to the x-axis can he known only to within an uncertainty Ap, where ApAx > till. Henderson-Hasselbalch equation An approximate equation for estimating the pH of a solution containing a conjugate acid and base. See also Section 11.2. Henry s constant The constant kH that appears in Henry s law. 

The buffer equation, which is often called the Henderson-Hasselbalch equation, is used to calculate the equilibrium pH of a buffer solution directly from initial concentrations. The approximation is valid as long as the difference between initial concentrations and equilibrium concentrations is negligibly small. As a rule of thumb, the buffer equation can be applied when initial concentrations of H j4 and A differ by less than a factor of 10. Example provides an illustration of the use of the buffer equation. 

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),... 

Fig. 3.2 Solubility profiles log S-pH. The dashed curves, representing uncharged precipitate in equilibrium with solution of the drugs, were calculated by Henderson-Hasselbalch equations. The dotted horizontal lines are estimates of the solubility of the charged form of the drugs, using either actual data (naproxen) or estimates based on the sdiff 3-4 approximation (atenolol and...

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

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 ... 

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... 

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... 

Electron microscopy was utilised by Henderson and Unwin 2301 to determine the arrangement of protein in the purple membrane. It was found to comprise seven, closely packed a-helical arrays roughly perpendicular to the plane of the membrane with bilayer regions occupying the remaining space. The seven helices were all 10-12 A apart and approximately 35-40 A long, the membrane itself being 45 A thick. These helices made up about 75 % of the polypeptide but the connectivity between them was not resolved. 

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

Similarly, rivers in Iceland (Vigier et al. 2002) showed a strong fractionation from the presumed-invariant isotopic composition of their solute sources (8 Li = +17.1 to +23.9, compared to approximately +4 for pristine basalts). Rivers draining terranes with the oldest exposed lavas yield low 5T i a positive correlation exists between dissolved Li concentration and 5T i in Icelandic rivers. These observations are consistent with the data of Pistiner and Henderson (2003) for naturally weathered, historically-erupted Icelandic basalt. The outermost 4 mm of the basalt erupted in 1783 was shown to have 5T i 2%o lighter than the interior of the sample, suggesting weathering over 200 yr, even in the arctic climate, lead to appreciable release of isotopically heavy Li to surface waters. 

The imidazole side-chain of histidine has a value of 6.0, making it a weaker base than the unsubstituted imidazole. This reflects the electron-withdrawing inductive effect of the amino group, or, more correctly the ammonium ion, since amino acids at pH values around neutrality exist as doubly charged zwitterionic forms (see Box 4.7). Using the Henderson-Hasselbalch equation, this translates to approximately 9% ionization of the heterocyclic side-chain of histidine at pH 7 (see Box 4.7). In proteins, plCa values for histidine side-chains are estimated to be in range 6-7, so that the level of ionization will, therefore, be somewhere between 9 and 50%, depending upon the protein. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

A full-scale Living Machine treats 32,000 gal of wastewater per day at a chocolate factory in Henderson, Nevada. The maintenance costs associated with the system are approximately 8000 per year for bacteria and 30 per day for power to run the pumps. In colder climates, electrical charges may increase during the winter months (D20916F D21941K). 

The very best buffers and those best able to withstand the addition of both acid and base are those for which [HA] and [A ] cire approximately equal. When this occurs, the logarithmic term in the Henderson-Hasselbalch equation disappecirs, and the equation becomes pH = pA. When creating a buffered solution, chemists therefore choose an acid that has a pK close to the desired pH. 

L. J. Henderson was a physician who wrote [H ] = /C0[acid]/[salt] in a physiology article in 1908, a year before the word buffer" and the concept of pH were invented by the biochemist S. R L Sorensen. Henderson s contribution was the approximation of setting [acid] equal to the concentration of HA placed in solution and [salt] equal to the concentration of A placed in solution. In 1916. K. A. Hasselbalch wrote what we call the Henderson-Hasselbalch equation in a biochemical journal.8... 

The Henderson-Hasselbalch equation (with activity coefficients) is always true, because it is just a rearrangement of the Ka equilibrium expression. Approximations that are not always true are the statements [HA] Fha and [A ] = FA... 

Figure 23-46 The photoreaction cycle of bacteriorhodopsin. After Bullough and Henderson.585 The subscript numbers indicate the wavelengths of maximum absorption of each intermediate and the approximate lifetimes are given by the arrows. Resting bacteriorhodopsin as well as intermediates J and O have all-frans retinal but K through N are thought to all be 13-cz s. A proton is transferred from L to aspartate 85 and then to the exterior surface of the membrane. A proton is taken up from the exterior surface via aspartate 96 to form N.

According to the Henderson-Hasselbach equation (3.6), an accurate approximation of the pK value is obtained from a semilogarithmic plot... 

Henderson-Hasselbalch equation An approximate equation for estimating the pH of a solution containing a conjugate acid and base. See also Section 11.3. 

Correct measured values for liquid junction potentials using the Henderson formalism and calculate ion activities according to the Debye-Huckel approximation. 

The conjugate acid of methylamine has a pXa of approximately 10.6. Based on the Henderson-Hasselbalch equation (Equation 9.1 in Chapter 9), calculate the percentage of methylamine that is not protonated (not in the form of its conjugate acid) at pH 7.4, the same as blood. 

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