Master variable

Table 16-2 also provides a list of 15 equations that can be solved simultaneously to yield the equilibrium condition (Taylor et ah, 1983). Furthermore, if the concentration of each species is calculated as a function of pH (the so-called master-variable diagram or Sillen diagram, named after Sillen (1967) who popularized the method, it is possible to examine various sensitivities in the system, e.g., to the addition of more solute (see explanatory box on Sillen diagrams). 

Figure 16-1 is a master-variable diagram corresponding approximately to the previous clean marine case, illustrating that HC03 derived from CO2 is only important at pH > 7, and that at equilibrium H", NH4, and SO4- are the dominant species. Figure 16-2 extends this approach to the small population of droplets without any SO - in them that are nucleated on particles of seasalt that is present. In this case, pH = 6.7 and the dominant cation is seawater "alkalinity" or Ak (alkalinity in seasalt is the sum of cation concentration due to dissolved... 

Fig. 16-1 Master-variable diagram of clean marine cloud at a model altitude of 875 m. Equilibrium occurs where Z[-r] = Z[ - ], i.e. charge balance. Input conditions are 0.2 fig/rn of aerosol, roughly half...

Fig. 16-5 Sillen (master variable) diagram for aver- Fig. 16-6 Sillen (master variable) diagram for sea-age" river water, using data from Stumm and Morgan water. Replotted from Sillen s own graph (Sillen,... 

Inorganic ligands in aqueous solutions, and in particular in natural freshwaters, include, in addition to H2O and OH, the major ions carbonate and bicarbonate, chloride, sulfate and also phosphate [29], The distribution of metal ions between these ligands depends on pH and on the relative concentrations of the ligands. The pH is a master variable with regard to the occurrence of hydrolysed species and to the formation of carbonate and bicarbonate complexes. 

Fig. 3.4a gives plots of charge resulting from surface protonation vs pH for various oxides. Dots represent experimental data from different authors (Table 3.1a) from titration curves at ionic strength I = 0.1 M (hematite = 0.2 M). It is interesting to note that the data "of different oxides" can be "normalised" i.e., made congruent, if we chose the master variable... 

The relative abundance of these species is commonly presented as a pie chart or as a phase-style diagram. In the case of the latter, the concentrations of each species is plotted as a function of a master variable such as pH or salinity. Examples are shown in Figure 5.4. 

Two master variables, pH and pe, can be used to define the limits of stability of the UO2 spent fuel matrix under repository conditions. The Pourbaix diagram depicted in Fig. 2 clearly indicates the stability space of UO2 and consequently the desired chemical conditions that ensure the correct performance of the waste matrix. 

The infinite dilution activity scale is useful for ionic equilibria in fresh waters, but for equilibria in sea water one gains precision by applying an ionic medium activity scale. Measuring pH in sea water gives less information than total alkalinity and total carbonate. Calculations on redox equilibria are simplified by introducing the master variable pE -----log e. ... 

The logarithmic form lends itself to graphical presentation. For example, in a system containing a number of acid-base systems of known total concentrations, the concentration of each individual species is a unique function of the master variable log [H ], which may be represented in a logarithmic diagram. In Figure 1, for a system with total phosphate... 

For other types of equilibria other master variables can be used— e.g., log [Cl ] or log [Br] for halogeno complexes, pE (see below) for redox equilibria, and log p(02) for equilibria between oxides and gas phases (12). 

Because so much of the behavior of suspensions is determined or modified by charge associated with the solid phases, ZPC may be inferred from a wide variety of experiments involving pH as a master variable. For example, coagulation and sedimentation rates are maximum at the ZPC, and anion and cation exchange capacities (measured with nonspecific, symmetrical electrolytes) are equal and minimum at the ZPC. More direct and less ambiguous are electrophoresis and streaming potential, in any of their modifications. One can estimate the IEP(s) by measuring adsorption of H+ and OH" if one is certain that no specific adsorption of other species occurs. 

Since graphical displays of physicochemical relationships are convenient to use, much of the data in this paper are represented by e vs. pH diagrams. Redox potential and pH have been chosen as master variables only for convenience this does not mean that e and pH always can be regarded as independent of each other. 

Two graphical methods described here, a master variable (pC-pH) diagram and a distribution ratio diagram, are extremely useful aids for visualizing and solving acid-base problems. They help to determine the pH at which an extraction should be performed. Both involve the choice of a master variable, a variable important to the solution of the problem at hand. The obvious choice for a master variable in acid-base problems is [H+] [equations (2.9)—(2.12)], or pH when expressed as the negative logarithm of [H+]. 

To graph the curves representing [HA] and [A-], a mathematical expression of each as a function of [H+] (a function of the master variable) is needed. The appropriate equation for [HA] is derived by combining the equilibrium constant for dissociation of a weak acid [equation (2.10)] with the mass balance equation [equation (2.13)] to yield... 

Point-by-point plotting of equations (2.15) and (2.16) produces the curves for the nonionized, 2,4-DB[COOH], and ionized, 2,4-DB[COO ], species in Figure 2.8. This approach can be expanded to generate master variable diagrams of more complex polyprotic systems (Figure 2.9) such as phosphoric... 

A second graphical approach to understanding acid-base equilibria is preparation of a distribution ratio diagram. The fraction, a, of the total amount of a particular species is plotted on the v-axis versus the master variable, pH, on the x-axis, where... 

Equilibria involving reductive dissolution reactions add to the complexity of mineral solubility phenomena in just the way that pE-pH diagrams are more complicated than ordinary predominance diagrams, like that in Fig. 3.7. The electron activity or pE value becomes one of the master variables whose influence on dissolution reactions must be evaluated in tandem with other intensive master variables, like pH or p(H4Si04). Moreover, the status of microbial catalysis under the suboxic conditions that facilitate changes in the oxidation states of transition metals has to be considered in formulating a thermodynamic description of reductive dissolution. This consideration is connected closely to the existence of labile organic matter and, in some cases, to the availability of photons.26... 

Plant Roots The Hidden Half, Third Edition, Revised and Expanded, edited by Yoav Waisel, Amram Eshel, and Uzi Kafkafi Handbook of Plant Growth pH as the Master Variable, edited by Zdenko Rengel... 

In biologically productive lakes which also develop a thermocline in summer, the bottom water may become depleted in oxygen, leading to a change from oxidizing to reducing conditions. As a result, the potential exists for remobilization of nutrients and metals from bottom sediments to the water column, one more example of how alterations in master variables in the hydrological cycle, in this case the redox status, can affect the fate and influence of pollutants. 

This chapter has discussed the fundamental importance of the master variables ps and pH in determining species distribution under the conditions prevailing in natural systems (pe —10 to - - 17 and pH 4 to 10). The equilibria contained in Sections 3.2.2 3.2.4 also provide the basis for many of the currently available geochemical computer models (e.g. WATEQ4F, PHREEQC, and MINTEQA2 ), which are being... 

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