Master variable diagrams

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,... 

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... 

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 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... 

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+]. 

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... 

Similar considerations apply to the plotting of equations SI and 52. Tlie sections having slopes of —2 or +2 are usually unimportant because they occur only at extremely small concentrations. Diagrams of the types given in Figures 3.3 and 3.4 are not only useful in evaluating specific positions of equilibrium, but permit us to survey the entire spectrum of equilibrium conditions as a function of pH as a master variable. 

The same information can be gained from an activity ratio diagram. The construction is very simple and is illustrated in Figure 7.14b. We again choose pH as a master variable and make our calculation for a given C7. In this figure we plot the ratios between the activities of the various soluble and solid species... 

A logarithmic concentration diagram is a plot of log concentration versus a master variable such as pH. Such diagrams are useful because they express the concentrations of all species in a polyprotic acid solution as a function of pH. This allows us to observe at a glance the species that are important at a particular pH. The logarithmic scale is used because the concentrations can vary over many orders of magnitude. 

Note that these expressions are analogous to the a expressions we wrote for polyfunctional acids and bases except that the equations here are written in terms of formation equilibria while those for acids or bases are written in terms of dissociation equilibria. Also, the master variable is the ligand concentration [L] instead of the hydronium ion concentration. The denominators are the same for each a value. Plots of the a values versus p[L] are known as distribution diagrams. 

Figure 18.9 shows Eh contours calculated from (18.60) drawn on a log fo —pH diagram. The fact that we can do this suggests that the true master variable is redox or oxidation state and that /o,/h2> Eh, pe and all other related variables are simply different ways of quantifying the same thing. 

Master variables other than pH and pOH can be used in log concentration diagrams. Diagrams using the negative logarithm of concentrations of other ions are often useful in the solution of solubility problems. As an example consider the carbonate salts of and... 

A block diagram of the spectrometer is shown in Figure 16. The master oscillator (MO) is frequency-stabilized (FS) to two lower frequency standards. A one-GHz signal (1 GHz) is produced by multiplying a 20 MHz oven-controlled crystal oscillator which has a frequency stability of 1 x 10 parts per day. A variable frequency oscillator (VFO) is also used for continuous frequency adjustment... 

Figure 12.7 Schematic diagram of DFLC-based variable optical attenuator (1 = polarization beam displacer, 2 = half-wave plate, 3 = master LC cell, and 4 = compensation ceU).

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