Ratio Diagrams

Continuing in this way, we can construct an approximate a diagram suitable for many purposes. 

Another useful approximation in this construction is that the maxima of a2 and a, occur over the crossings of the adjacent species. [This can be proved by differentiation of the a expressions, set (5-4).] If the maximum of X2 occurs over the crossing of and aj, we can set equal the expressions for R3.2 and Rj,2 [set (5-5)] to find simply that H = (KiK2y. Next we substitute this H in the a2 expression in set (5-4), omitting the final term, which is the small contribution from the Uq. This gives 

Putting in the K values for citric acid, we find 2max = 0.72. Similarly, ax max occurs approximately at pH 5.05 at 

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The first step in data analysis is the selection of the best filling probability function, often beginning with a graphical analysis of the frequency histogram. Moment ratios and moment-ratio diagrams (with p as abscissa and as ordinate) are useful since probability functions of known distributions have characteristic values of p, and p. ... 

Figure 9.9 Bivariate lead isotope ratio diagram for copper ores from some Aegean and Anatolian deposits, as defined by the Oxford group (Gale and Stos-Gale, 1992 Figure 13). (Reproduced with permission from Proceedings of the British Academy, vol. 77, New Developments in Archaeological Science. The British Academy 1992.)...

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... 

The construction of an activity-ratio diagram can be summarized in the following four steps ... 

As an example of these ideas, an activity-ratio diagram for control of Al(III) solubility by secondary minerals in an acidic soil solution will be constructed. The Jackson-Sherman weathering scenario14 indicates that when soil profiles are leached free of silica with fresh water, 2 1 layer-type clay minerals are replaced by 1 1 layer-type clay minerals, and ultimately these are replaced by metal oxyhydroxides. This sequence of clay mineral transformations can be represented by the successive dissolution reactions of smectite, kaolinite, and gibbsite ... 

Equation 3.18 can be used to construct an activity-ratio diagram in respect to A1 solubility as influenced by the leaching of silicic acid (H4SiOj). The equations for log[(solid)/(Al3 )] are as follows ... 

Fit . 3.5. Activity-ratio diagram for smectite (montmorillonite), kaolinite, and gibbsite based on Eq. 3.22. Solubility windows for kaolinite and gibbsite are shown bounded from below by dashed lines. The range of silicic acid activity expected in soils is indicated by the two short vertical lines labeled Si(),(am)" and "quart/."... 

The activity-ratio diagram resulting from Eq. 3.20 is shown in Fig. 3.5. 

The application of the GLO Step Rule and, for that matter, the interpretation of activity-ratio diagrams in general are influenced by the existence of varying degrees of crystallinity or of particle size in soil minerals, with a corresponding variation in their solubility.15 For example, in the case of Fig. 3.5, very poorly crystallized forms of gibbsite and kaolinite, alluded to above, would require... 

Kl< . 3.6. Activity-ratio diagrams at pi I 5 and 6 for the same set of solid phases as described in Fig. 3.5. Open circles represent initial stales, whereas filled circles represent the subsequent stales after did days of equilibration (dotted "reaction paths" link these two kinds of data point). (Data from May et al.1 )... 

The effect of Al-goethite on A1 solubility can be illustrated through a reconsideration of the activity-ratio diagram in Fig. 3.5, but with the system simplified to comprise only kaolinite and gibbsite in addition to ideal Al-goethite. For (H20) = 1.0, Eqs. 3.21b and 3.21c yield the activity ratios for kaolinite and gibbsite ... 

Prepare activity-ratio and predominance diagrams for the Al(III) minerals whose dissolution reactions are described in Eqs. 3.18-3.20. Set pH = 6 for the activity-ratio diagram, but otherwise use fixed activity data as given in connection with Figs. 3.5 and 3.7. Repeat your calculations for (H20) = 0.5 instead of unit water activity. What is the effect on mineral stability (Answer. 

Solubility, Predominance, and Activity Ratio Diagrams. From thermodynamic information, diagrams can be constructed that circumscribe the stability boundaries of the solid phases. Depending on the variables used, different kinds of predominance diagrams can be constructed. 

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... 

At any pH the ordinate values on the activity ratio diagram give the activities for the various species on a relative scale. Thus, in Figure 7.14a at pH = 12, Fe(OH)2(s) has the highest relative activity. This solid phase will precipitate at this pH as a pure solid its activity will be unity, and FeC03(s) must have an activity of much less than unity and cannot exist as a pure solid phase. Figure 7.14b shows that, for Cy = 10 M, FeC03(s) is stable below approximately pH = 10. 

Establish stability domains as a function of the variables Cy, pH, and Pco2-In order to obtain a survey of the stability relationships, we construct an activity ratio diagram using Mg as a reference state. 

These equations are plotted in Figure 7.15a for an assumed value of log Ct = -2.5. It is convenient to start to plot these equations at high pH values where log 2 = 0. In the pH region pATi < pH < pAT2, d log a2/d pH has a slope of +1. Our activity ratio diagram postulates that, for log = —2.5, magnesite is stable below pH = 10.7 nesquehonite is less stable than magnesite brucite becomes stable above pH = 10.7. 

In order to gain insight into the predominant solid phases and soluble species, it is expedient first to construct an activity ratio diagram. Taking Cu as a... 

Figure 7.16. Solubility of Cu(II), (a) Activity ratio diagram, (b) Solubility diagram. The solid line surrounding the shaded area gives the total solubility of Cu(II), which up to a pH value of 6.96 is governed by the solubility of malachite [Cu2(0H)2C03(s)]. In the low pH region, azurite [Cu3(OH)2(C03)2(s)] is metastable but may become stable at higher C-j- Above pH 7, the solubility is controlled by the solubility of CuO (tenorite). The predominant species with increasing pH are CuC03(aq), Cu(C03)2 , and...

HOCl is a more stable oxidant than Cl2(aq). (If any doubt should arise about which species predominates thermodynamically, an activity ratio diagram either at a given pe or at a given pH can immediately clarify the stability relations.) Equations vi and vii have slopes of —0.5 and — 1, respectively, in the graphical representation the lines intersect at pH = pK of the hypochlorous acid. 

Appendix 8.1 Activity Ratio Diagrams for Redox Systems 513... 

Activity ratio diagrams provide a simple way to gain a first-hand impression on the meaning of equilibrium data with respect to the relative stability of phases. We illustrate here the approach with examples of stability relations of iron and manganese compounds. 

Example A8.1. Stability Relations of Iron and Manganese Compounds at pH = 7 Discuss the stability relations of Fe and Mn compounds with the help of activity ratio diagrams for the conditions pH = 7, HCO3" = 10 M, and S04 = 10" M. The equilibrium constants at 25°C can be obtained from the data on the free energy of formation in Appendix C at the end of the book. 

For construction of the activity ratio diagram we take Fe as a reference. The following equations can be derived ... 

Such activity ratio diagrams illustrate that Hg°(aq) becomes the major dissolved inorganic Hg species below the pe range 4-6 for both seawater and fresh water above this pe range, Hg(OH)2(aq) and HgCl4 are the preponderant species in fresh water and seawater, respectively. 

Fig. S-20. (a) Frequency distribution of the sum of methane homologs in 3,500 samples from different types of reservoirs (from Nikonov, 1971). Gas, oil and condensate surveys (b) location frequency distributions of hydrocarbons in soil gas over different basins, (c) methanerethane ratio, (d) propane methane ratio, (e) methane content, (0 Pixler ratio diagram (Pixler, 1969), (g) soil-gas data plotted on Pixler diagram, (h) Reservoir gas analyses of Verbanac and Ounia (1982) plotted on Pixler diagram.

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