Logarithmic distribution diagrams

Non-logarithmic distribution diagrams — due to their clear representation — are used for orientation search for significant and negligible species in the system in question (particularly the solution of the complex-forming equilibria). 

The purpose of construction of logarithmic distribution diagrams consists in the representation of linear dependences within the widest concentration range. Again considering the reaction MA M A A, the following relations can be derived for the molar concentrations of MA and M  

Taking the logarithms of these relations the following expressions are obtained  

These are linear dependences with the intersection point S. Concentrations of both MA and M components at the given concentration of the predominant variable A are found from the point of intersection of a normal line, erected for the considered A, with the lines MA or M (Fig. 3.16). 

Logarithmic distribution diagrams are constructed with an accuracy of 5%. Their advantage consists in their easy construction and in the possibility of a quite precise determination of low concentrations of individual components. After subtracting these values from the total concentration it is possible to calculate the higher concentrations of the remaining principal species. 

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The construction of non-logarithmic distribution diagrams is based on the equation of analytical concentrations and equations for the corresponding equilibrium constants. In the case of the reaction MA M + A with an equilibrium constant K it is necessary to take into consideration two components, i.e. MA and M, whose concentrations depend on the concentration of the predominant species A. When solving the two equations the following relations are obtained ... 

One of the most important aspects of hydrochemistry is the determination of the relative representation of the particular components of a protolytic equilibrium involving distribution coefficients 6. Graphical illustration of the relationship S = /(pH) can be expressed by either non-logarithmic or logarithmic distribution diagrams. Using these diagrams it is possible to read off the values of the concentrations of the particular species. 

Fig. 3.18. Non-logarithmic distribution diagram of a system of mercury chloro complexes Hg +, HgCl+, HgCl2, HgCIj, HgClJ-...

Figure 4.4. (Top) Logarithmic equilibrium diagram for seawater (10 C). Because seawater contains 4.1 x 10 M boric acid and borate [H3BO3 or B(OH>3 and B(0H)4 ], the distribution of these species is also given. (Bottom) Buffer intensity of seawater. Note that seawater has its minimum buffer intensity in the slightly alkaline pH range (end point y) approximately half a pH unit lower than fresh water. The H3B03 B(0H)4 couple does not contribute significantly to the total buffer intensity (jSg for the contribution of aqueous B to the buffer intensity).

The logarithmic concentration diagram applies only for a specific acid and for a particular initial concentration of acid. Such diagrams can be readily obtained from the distribution diagrams previously discussed. The details of... 

We can draw a logarithmic concentration diagram easily by noting the relationships just given. An easier method is to modify the distribution diagram so that it produces the logarithmic concentration diagram. This is the method illustrated in Applications of Microsoft Excel in Analytical Chemistry , Chapter 8. Note that the plot is specific for a total analytical concentration of 0.10 M and for maleic acid, since the acid dis.sociation constants are included. 

For the single, monoprotic acids and bases in the above example, the distribution and logarithmic concentration diagrams are rather simple, yet they clearly show the relative and absolute concentrations respectively of the various species present. Such diagrams become all the more useful when we consider more complicated systems, such as polyprotic acids and bases, where it otherwise becomes increasingly difficult to envision what happens as a function of pH. We will do so in exercises 4.5 and 4.6. 

In this chapter we have encountered the most important analytical aspects of acids and bases (a) their individual speciation, as described by the mass action law, and as reflected in the distribution and logarithmic concentration diagrams, (b) their buffer action, and (c) their neutralization, as exploited in acid-base titrations. 

A log concentration diagram is a log-log plot where the x axis is the same as for the a-distribution diagram, but the y axis is the logarithm of the concentration of the species of interest. Because the y axis indicates specific concentrations, we can indicate such species as H and OH , in addition to those of the various forms of the acid. 

The logarithmic division of the horizontal axis of a particle size distribution diagram offers the advantage that the finer sizes are characterized more prominently, which is appropriate because of their greater importance than the coarser ones in determining the overall surface area of the comminuted material concerned. 

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+]. 

The coefficients of the model are calculated by linear regression (the logarithm of the particle size was used here) and then plotted as a cumulative distribution of a normal plot (Fig. 2). The important coefficients are those that are strongly positive or negative, for example, the spray rate and the interaction between atomization pressure and inlet temperature 35. Others not identified on the diagram are not considered significant and could well be representative mainly of experimental error. The equation can thus be simplified to include only the important terms. However, if interactions are included, their main... 

Figure 4.1 illustrates the equilibrium distribution of the carbonate solutes as a function of pH (cf. Sections 3.6-3.9). The constmction of the double logarithmic diagram has been explained in connection with Figure 3.4. The equations 5, 6, 7, and 8 of Table 4.2 can be drawn graphically as linear asymptotes in different pH ranges. For example, for the equations (see 5 and 6 of Table 4.2) ...

Figure 5.8a gives the proportions of SO2 in the gas and aqueous phase as a function of pH. For pH < 5, sulfur dioxide occurs mainly in the gas phase for pH > 7, it occurs mainly in the solution phase. The fraction of SO2 in the aqueous phase is given in Figure 5.8b as a function of q (water content) for a few pH values. The double logarithmic graphic representation is particulariy convenient to plot the equilibrium distribution of the aqueous sf>ecies (Figure 5.8c). For a sketch of this diagram it is convenient to recall the following ...

In the presence of a solid phase the distribution logarithmic diagrams are called solubility log concentration diagrams. From these diagrams it is possible to find the liquid phase composition (distribution of complexes and free-ion form) of the areas with predominating existence of the particular forms, total and minimum solubility of the solid phase and pH, or precipitant concentration required for separation of the solid phase at a given pH... 

The first type of graphic representation is that of distribution and logarithmic diagrams, representing species fractions (in linear or logarithmic form) as a function of composition variables of the system [14,15]. The second class is that of the reaction prediction diagrams, and to... 

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