Logarithmic diagrams

All the previous considerations can be shown in a logarithmic diagram. This is a plot of pH as a function of —log C (Fig. 5.1). The dotted curve recalls the activities problem when the concentration is higher than about 10 mol/L. Then, ah30+ differs markedly from H3O+. 

Let s consider the strong base B, which reacts quantitatively with water according to 

The general relation permitting the pH calculation is found by expressing the water ion product and the charge and mass balances of the solution. If C is the analytical concentration of the base, the following system of equations holds  

It becomes simplified when the concentration is higher than about 5 x 10 mol/L since H3O+ is then negligible. The pH value is given by the following reactions  

In very diluted solutions (C 10 mol/L), pH = 7. In the intermediate concentrations region, no simplification is possible. The general equation must be used. Hence, the pH calculation of a strong base s solution is quite analogous to that of the pH of a strong acid s solutions. In particular for concentrations higher than 10 mol/L, Eq. (5.11) is only an approximation because of the ionic strength. 

---

Vincente-Perez, S. Durand, J. S. Alvarez, M. D. 1992. Limitations of complexes logarithmic diagrams as a function of the ligand concentration-diagrams of conditioned variable. An. Quim. 88(7-8) 683-688. 

Figure 2. The rate constants k2 for some electrolytes following the parabolic rate law, vg = k2

A) Logarithmic diagram of Al(III) solubility as function of Al concentration and pH, derived from ther-modyn. equilibrium constants. (B) Extent of Al-hydrol-ysis as function of pH. (C) Variation of the coagulation rate, expressed as collision efficieny factor, with pH at constant Al dosage (a values determined from Equation 3)... 

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... 

FIGURE 3-1 Logarithmic diagram for 10 M acetic acid (pK, = 4.7). The system point is denoted by . 

Logarithmic diagrams (Section 3-1) are helpful devices for drawing conclusions about the predominant species in solution at various stages in titrations. Johansson described their application to complexation reactions. 

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) ...

This line in the double logarithmic diagram (equal abscissa and ordinate scales) shows for log [HCOf ] versus pH a slope of +1 and the line intercepts at pH = pAT) with [H2CO ]. 

Example 4.3. Seawater in Equilibrium with the Atmosphere (pco2 — 10 atm) Constmct a double logarithmic diagram—similar to that of Figure 4.5 for seawater (35%o salinity). At 20°C the following constant can be used ... 

Logarithmic diagrams for describing multiple equilibria species (the preparation of these using spreadsheets is now introduced)... 

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... 

Logarithmic diagrams of protolytic equilibria can be employed for the assessment of pH of acids, bases and their salts. In the proton balances only those components are considered whose concentrations are important in the context of the particular study. 

The measured individual times until the supercooled droplets freeze at a certain temperature are converted into a nucleation rate by plotting the cumulative freiction of unfrozen droplets against the product of droplet volume and time in a logarithmic diagram as shown in Fig. 6. The negative slope of the curve gives the nucleation rate J at that temperature. 

After Thiesen (1923), it seems favorable to plot the enthalpy of vaporization vs. the difference of the critical temperature and the temperature 7 in a double logarithmic diagram (see Fig. 2.1-4). A straight line results as long as the critical point is not very close. This diagram also shows that the enthalpy of phase transition changes at the triple point. At the triple point the enthalpy of vaporization and the enthalpy of melting add up to the enthalpy of sublimation ... 

It is useful to visualize the way in which the electrode potential of a redox couple varies with the concentrations of the reaction components. A logarithmic diagram analogous to the type used in acid-base systems can be constructed by using the Nemst equation written in the format of Equation 7-1. 

Figure 2 Logarithmic diagram for the formic acid-formate ion system fortotai concentration C=0.1 moii f...

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... 

R may not be constant with time (as in Figure 5.32 (B)) and it may be useful to make the plot in a double logarithmic diagram [392] to identify the nature of development. The typical situation results in nearly constant values of R after an initial period with faster deactivation. Typical values... 

Figure 3.11. Logarithmic diagram (In (p/n), n) of adsorption equilibria data of (COj, CH4, CO, N2) on AC NORIT R1 EXTRA at T = 298.15 K, [3.27]. Data have been correlated by an AI of generalized Langmuir type, eq. (3.38).

Sketch a Brouwer diagram (double logarithmic diagram of defect concentrations vs P02) for an oxide Mi+xO doped with a substitutional lower-valent dopant. 

When the R(t) t relationship is plotted in a double logarithmic diagram, it can be observed that during the stable evolution the R(t) t relationship can be fitted with a straight line. The index numbers of the fitting lines are 0.369 and 0.341, for the cases of C3 = 0.60 and C3 = 0.30, respectively. These index numbers are close to that in the well-known relationship of R(t) oc P reported elsewhere regarding numerical simulations and experiments [75,76]. The R(t) value of the condensed system is larger than that of the diluted system, and it is clear that with the... 

---