Distribution, generally ratio diagram

Figure 6.4. Hydrolysis of metal ions, (a) Predominant pH range for the occurrence of aquo, hydroxo, hydroxo-oxo, and 0x0 complexes for various oxidation states. The scheme attempts to show a useful generalization, but many elements cannot be properly placed in this simplified diagram because other factors, such as radius and those related to electron distribution, have to be considered in inteipreting the acidity of metal ions, (b) The linear dependence of the log,o of the first hydrolysis constant Ki = MOH H / M on the ratio of the charge to the M-O distance (z/d) for four groups of cations (25°C). (Note change of abcissa zero for different groups.) (From Baes and Mesmer, 1976.) (c) Hydrolysis constants of some important metal ions.

A last important but mathematical interpretation of probability is that it is a measure of a set. This can be understood by reference to the Verm Diagram in Fig. 5.1. If one imagines points x randomly but uniformly distributed over the sample space S and a summation of the indicator function values is made (1 if in the set, 0 if not) then the total will be the ratio of the area of the set A to the sample space. In general if the point is not uniformly distributed but each point X has a probability of occurring p(x) then... 

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