## Q Ca [Hm10

where m is the total amount of solid divided by the total volume of the suspension, and Ca is the concentration of the added acid.

By the same token, if the titration is done with a monobasic hydroxide (e.g., NaOH), at each point of the titration one has

where C& is the concentration of the added base.

From equations 1 and 2 it is clear that the total concentration of surface sites, {S}tot is the sum of the three types of possible sites,

{5}tot can be experimentally determined by measuring for example the amount of cations removed from solution by the target solid suspended in it (see Davranche, 2003).

In order to determine the values of the surface acidity constants (equations 5 and 6), one can discriminate between the following two cases:

(a) pH values below the pzc

Here the protonated sites predominate and {MO-} is negligible. Then,

Because {MOH} is neutral, the total surface charge is due to MOHj only:

Then, by substitution in equation 5 one can obtain Ka 1 in terms of measurable quantities:

= ({5}to. - {MOH+})[H+]/(2 = ({5}to, - <2)tH+]/e (15)

(b) pH values above the pzc

Here the unprotonated sites predominate and {MOHj } is negligible. Then,

Because {MOH} is neutral, the total surface charge is due to MO" only:

Then, by substitution in equation 6 one can obtain Ka2 in terms of measurable quantities:

(note that the calculation of Ka\ and of Ka2 requires the absolute value of Q).

The values of Ka\ and of Ka2 vary with pH in an approximate linear fashion, and the following relationship has been inferred for the corresponding pK values:

where p^int) represents the intrinsic value of this quantity (i.e., the pKa at the pzc) and (3 is the slope.

From the values of pKa the value of pzc can be calculated (see a related problem in the Postlaboratory questions and problems).

We present next a very simple experimental method to obtain the approximate pzc for several environmentally-relevant, simple metal oxides (Ti02, Si02, A1203, ZnO, and MgO). These experiments have the objective of facilitating the comprehension of the phenomena involved in many surface processes involving simple oxides.

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