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The next part of the equation, [Ksp,Al(OH)3 {H+ }3]/TaK , should be easy to derive and will no longer be pursued. Note that Kw came from the ion product of water that reads:

This ion product is an equilbirium constant for the dissociation of water:

Equations (20) to (25) may now be substituted into Equation (19). This produces Equation (32), where, now, the complexes are eliminated and the equation only expressed in terms of the K's and the hydrogen ion concentration. This equation may then be differentiated and equated to zero to obtain the optimum pH. We will not, however, complete this differentiation and equate to zero in this chapter, but will do this in the unit processes part of this book.

^sp,ai(oh),7h[H ] kai(oh)CKSP,ai(oh),7h[H ] [ sp Al ] = -5- + -5-

7aiK w 7Al(OH) cKw

7Al7( OH)17cKw

13KAl13(OH)34cKsp,Al(OH)37H[H ] KAl(OH)4cKw

7Al2(OH)2cKw

Still another important application of the concept of K equilibrium constants is the coprecipitation of FePO4 and Fe(OH)3 in the removal of phosphorus from water. As in the case of coagulation using alum, it is desired to have a final equation that is expressed only in terms of the constants and the hydrogen ion. Once this is done, the equation can then also be manipulated to obtain an optimum pH for the removal of phosphorus.

In phosphorus removal, the phosphorus must be in the phosphate form and, because the reaction occurs in water, the phosphate ion originates a series of reactions with the hydrogen ion. The series is as follows:

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