Al(OH)3(s)(fresh precipitate) ^ Al3+ + 3OH- Ksp AKOHh = 10-33 (12.8) Al(OH)3(S) + OH- ^ Al( OH)- Kakoh)4c = 10+° (12.9)

2Al3+ + 2H2O ^ Al2 (OH )4+ + 2H+ Kai2(oh)2c = 10-" (12.10)

The equilibrium constants apply at 25°C:

Note: From the number of aluminum atoms they contain Al7(OH)4+, Al13(OH)^+, and Al2(OH)4+ are polynuclear complexes.

Also, the H+ and the OH- are participants in these reactions. This means that the concentrations of each of these complex ions are determined by the pH of the solution.

In the application of the previous equations in coagulation treatment of water, conditions must be adjusted to allow maximum precipitation of the solid represented by Al(OH)3(s). To allow for this maximum precipitation, the concentrations of the complex ions must be held to a minimum.

12.5.1 Determination of the Optimum pH

For effective removal of the colloids, as much of alum should be converted to the solid Al(OH)3(s). Also, as much of the concentrations of the complex ions should neutralize the primary charges of the colloids to effect their destabilization. Overall, this means that once the solids have been formed and the complex ions have neutralized the colloid charges, the concentrations of the complex ions standing in solution should be at the minimum. The pH corresponding to this condition is called the optimum pH.

Let sp Al represent all the species that contain the aluminum atom standing in solution. Thus, the concentration of all the species containing the aluminum atom Al(III), is

[ sp ai ] = [ Al3+] + [ Al( OH) 2+] + 7 [ Al7( OH )£] + 13[ AlB (OH )£]

All the concentrations in the right-hand side of the previous equation will now be expressed in terms of the hydrogen ion concentration. This will result in the expressing of [spAl] in terms of the hydrogen ion. Differentiating the resulting equation of [spAl] with respect to [H+] and equating the result to zero will produce the minimum concentration of spAl and, thus, the optimum pH determined. Using the equilibrium reactions, Eqs. (12.5) through (12.10), along with the ion product of water, we now proceed as follows:

[Al3+] {A^} ^sp^lCOH), K*sp,Al(OH)3{H } ^sp,Al(OH)37H[H ]

7ai YaI(OH-}3 YaK YaK

raunm 2+1 { Al (OH)2+ } K m( oh) c { Al3+} K ai(oh) cKsp,Ai( oh)3yh[ H+f

[ Al7(0H)171 Kai7(0H)17c{ Al +} KAl7(0H)17cYA7l[ Al 1

KAl7(QH)17cKsp,Al(QH)3YH[H I (1214)

YAl7(QH)17cKw rA1 ^m5+n { Ali3( OH ) 3+ } KAli3(QHV { Al3+}13 KAli3(0HVYA3l[ A^+f

= Ka113(0h)34Ap1a110h)3Yh[H 1 (12 15) YAl13(QH)34cKw

4, {Al2(0H)2+} KAl2(QH)2c{Al3+}2 KAl2(QH)2cY^l[Al3+]2


YA1, Yh, Yai(oh)c, Yai7(oh)17c, yai13(oh)34c, Yai(oh)4c, Yai2(oh)2c are, respectively, the activity coefficients of the aluminum ion and the hydrogen ion and the complexes A1(0H)2+, A1.(0H)4+, A1B(0H)34+, Al(OH)-, and Al2(0H)4+. K^q^ is the solubility product constant of the solid Al(0H)3(s) and Kw is the ion product of water. K Al(0H)c, K Al7(0H)17c, K A113(0H) c K ai(oh)4c , and ^ai2(oh)2c are, respectively, tlie equilibrium constants of the complexes Al(OH) , A17(0H)17, Al13 (0H)534+, Al(OH)-, and A12(0H)2+.

Remember that the equilibrium constants are a function of temperature. To obtain the corresponding values at other temperatures, the Van't Hoff equation should be used. The use of this equation, however, requires the value of the standard enthalpy AH°98. At present, none are available for the aluminum complexes. Research is therefore needed to find these values.

Equations (12.12) through (12.17) may now be substituted into Equation (12.11) to produce

Ksp,Al(OH)37h[H ] KAl(OH)cKsp,Al(OH)37H[H ] [ sp Al ] = -3- +


Ykl1(OH)„cKw 7Al13(OH)34cKw

KAl(OH).cKw 2KAL(OH),cKsp,Al(OH),YH[H ] „„,„,


To obtain the optimum pH, differentiate [spAl] of Equation (12.18) with respect to [H+] and equate the result to zero. Doing the differentiation, rearranging the resulting equation, and calling the resulting solution for [H+ ] as [H+pt ], obtain the following equation:

f2KAl(OH)cKsp,Al(OH)37H 1 + 3 f 3Ksp,Al(OH)3?H 1 + 4 i-p-H Ropt J + i--5— H Hopt J

f 28 KAl1(OH)11cKip,Al( OH)37H 8KAl2(OH)2cKsp,Al( OH)37H 1 + 5

L 7Al1(OH)11cKw YAl2(OH)2cKw J

f65KAl13(OH)34cKSp,Al(OH)37H 1 + 6 KAl(OH)4cKw ... ...

L YAl13(OH)34cKw J 7Al(OH)4c7H

By trial and error, Equation (15.19) may now be solved for [H+pt ]. Thus, the optimum pH is pH = -log {H+pt} = log (YH[H+pt] ) (12.20)

Example 12.1 A raw water containing 140 mg/L of dissolved solids is subjected to coagulation treatment using alum. Calculate the optimum pH that the operation should be conducted. Assume the temperature of operation is 25°C.


f2KAl(OH)cKsp,Al(OH)3 YH 1 + 3 f 3Ksp,Al(OH)37H 1 + 4 i -- f[HoptJ +i --3- HHoptJ

f 28 K Al1(OH)11cKsp,Al( OH)3 YH 8K Al2(OH)2cKsp,Al( OH)3YH 1 + 5

L YAl1(OH)11cKw YAl2(OH)2cKw J

+ i ^-^Al^OH^c-^sp^l-OH);, YH h+ j6 = KAl(OH)4cKw YAl,3(OH)34cK W9 J 00 Ya1(oh)4CYh

K ai(oh) c — 10_5 KSp,Al(OH)3 — 10_33 V — 2.5( 10_5) TDS 7 — 10 1+U4(-/v)

V — 2.5( 10-5)(140) — 3.5( 10-3) 7h — 10 — 0.94


2K ai(oh) cKsp ,ai(oh)3 yH — 2 (10-5)(10-33)(0.94 )2 — 2 30 (104)

7aiK w


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