pH-dependent dissolution of magnesium hydroxide can be formulated in terms of the degree of surface protonation, or extent of proton
adsorption at surface >Mg-OH sites. If adsorption of a single proton is sufficient to bring about release of one magnesium ion into solution, the concentration of activated complex should be proportional to
{>Mg-OH 2 }, and (2.21) would be:
+1
r = k1 {>Mg-OH 2 }
+(2.22)
where k1
= - y+is the activity coefficient for the surface species >Mg-OH 2
+• If more than one adsorbed proton is involved in forming the activated complex, then the probabil ity that adsorbed protons will occupy adjacent sites must be considered. The formul at ion is analogous to that for reactions involving surface catalysis (Boudart, 1975; Hill, 1977). Figures 2.6 and 2.-7 illustrate possible physical pictures for adsorption of two protons at adjacent sites on a hydroxide surface. As noted above, each surface hydrox ide is a potential proton
adsorption site, each underlying magnesium is coordinated to three
surface hydrox ides and each surface hydrox ide is coord inated to three
underlying magnesiums. figure 2.6 is a plan view, showing an edge, or
"-"D--
I ,
I ,
(J-
I, ,
\ \ IFig. 2.6. Plan view of magnesium hydroxide surface illustrating release of magnesium resulting from adsorption of protons at adjacent sites; magnesium. , hydroxide (), adsorbed proton
+ ,() water molecules.
' .... _'"'
'-./
a)
b)
c)
" /
HO-Mg-OH )< >(1 HO-Mg-OH
' / ...( I HO~Mg~OH
/ "I
HO OH
/
" / I
HO-Mg- OH
X X.
HO~M ~OH+
' / 9 / I 2
HO~'M4' OH
+HO/ "6H 2
/
Complexation
, / I
HO-Mg-OH
X XI
+HO-Mg-OH2
X Xl
HO ~Mcf - QH 2
+/ " I
HO OH
/
" / I
HQ- Mg-OH HOXM ><6H
+" 9 Ii 2
HO< ~/OH
+~ / \ 2 /Mg,\
HO 'OH
Activated Complex Formation
~Detachment
Fig. 2.7. Schematic of chemical steps for release of one magnesium from chrysotile surface; a) complexation of protons at adjacent surface sites, b) formation of activated complex involving lengthening of bonds between magnesium ion and
structural oxides, and c) detachment of magnesium from surface
and coordination with water molecules.
a change in the position or bonding of that cation in the solid lattice and eventually in release of a Mg2+ ion into solution. This is depicted in the schematic of Figure 2.7. An analogous picture could be developed for adsorpti on of three protons on adjacent sites.
Following Boudart (1975) and Hill (1977), the probability of two adjacent sites being occupied is proportional to the square of the probability that one site is occupied, where the probability of
occupyi ng anyone site is independent of the probabil ity of occupyi ng any other site. In the present case, the probability that anyone surface >Mg-OH site is ocupied by a proton is {>M9-OH
2+} Sm- 1 , the fraction of surface sites that are protonated. The total concentration (mol/cm2
) of pairs of sites on the surface is 1/2(Z1Sm)' where z1 is the number of nearest neighbors to any given site, a constant. The factor 1/2 accounts for each pair being counted twice. For the concentration of pairs of protonated sites being C
2, the fraction of pairs of sites that are protonated is C2(1!2(z1Sm))-1. This is equal to the square of
+
2 -2
the fraction of sites that are protonated, {>~g-OH2 } . Sm • This gives
(2.23)
An analogous argument for three adjacent sites would give:
(2.24)
where z2 is the number of nearest neighbors to each pair of sites.
Substituting
(2.23)into
(2.21)for the case of the activated complex involving two adjacent protonated sites gives:
where
1 +
2
r = k2 Sm {>Mg-OH2 }
y+ • 2
(2.25)
Analogous expressions for the influence of adsorbed protons and anions in the dissolution of aluminum oxide have been presented (Stumm et al.,
1983).Substitution of
(2.1)and
(2.2)into
(2.22)gives an expression in tenns of hydrogen ion concentration:
r
=
k S 1m (2.26)
Simil a r expressi ons woul d resul t for the cases of two (2.25) or th ree proton adsorption. Over a 1 imited pH range an approxlmation of the fall owi ng fonn can be used (Stumm et al.,
1983):(2.27)
n is an integer, likely one, two or three and kl incorporates k
1, KI, Sm and othe r constants, dependi ng on n.
Figure 2.8 shows the expected dissolution rate for values of n of 1 (2.22), 2 (2.25), and 3, usi ng the same equi1 ibrium constants as on Figure 2.3. The curves are nonna1ized to zero at pH 9 to pennit comparing the expected slopes for these three cases. The slopes (an values) are 0.3,0.6, and 1.0 for n
=
1,2, and 3, respectively'.2.2.3.3. Observated Kinetic Behavior of Other Minerals Two sets of data are reported in the literature in the framework outlined above. For dissolution of cS-A1
203, Stumm et ale (1983) reported a rate expression of the fonn of (2.28) with an = 0.4 and n = 3 over the pH range 3.5-6. Pulfer et ale (1983) reported an = 1 for dissolution of bayerite (y-A1(OH)3) over the pH range 3-4.
Interpretation in the same manner implies n > 3, which seems unlikely based on su rface geomet ry.
Wirth & Gieskes (1979) investigated dissolution of vitreous silica from pH 6-10, with an
=
-0.5 observed. An evaluation in tenns ofsurface charge, with >Si-O- being the species of interest, suggests that n = 2, or two OH- i nv 01 ved in the rate-dete nni ni ng step.
-
Q) 0a::
0 C\
..J 5 4
3 2
0 -1
-2 -3
3 4
{>M9-OH 2+}3
{>M9-OH 2 +} 2
5 6 7
pH
8 9 10 11
Fig. 2.8. Expected dissolution rates for brucite
outer layer for adsorption of 1, 2, or
3 protons per Mg2+ released; rates
normalized to zero at pH 9.
Both detenninistic (O'Melia, 1980; Lawler et al., 1980; Hunt, 1980) and stochastic (Valioulis, 1983) models of coagulation have been applied to predict particle removal and
particle-size distributions in dilute suspensions. The fanner method uses an empi rical coagulation-efficiency factor (a) to account for particle interaction and non-ideal behavior. The latter incorporates functions to describe hydrodynamic interactions and uses linear
approximations to describe electrostatic forces between interacting double layers. Both methods treat spherical particles. In principle, either could be modified to describe behavior of non-spherical particles and could express coagulation efficiency in tenns of fundamental
chemical properties of a system.
For a detenninistic model, the rate of disappearance (L-1
s-1) for a given size and type of particle (indicated by 1) is given by:
(2.29)
where ki is the coagulation rate constant (Lis) and ni is the concentration (LO-1
) of size i particles. n1 is the concentration of size 1 particles. This is a simplification of the general equation used to predict size distributions (Friedlander, 1977) that considers
fonnation of size 1 particles fran aggregation of smaller particles as we 11.
a
should be a function of particle type and size, but in practice is generally observed only for a suspension of different particle sizes and assumed to be constant for given chemical conditions (Lawler, et al., 1983). For a fully destabilized particle,
a= 1. Stumm (1977) suggests that
amay be from 0.1-1.0 in seawater, but that lower values
characterize estuaries (0.01-0.1), rivers (10- 4_10-3) and lakes
(10-5-10-4 ). The highest
avalues measured in laboratory experiments are on the order of 0.5 (Birkner
&Morgan, 1968), but values on the order of 0.01-0.1 are more commonly reported (Edzwald et al., 1974; Hahn
&
Stumm, 1968; Eppler, 1975; Gibbs, 1983). Many of the higher values
are for estuarine waters and seawater.
Calculated values for ki for three coagulation mechanisms
Brownian diffusion, fluid shear and differential sedimentation -- are shown on Figure 2.9.(a) From these calculations it is clear that in the
(a) Equations used to calculate ki are as follows: for Brownian diffusion, k; r. '
1where f
i=
3TI~d;(sphere)
1
8TIllal
and fl = ln2p+0.S ' the Stokes friction factors for flow parallel and perpendicul ar to acyl inder axis
(Happel
&Brenner, 1973; the average friction factor (f
ll
/3 + 2f
1/3) appl ies to random orientation and was derived by Fuchs (1964) for a prolate ellipsoid. For fluid
(d1 + d i)3G
s he ar,
k i=
6 ;for 2 2 2
= TIg(pp-p)(d 1+d i ) (d i -d1 )
ki 7211
differential sedimentation,
-9
-10
en -II ...
-'
~
-12
0' o
Brownian Oi ifusion -13
-14
-150~---L---~----~~----~---50~----~60
10 20 30 40
DIAMETER,
f-Lm
Fig. 2.9. Rate constants for coagulation of chrysotile fibers (0.05
~mdia. by 0.5
~mlong) with larger spherical particles in water at 20 C with low fluid shear (G = 1 s-l) and high density (pp= 2.6 g/cm3); adapted from Hunt (1980)
and OIMelia (1980).
presence of larger particles, removal of submicron fibers by differential sedimentation dominates.
In this first-order estimate, the coll iSlon radius is assumed to be fiber length rather than diameter. This assumption is val id if fiber rotation is fast rel ative to transl ation; otherwise coll is ion rad ius should be based on the preferred orientation or distribution of particle orientations. The orientation of axisymmetric particles is determined by the combined effects of shear-induced rotation and rotary Brownian motion, which in turn affects transl ational movement (Cerda
&van de Ven,1983). For a spherical particle, the distance a point on the surface moves by
translation~and
rotation~diffusion are
approximately equal (Lyklema, 1976). In the absence of shear flow and for long times, Brown ian rotation resul ts in equal probab il ity for any given orientation of a non-spherical particle, i.e. random orientation
(Fuchs, 1964). The preferred orientation of a rod-shaped particle under weak Brownian motion and in the presence of a simple shear is along the flow (Leal
&Hinch, 1971).
k = 1.4 x 10-16 g·cm 2s-1 , Boltzmann's constant; T = 293 K;
].l= 0.01 g·cm- 2s- 1, viscosity; G = 1 s-l, fluid shear rate; g = 980 cm·s- 2; Pp =
2.6 g·cm- 3, particle density; p = 1.0 g·cm- 3, water density; a
1 = 2.5 x 10- 6 cm, fiber radius; d 1 = 5 x 10-5 cm, fiber length; p = d/2ai = 10;
di = size of type i particle.
random and rotational d iffus ion was assumed to be fast rel ative to translation. The effect of having fibers al igned in a flow would be to reduce the coll ision radius and thus the rate constant. The magnitude of this uncertainty in rate constant is unimportant (small) in the current context (southern Cal ifornia surface waters) for two reasons.
First, coagul ation rates are greater at d· 1 arge, in which range
1