Kinetics for the aquatic environment In some cases, kinetics calculations are more appropriate than equilibrium calculations when modeling natural water chemistry phenomena. It is important that the correct choice be made between these two types of models
James F. Pankow Oregon Graduate Center Beaverton, Ore. 97006 James J. Morgan California Institute of Technology Pasadena, Calif. 91 I25 The applicability of equilibrium models to natural water systems requires that the various reactions within the model occur either very quickly relative to the time frame of interest or extremely slowly. It is the reactions of intermediate rate that pose problems for equilibrium models. Clearly, fast reactions will promote equilibrium. The extremely slow reactions and their associated equilibrium expressions may simply be left out of the model altogether. In the latter case-for example, when considering river or lake water chemistry-one need not be concerned that molecular nitrogen at ambient levels is not at equilibrium with respect to oxidation by oxygen. The chemical oxidation of N2 by 0 2 is simply so slow that this reaction need not be considered. Neglecting the kinetics of this reaction does not impair the quality of the results; in fact, it reduces the complexity of the model. (The validity of this assumption is perhaps best reflected in the surprise many express when they first learn of this instability of ambient molecular Nz.) With reactions of intermediate rate, on the other hand, neither the assumption of complete equilibrium nor that of unimportance is correct. Complete equilibrium is, therefore, not the criterion for the applicability of an equilibrium model. Rather, its ap1306
Environmental Science & Technology
plicability rests upon the correctness of the assumption that those species considered to be reactive with one another, are at equilibrium with one another. In our first paper in this series (1 ), we discussed some general kinetics points, as well as the specific kinetics that might, under certain conditions, accompany the hydrolysis of tripolyphosphate and the oxidation of ferrous iron in the presence of a stabilizing ligand. The two main issues in that paper were concerned with the manner in which varying the system parameters-such as concentrations and rate constants-affected: 1) the overall reaction rates; and 2) the degree to which a pair of opposing reactions within an overall system is maintained at equilibrium. In this second paper, we extend the discussion with three different examples: the autocatalytic oxidation of Mn(I1) in both the presence and absence of ligand, and the adsorption of a metal ion on the surface of suspended particulates in the presence of a ligand.
Mn(I1) oxidation with no ligand As is the case for iron ( I ) , manganese is at least partially present in the (11) state in anoxic groundwaters and in the hypolimnion of eutrophic lakes. When such groundwaters are brought to the surface for use, and when the hypolimnetic waters undergo the spring and fall overturns, the concomitant exposure to oxygen initiates conversion of Mn(I1) to Mn(1V). As with the conversion of Fe(I1) to Fe(III), this oxidation is accompanied by a reduction in pH and dissolved oxygen concentration and, at sufficiently high total Mn levels, the formation of a precipitate ( I ) .
The addition of a metal oxide to a natural water system is of interest due to the adsorptive properties that such surfaces exhibit towards trace metals. The particularly important role manganese plays in this regard is exemplified by the formation of the mineral-rich manganese nodules on the ocean floors and in many lakes. The rate equation for the conversion of Mn(I1) to Mn(1V) includes an autocatalytic term in addition to a term proportional to the reduced metal ion concentration. Morgan ( 2 ) found that the autocatalytic term involves the product Mn(1V) oxide, which, because of its adsorptive properties, accelerates the removal of Mn(I1) from solution and thereby causes the mixed oxide [Mn(II), Mn(IV)]O, to form ( x = 1.4-1 $ 6 ) .Although it is believed that the MnO, surface then catalyzes the oxidation of a portion of the adsorbed Mn(II), the nature of this reaction is not understood. This sequence of events may be abbreviated:
02, M ~ o , (2) Morgan also concluded that the magnitude of the available surface area was proportional to the concentration (mol/L) of precipitated product MnO,. Therefore, the equation describing the time rate of change of [Mn(II)] may be written:
- k3 [Mn2+][MnO,]
(3)
The rate constant k2' has been primed
0013-936X/81/0915-1306$01.25/0 @ 1981 American Chemical Society
FIGURE 1
The oxidation of Mn(I1) for varying values of k2' and kl' 10-5
10-7
10-8 0 1 2 3 4 5 6
7 8
91011
Time (hours)
10-5
(e) k2' = 5.0 x 10.- s-' k i = 5.0 M-1 ?I-I
kr = 50.0 M-' s-'
10-8
10-0 0 1 2 3 4 5 6 7 8
91011
0 1 2 3 4 5 6 7 8 91011
Time (hours)
Time (hours)
to emphasize its difference from the kz in the Fe(ll) oxidation case discussed in the first paper of this series ( I ) . Because the solubility of Mn(lV) is minimal at most intermediate pH values, and because the Mn(ll) hydroxo complex formation constants are small [ K , = 2.5 X IO3 M-' (3)l. Mn2+ and precipitated MnO, will be assumed to be the only important manganese species. Under these conditions, equation 3 may be integrated analytically:
concave up, respectively. On the other hand, if the autocatalytic term becomes important (has a large enough k3), a log[ Mn2+] vs. time plot will not be linear and the [MnO,] and [Mn2+] vs. time plots will both be sigmoidal in character. In both Figures l a and Ib, therefore, the absence of simple first-order kinetics is immediately evident. When these data are plotted in a linear-linear format (Figures 2a and 2b), the [MnO,l and [MnZ+l vs. time curves
Sung ( 4 ) recently reexamined the data obtained by Morgan (2) at a po2of 1 .O atm and pH of 9.5. His analysis suggests values of 7.0 X s-I and 4.4 M-I s-I for k; and k3, respectively. Effect of kz' and k,. Figure 1 presents the results of a series of calculations using different sets of kz' and k3 values. An examination of these curves reveals that increasing the value of k i to 5.0 X s-I doer from 5.0 X more to accelerate the oxidation kinetics than does increasing the value of k3 from 5.0 to 50.0 M-' s-I. (It is important to note here, however, that increasing k3 by an order of magnitude could have had the greater imDact if the Figure l a valueof k3 of 5.0 M-' s-I had been substantially larger.) For simple first-order kinetics, a plot of log[MnZ+] vs. time will be linear and plots of [MnO,] and [Mn2+]vs. time will be strictly concave down and
have a sigmoidal shape in 2b but not in 2a. The k3 term is large enough to accelerate the reaction as the MnO, is produced in Figure I b (2b). but it is too small to do so in Figure l a (2a). Even in Figure I b (2b), however, the rate must ultimately slacken as the Mn2+ is depleted. Since any acceleration must be accompanied by an inflection point, that is, a zero value for the second derivative of both [MnO,] and [MnZ+] with time, the possible presence of sigmoidal character in these plots may be predicted. From equation 3, and since d[MnOx]/dt = -d[MnZ+]/dt: ~. d2[Mnoxl= /d[Mn2+1),va, dt2 \ dr - 2k3[Mn2+] k 3 M n ~ ) ( 5 )
+
We also note that dZ[Mnz+]/dtZ= -d2[Mn0,]/dr2. These two derivatives will equal zero when either of the
two factors in equation 5 equals zero. Since d[Mnz+]/dt = 0 only at r = a, this case is of lesser interest than if (kz' - 2k3[Mn2+l k 3 M n ~ )= 0, or:
+
For the conditions in Figure l a (2a). M which is [Mnz+]i.r = [Mn2+],. Thus, there will be no inflection point between r = Oand I = -. For the conditions in Figure I b (2b). [Mn2+]i.r = 5.5 X and there will be an inflection point between t = 0 and I = m . For Figure IC (2c). [Mnz+]i,, = 5.5 X IO+. Since this is physically impossible, we see that a large k2' value will not allow an inflection point due to the essential dominance of the first-order term. In the case of Figure IC, the plot seems to indicate first-order kinetics even though k3 was not zero. The explanation here is simply that the value of kz'[MnZ+] is always substantially larger than k3[Mn2+][Mn0,]. The fact that the autocatalytic term is never of major importance during the course of the reaction described in Figure I C is evident from an examiof the ratio kz'[Mn2+]/ &on k3[MnZ+][Mn0,] = kz'/k3[MnOX]. Since the maximum value which [MnO,] assumes is M, the minimum value for the above ratio would be IO, a condition that assures the continued dominance of the first-order rate term in equation 2. If the plot presented in Figure I C (2c) were experimental data, some modelers might be tempted to assume that an equation
Volume 15. Number
11. November 1981 1307
of the type:
sumed that the production of oxidized product is an irreversible process. Considering that at pH = 8.0 andpo2 = 0.2 atm the equilibrium [Mn2+]will M , the asbe approximately sumption of irreversibility will remain valid for a considerable time, as in the oxidation of Fe(ll). Although the rate law for the ligand free oxidation of Mn(l1) is more complicated than that for Fe(I1) (I), it may still be integrated analytically. A second difference between these two systems is the time required for substantial oxidation-at neutral to basic pH, it would takeseconds to hours for Fe(I1) (I), hours todays for Mn(ll). The value of MnT and the relative magnitudes of the rate constants kz’ and k, will determine the degree to which the manganese concentration plots exhibit the effects of autocatalysis. Under certain conditions, the plots of [MnO,] and [Mn2+]vs. time will possess inflection points. A simple
describes the oxidation kinetics of Mn(ll) adequately. If, however, a modeler were to apply these data to a system wherein the [MnO,]. were of the order of M, serious error would be incurred due to the neglect of the k, term. The order of magnitude of the error may be determined from a comparison of Figures 3a and 3b. Figure 3a is an expanded scale version of 2c, and 3b was prepared using an [MnO,], level of M . Thus. as in the application of equilibrium constant data, caution must be used when extrapolating kinetic data to conditions (e.g., reactant and product concentrations, pH, ionic strength, etc.) other than those under which they were obtained. Conclusions.As with our analysis of the oxidation of Fe(II), we have asFIGURE 2
I
I
r
r’
I
I
r
I’
10-5
M.
Mn(l1) oxidation with ligand The addition of a ligand L to the above system complicates the kinetics considerably. As with the oxidation of Fe(ll) in the presenceof ligand (I), we will assume that the complexed species is stable with respect to oxidation. The process is: ki
MnL2+ e Mn2+ k-i
+L
Mn2+ % MnO,
(8) (9)
02, MnO, (10) As an extension of the previous case, assume that MnL2+, Mn2+.and precipitated MnO, are the dominant manganese species. With L, this gives a total of four unknowns. With two mass-balance equations (one each for total L a n d total Mn), twodifferential equations and a set of initial conditions are needed to solve the problem ( I ) . For the Mn2+ and MnL2+ pair, the two coupled. nonlinear differential equations that must be solved are:
The oxldation of Mn(II) for varying values of k2‘ and kla I
procedure for locating the position of the inflection points in the kinetic plots was presented and employed. For the types of values obtained by Morgan at pH 9.5 and po2= 1 atm [kz’ = 7.0 X s-I and kz = 4.4 M-I S-I (4)], substantial autocatalysis will be evident, and by equation 6, an inflection point in the plots of [MnO,] and [Mn2+]vs. time will be present if the value of MnT is greater than 1.5 X
I
k i = 5.0 x IO-’ 1-‘ k j = 5.0 M-’ S-’
- k-I[MnZ+](LT - [MnL2+]) 0
1
2
3
4
5
6
7
-_
8
9
10
-
10
I
. ‘I
8
+ k-l[Mn2+](L~- [MnL2+])
8 6
-
4 -
2
0 -
0 0
1
2
3
4
l i m e (hours)
1308 Environmental Science h Technology
5
0
1
2
3
lime (hours)
(11)
I
8 x
- k2’[MnZ+] - k,[Mn’+](Mn~ - [Mn2+] - [MnL2+])
Time (houn) 10
11
4
(12) Using a method described in our previous paper (I), we carried out the numerical integration of equations I 1 and 12 for a variety of initial conditions. As in the FeLZ+/FeZ+/L case (I), the timedifferentials for the free ligand and the oxidized product may be deduced from equations 1 1.12, and the time derivatives of the mass balance equations on Land total Mn, respectively. Effect of h.As in the oxidation of Fe(1l) in the presence of ligand (I), adding L to this system lowers the free Mn2+ concentration, thereby decelerating the oxidation rate. A comparative examination of Figures l a and
4a-d reveals the magnitude of this effect for four different values of LT, and kl and k-l values of 1.0 s-I and IO6 M-I s-I , respectively. For LT = M ,almost half of the initial Mn(ll) remains after 14 h. I n the absence of any L, the same amount of time effects a 99% reduction in the Mn(ll) level. The approach used tocalculate QSt+ for the MnL2+/Mn2+/L system IS identical to the one we presented for the FeL2+/Fe2+/L case ( I ) . The equation for Qrtab is the same as equation 30 of that paper, with a = k2/ k,MnT. In contrast to the iron case, however, very littledisturbanceof the metal-ligand equilibrium occurs. This is a direct result of the much slower Mn(ll) oxidation kinetics. I n thecase of Fe(ll), the oxidation rate term multiplying the reduced iron concentration (k2[OH-I2po2) is 0.26 SKIat pH 8 and 0.2 atm po2 ( I , 5 ) . For Mn(ll), on the other hand, the corresponding term (k2/ k3[MnOX]) is much smaller, only 9.2 X s - I at apH9.5,l.Oatmpo2,andSX 10-"M [MnO,] ( 2 , 4 ) .Nevertheless, in order for the reaction to proceed at all we must have some disturbance of the MnL2+/Mn2+/L equilibrium, and equation 30 of our preceding paper ( I ) may be used to calculate a Qrtvb of 1.0000166 X IO6 M-' for Figure 4b. Since the values of Q remained very close to K throughout the course of the various reactions analyzed for this system, the plots of Q vs. time have not been included in the figures. Effect of kl. For a constant value of k-1, decreasing kl will decelerate the oxidation because the level of free oxidizable Mn2+ is lowered. Figures Sa and b were calculated using kl values that differ by an order of magnitude. On first inspection, it is somewhat surprising that the oxidation is not markedly faster with the larger kl. Indeed, for a IO-fold increase in kl, the time required to oxidize one-halfof the M n ( l l ) is not even halved (6 h vs. I O h)-the stoichiometric excess of Mn(ll) limits [MnL2+], to being no more than 5.0 X M in both cases. Therefore, changing kl does not greatly alter the initial concentrations of Mn2+ and MnL2+. Since the oxidation kinetics depend upon the speciation (viz., free vs. bound Mn(ll)), the general kinetic profiles appear similar. Had the ligand been in stoichiometric excess, however, the effect of changing kl would have been greater. Consider, for example, the consequences of MnT and LT values of 1 .O X and 2.0 X IO+. For kl values of 0.1 and 1.0 s-l, the [Mn2+], values would be 8.4 X IO-' and 9.8 X
+
+
M. respectively, and therefore the initial oxidation rates would differ by an order of magnitude. Since the L/ M n ( l l ) ratio is continuously growing in Figures Sa and b, then these two cases will require increasingly different time periods to achieve the same percentage of oxidation. Effects of k2' and k3. The presence of ligand will dampen the accelerating effects of increasing either k2' or k,. Nevertheless. as was the case in the absence of ligand, increasing k2' from s-I has a 5.0 X IO-' to 5.0 X greater impact on the kinetics than does increasing k, from 5 to 50 M-' s-I (see Figures 5c and d). Under some conditions, however. the first-order-plus autocatalytic kinetics may still produce an inflection point in the plot of [MnO,] vs. time. The location of the inflection point may be determined from an analysis similar to that carried out for M n ( l l ) oxidation in the absence of ligand.
Earlier, we discussed how plots of data should appear in first-order and firstorder-plus autocatalytic kinetics, and also how varying k2'and k, can affect the nature of those plots. In the presence of ligand, the linearity of the log[Mn2+]vs. time plot will now only be preservcd when k, is small and either the stability constant or LT is very small. The latter condition requires, in effect, that no effective ligand be present. Conclusions. As with Fe(ll) ( I ) , the presence of ligand decelerates the oxidation rate. Since the oxidation of M n ( l l ) is considerably slower than that of Fe(ll), however. the metal/ complex equilibrium is disturbed only slightly for the range of values of the rate constants studied here. (If a very small value for k-1 were selected, however, a substantially greater disruption would occur, as predicted by the equation for Q I l n b ( I ) . ) In thecase of the Fe(ll) oxidation in the presence
FIGURE 3
The oxidation of Mn(ll) with two different levels of initial MnOx
-
10
5
8 -
x
6 c
.-0
z
I
E
0 m
4 -
c
s
2 -
0 0
0.5
1.o
15
20
Time (hours)
\ (b) [MnO,I,
0
0.5
1.o
I
lW'M
1.5
2.0
2.5
Time (hours)
Volume 15. Number 11. November 1981
1309
FIGURE 4
The oxidation of Mn(g in the presence of varying levels of total ligand LTa 10-5
Ec
--m
.-0
10-6
c
g
s
lo-'
R 1 - -4 MnL"
(a) LT = 0.1 x IO-'M
10-0
lo-'
(b) LT = 0.5
IO-5 M
X
10-8
0
2
4
6
8
0
I 0 1 2 1 4
2
6
4
8 1 0 1 2 1 4
MnL"
MnOx
0
2
4
6
8 1 0 1 2 1 4
lo-' 10-8 0
(d) LT = 1.0 2
4
6
10-IM
X
8
10
1214
Time (hours)
Time (hours)
FIGURE 5
The oxidation of Mn(ll) In the presence of ligand with varying rate constants'
.-0
I
m c
s
kr'
= 5.0 x I O - I
8.'
lo-' (b) kl = 0.1
k i = 5.0 M-' 8.' 10-8
Metal adsorption with ligand Although we have emphasized oxidation reactions in this paper, other processes provide very interesting examples of kinetics problems. The competition arising among different complexing species for metal ions is of particular importance. Because of the wide variety of adsorptive surfaces and ligands possible in natural waters, there are many instances in which new materials of this type are suddenly present in a given system and are therefore in competition for the system's metal ions. Considerable evidence supports the view that the adsorption of metal ions on oxide surfaces (e& silicon, aluminum, and iron oxides) takes place at specific sites. We will therefore assume that the "concentration" of adsorptive sites may be expressed in units of molarity. Provided that the adsorption reactions a t the surface do not encounter any diffusional resistance (Le., transport to and from the particle or within the particle), a metal/ligand system (M/L) will respond in thesame manner to the introduction of either: 1 ) an adsorptive surface, S ; or 2) a second ligand. In addition, the mathematics of the two cases will be entirely analogous. Because of the current interest in adsorption of metals on oxide surfaces, this analysis will be carried out in the notation for the adsorptive surface. The overall process may then be abbreviated:
- 1-i 10.6
10-8
of ligand, once Q reaches Qstilb, the continued integration was simplified enormously. The same conclusions apply here. In this case, since Q remains so close to K, it is not necessary to calculate QStrb to take advantage of this, but we must wait until [L] becomes essentially equal to LT. When LT is large with respect to MnT. this simplifying feature may be exploited ffom time zero. Finally, changing the value of a certain rate constant (such as kl in Figures Sa and b) may not alter the reactant profiles substantially if its effects are limited by the concentrations of certain of the reactants.
0
2
4
6
8
0
I 0 1 2 1 4
2
4
6
8
8.'
I 0 1 2 1 4
10-5
10.6
ki
M L e M + L
10-1
k-i
(d) ka = 50.0 M-'
8.'
k-2
M+S+MS kz
0
2
4
6
8 1 0 1 2 1 4
Time (hours)
1310
Environmental Science 8 Technology
0
2
4
6
8
I01214
Time (hours)
(13)
(14)
With five unknowns and three massbalance conditions (one each for total metal, ligand, and surface), two differential equations and a set of initial conditions are needed. One set of several possible pairs of coupled, nonlin-
ear differential equations for this problem is:
+ ~ - ~ [ M ] ( S-T EMS])
(16) The equations for d[S]/dt and d[M]/dt may be obtained either directly from the examination of equations 13and 14orfrom 15.16,andthe differentials of the constants ST and MT with respect lo time. As with similar problems we have discussed, equations 15 and 16 may be integrated numerically. We have carried out such integrations for a variety of initial conditions and rate constant values. In all cases, equilibrium is assumed to exist initially for the ML/M/L system. The surface S is added at time zero, and the MS/M/S system seeks to attain its equilibrium while disturbing that of the former system: The overall system then approaches equilibrium. Effect of LT. Figure 6 presents the concentration profiles for three different values of LT with ST. MT, kl, M, kLl, k2. and k-2 values of 10-5 M, 1.0 S-1, 1.0 x 108M - 1 S-1, 10-2 s-l, and IO4 M-' s-l. respectively. The relatively large value of S was chosen to simulate high mass concentrations of particulates with high specific surface area. For the very simple complexation/dissociation involved here, KML= k-l/kl = IO8 M-I
and K M =~ k-2/k2 = IOb M-I. The relatively large KMLvalue was selected to allow the ligand L some margin of competition with the high S concentration. Possibly the most striking feature of Figure 6 is that raising LT tends to increase the length of time required to achieve equilibrium in the MS/M/S system. This is a result of the fact that increasing LT increases the absolute magnitude as well as theproportion of metal M, which must be freed from the M L complex before it can form MS. These quantities may be represented by ([MLI, - [ M L l d and ([MLI, [ML],) X IOO/[MS]e, respectively. The subscript "e" refers to thefinal equilibrium state. For Figures 6a-c, the former quantity equals 9 X M, respec7 X IO-', and 2 X tively, and the latter quantity equals 1, 12, and 89%. respectively. The disruption of the M L / M / L system that occurs in Figure 6 is so slight that it cannot be detected on the scale used in these plots. The QML plots for Figures 6a-c have therefore been drawn on a much expanded scale in Figure 7. For reasons completely analogous to those of the oxidation of Fe(ll) in the presence of ligand (I), increasing LT reduces the maximal degree of disruption. However, increasing LT also extends the time over which the disruption takes place. This latter observation is directly related to the fact that raising LT prolongs disequilibrium in the MS/M/S case; this requires that the M L / M / L system also remain out of equilibrium for a
-
greater length of time. Effect of kl and k-1 with constant KML. Figure 8 presents concentration profiles for four sets kl and kLl values, g with k-l/kl = K ~ ~ e q u a l i n108M-' in each case. The fact that KMLis a constant implies that all four cases have the same initial and final equilibrium states, though we naturally expect the time required to reach that final state to increase as kl decreases. By the same token, for reasons that should now be clear, decreasing kl tends to increase the disruption of the M L / M / L equilibrium. Interestingly, the Figure 8d value for kl of IO-'s-I is small enough to cause [MI to drop below [MI, during a substantial portion of the reaction time. Indeed, even after 1 min, the (MI value is 1.58 X M , while (MI,equals 2.75 X M. Even if KMLis held constant, we see that if k-1 were reduced to values expected for highly inert complexes, the time required to achieve equilibrium could easily approach hours to days. Conclusions. The types of conclusions we have drawn in thisdiscussion are similar to those presented in earlier sections and in our earlier paper (I). The concept of a Qrtab does not apply in this case because of the complete reversibility of this sytem, Le., the inclusion of a nonzero rate constant (k2) for the back reaction from the product
MS. Kinetics vs. equilibrium In view of the complexity of aqueous systems, equilibrium calculations offer
FIGURE 6
Metal adsorption In the presence ot varying total ligand L,'
%
I
10-4
!-0.5
-:-o.o
MS
105
MS ML
-2.0 ML
5 10.6 (a) LT = 0.1 x IO-SM i-1.5
(b) L r = 0.5 x lO-'M QML
i-1.0lo-'
~
10
20
30
40
Time (s)
50
60
10-8 0
10
20
30
40
Time (s)
50
1
Time (s)
Volume 15. Number 11. November 1981
1311
FIGURE 7 QML
plots for Flgure 6 conditions
,
1.020
, , , ,
I
I
1.Ol8l-LT = 1.0 x 1
0
m
1.016 9) 1.014
2
-
x 1.012
7 1.010
2
2 1.008
0
1.006 LT = 0.5 x IO-'M
1.004
much in the way of simplicity, and we clearly want to apply them whenever possible. On the other hand, we do not want to neglect kinetics considerations if they are important. Whether kinetics- or equilibrium-based modeling is more appropriate in a given situation will depend upon two things: the initial conditions and the time scale of interest. For example, consider the sudden admixture of deep anoxic lake waters containing Fe(ll) with overlying oxic waters (viz., an autumnal overturn). A calculation of the equilibrium composition would indicate an essentially quantitative conversion of the Fe(ll)toFe(lll). IfthepHisequalto approximately 7 and the conditions (LT. ligand stability, ligand lability, etc.) are similar to those present in Figure 4a of our preceding paper ( I ) , kinetic models will be required if chemical changes occurring ,within
hours (day to weeks at pH values much less than 7) are important to the modeler. On the other hand, when pH values are greater than 8 and oxidation is very rapid, the equilibrium-predicted Fe(1ll) levels will be achieved shortly (minutes) after the mixing. ( I n both the low and high pH cases, however, thequestion of how quickly the Fe(ll1) species-e.g., solid, aqueous mononuclear hydroxo complexes, aqueous polynuclear oxohydroxo complexes, complexes with organic ligand, e x come toequilibrium withoneanother is another matter altogether.) The foregoing remarks and the examples discussed have assumed a system into which a reactant is added instantaneously (tripolyphosphate, oxygen, and surface) and which is closed immediately thereafter. The system then proceeds at a velocity determined by the specific kinetic rate laws and a set of initial conditions toward whatever equilibrium state they prescribe. This approach is adequate provided the initial chemical addition (or removal) occurs on a time scale that is short with respect to the period of interest. Many natural processes (e& algae blooms and sediment remineralization) and anthropogenic inputs, however, occur over time scales of the same order of magnitudeas thoseof interest. Such systems must be considered open. I f they are assumed to be well-mixed, the chemist is again faced with deciding between equilibrium- and kinetics-based modeling. This situation is similar to the cases analyzed in this series that involved an initial equilibrium that is disrupted by a new reaction term. The input (sink) term in the open system is analogous to the newly possible reaction term in the closed system, and the analogue of the degree of equilibrium in the total, open system is the degree of equilibrium maintained within one opposing reaction pair in theclosed system. We have shown how the latter condition is preserved as long as the magnitude of the kinetic terms that must keep pace are large with respect to the rate of the disrupting reaction. I n the open system, the equilibrium approach would be to consider the inputs or sinks as spike increments and assume total equilibrium along the way. This is adequate if the rates of reaction are large with respect to the input (sink) rates. This assumption is often made in natural water chemistry. I f an examination of the various rates indicates that equilibrium will not be maintained, a kinetic model must be used. This will require the inclusion of
1.002 1.000 0
1
2
3
4
5
Time
(s)
6
7
8
FIGURE 8
Maal adsorption in the presence ot ligand with varying kl and k-c but constant K M ~ 10-3
- lo-' 3 -5m -
0.5
.- 10-5
0.0 1r-5 2.0
c m
10.6
.._,
0
10-6
-
...
1.5
ma
"-1
=
IV.
I n
' S
QYL
1.0 10-7
10-7
10-1 0
7 2
'
1.5 1.0
-M
i
0
10-8 10
20
30
40
50
60
z ML L ..
~~
.
.
1312 Environmental Science 8 Technology
r'
0
IO
20
30
40
50
60
the appropriate input or sink terms in the differential equations as well as running mass-balance equations, but the general mechanics of the solution process will be similar. (A recent paper (6) discusses the existence o f steadystate solutions o f some simple open systems.) In conclusion, the required mathematical (both analytical and numerical) methods are available as tools for carrying out the kinetic modeling discussed in this paper. This i s true for both open and closed systems. What is often needed is the kinetic data concerning the reactions that may be limiting, as well as the motivation to examine such data with regard t o the relative appropriateness of the kinetic and equilibrium models.
References ( I ) Pankow. J . F.: Morgan. J. J. EnuirmSci. Techno/. ‘**‘ 1981,/S(II). I 1155. (2) Morga Morgan, J. J. Ph.D. Thesis. Harvard. Cambridge. Mass.. 1964. (3) Smith. R. M.: Martell. A. E. “Critical Stability Constants”: VoI. 4 “Inorganic Complexes”; Plenum Press: New York,
..
1961.53. 143-146. (6) Iioffman. M . R. Enciron. Sei. Tmhnol. 1981. 15. 345-353.
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Dr. James F. Pankow ( l e j i ) is an assistant projessor o/Enrironmental Science at the Oregon Graduate Center. His research interests include the equilibrium and k i netic aspects o/natural water chemistry as well as the transport o/organicpollurants through the entiironment. O/ particular interest are the mechanisms by which precipitation scavenges organic pollutants, the transport of organic pollutants across the airfsea inter/ace. and the development of new techniques/or analyzing trace organic compounds. He received his B.A. in chemistry in 1973 and his Ph.D. i n environmental engineering science /ram the Calijornia Institute a/ Technology i n
1978. Dr. James J. Morgan ( r i g h t ) isprofessor of environmental engineering science and vice president /or student a//airs at Caltech. His research interests are i n thefield o/aquatic chemistry andsur/ace chemistry in water treatment. His major research interests involve solution and inter/acial processes governing the transport and removal o/aquatic pollutants. He was the first editor o/ ES&T.
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