Kinetics for the aquatic environment Most natural waters never reach complete equilibrium and are controlled by the chemical kinetics of the system. The kinetics of tripolyphosphate hydrolysis and F e ( I I ) oxidation are discussed in the first part of a two-part article
James F. Pankow Oregon Graduate Center Beaverton, Ore. 97006 James J. Morgan California Institute of Technology Pasadena, Calif. 91 125
Over the last several decades, scientists from many different backgrounds have made substantial progress towards understanding, in a quantitative sense, many aspects of what is collectively called the aquatic environment. Progress has, nonetheless, been slow. In this field, as in others, obtaining an understanding of a system depends heavily upon the successful construction of models which incorporate the most important parameters. Upon variation of these parameters, the model should predict what is observed under natural conditions. Due to the extremely complex and interdependent nature of the many processes occurring in natural waters, construction of comprehensive working models that describe natural water chemistry has to date proved impossible. Consequently, broad assumptions have often been made to simplify things. For example, in his classic thermodynamic (i.e., equilibrium) model of seawater, L. G. Sill& entirely neglected biological, fluid mechanical, and chemical kinetic considerations, and designed his model so that it would be valid only under conditions of complete chemical equilibrium ( 1 ). Using well-known thermodynamic equations describing solution and multiphase equilibria as the model’s
foundation, S i l l h was then able to make several important, albeit somewhat qualitative, statements regarding the identities and concentrations of some of the oceans’ dissolved species. Sill&nalso used his model to predict which solid phases might be present under equilibrium conditions. The calculations regarding dissolved species and solid phases are, of course, inseparable. With the advent of modern computer techniques, SillCn’s approach was extended greatly, and a number of general computer programs have been developed that can quickly produce numerical solutions to the large number of coupled equations obtained in any reasonably complicated aqueous system ( 2 , 3 ) .However, it is important to remember that the accuracy of results so obtained depends not only upon the presence of chemical equilibrium between the interreacting components, but also upon the reliability of the literature values for the thermodynamic constants used in the computer program. Such computer programs are capable of accurately predicting many characteristics of natural waters, such as pH; pc; the concentrations of many mono-, bi-, and trivalent metal ions; as well as the concentrations of many singly or doubly charged anions. In addition, the presence or absence of a respectable number of solid phases often can be predicted accurately. When this type of success is achieved for any natural water, it is strong evidence of equilibrium between at least some of the system’s chemical components. However, a natural water system that is in a chemical equilibrium with
0013-936X/81/0915-1155$01.25/0 @ 1981 American Chemical Society
all of its components (both dissolved and solid) would probably be difficult to find. Most natural waters probably never attain complete chemical equilibrium. Indeed, since the earth can be viewed as a solar engine, there is a relatively steady flow of thermal, kinetic, and chemical (through photosynthesis) energy through the hydrosphere. The flow of such energy through natural waters, as well as the introduction of thermal energy and chemicals of geologic and anthropogenic origin, serves to continually frustrate the attainment of chemical equilibrium. Even when such disruptions directly affect only selected chemical equilibria, these equilibria are likely to be coupled to others, which will then be disturbed in turn. Consider, for example, an algal bloom. As a result of the initial uptake of CO2, the pH will rise. Since many reactions that occur in natural waters are affected by pH, they will be forced to adjust their equilibrium positions accordingly. Some of the reactions possess only slow kinetics (e.g., certain adsorption, redox, and metal hydrolysis reactions) and will not be able to respond to the change as quickly as those which possess fast kinetics (e.g., most proton transfer reactions). A fast reaction will quickly attain its own equilibrium position. In the unlikely event that such a reaction is completely uncoupled from any of the kinetically hindered components, this position will remain unchanged while the remainder of the system approaches equilibrium. The other alternative is more likely: The kinetically fast components are linked to those which are kinetically slow and are thereby forced to pass through a series Volume 15, Number 10, October 1981
1155
of equilibrium positions, or more properly, near-equilibrium positions as the system pursues the state of total equilibrium. These thoughts lead to three questions: What do the terms “fast” and “slow” kinetics mean within the context of natural water chemistry? Which system parameters affect the kinetics of the reactions comprising this chemistry? What time frames separate the appropriate application of kinetic and equilibrium models? The purpose of these two papers is to address these questions within the context of several examples of interest to the natural water chemist.
Fundamentals of kinetics Many texts provide detailed discussions of the various topics in modern chemical kinetics (e.g., 4 and 5 ) . The review article by Brezonik (6) provides an excellent introduction to how some of these concepts apply to natural water chemistry. One of the most useful concepts is that of the rate law, i.e., the expression which describes the time-dependent velocity at which a reaction proceeds (viz., reactants consumed or products produced). The terms “zero order,” “first order,” and “second order” refer to the sum of the powers of the concentration terms that appear in the rate law. Thus, the expression:
ql[Al = -kl[A] dt
(e.g., A
+
B)
would be an example of a first-order rate law with rate constant k1; examples of second-order rate laws are: (e.g., 2A
-
B) (2)
ent reactions will be discussed in these articles, and the reactions to which a given kl or k2 refers will be made clear in the immediate context within which it appears. If we assume that there are no back reactions, Le., no reactions which convert B back to A, the integration of (1) and (2) may be carried out analytically, and for the initial conditions [A] = [A], when t = 0, the expressions: 1 n L - klt [AI and
are obtained, respectively. The integrated form of (3) will be identical to ( 5 ) if the reaction stoichiometries of A and B in (3) are equal. These equations may be used to derive what kineticists refer to as tip, the time required to consume one-half of the reactants, or “the reaction half-life.” For first-order reactions, t 112 is dependent only on the rate constant: t1/2 = ln2/kl = 0.69/kl
ki
A+B k-1
d[Al = -k3[A][B] dt (e.g., A
+B
-
(6)
For second-order reactions of type ( 2 ) (and (3)), and for the initial conditions [AI = [AIo (= [BIOI,when t = 0, t1/2 = 1/[A],k2 (= l/[B],k3). Figure 1 provides an overview of how the magnitude of the rate constant kl affects the half-life for a number of first-order reactions of interest to the aquatic chemist. Figure 2 tells a similar story for second-order reactions, but also demonstrates the effect of varying the reactant concentration. Other papers provide reviews of additional data (6, 29). In reaction (l), the inclusion of a back reaction with rate constant k-1 may be represented by:
and
(7)
.
and the differential equation for both A and B is:
C)
(3) Beside each rate law, we have provided an example of a reaction which could give rise to that rate law. (In these simple cases, just one differential equation is required to describe the kinetics. If more than one reaction is occurring simultaneously, a series of coupled differential equations (rate laws) must be written.) Rate constants are generally strong functions of temperature and may be dependent upon other variables, such as pressure and ionic strength ( 4 ) .A variety of differ1156
(4)
Environmental Science & Technology
-d[A1 - -kl[A] dt
+ k-l[B]
=
-- [B1 (8)
dt Inclusion of back reactions in (2) and (3) yields equations similar to (8). Schemes such as these are among the simplest cases involving opposing reactions. As a direct result, their analytical solutions are well known ( 5 ) . An example in which a pair of opposing reactions is succeeded by an irreversible reaction is:
Extreme caution must always be exercised when relating rate constant data to equilibrium constants; however, in the simple kinetics depicted in (9), we may assume that the stability constant for compound A is equal to k-l/kl. Whether or not reaction (9) will remain close to equilibrium as B is converted to D depends upon the magnitudes of the rate constants and the concentrations of the reacting species. There is a large variety of chemical situations in which preexisting equilibria are disturbed by newly possible reactions. We will investigate several of interest to the natural water chemist and include discussions of how the degree of equilibrium disruption is influenced by these two factors. With each addition of either a forward or a backward reaction, however, the attendant mathematical complexity of the differential equations may prevent an analytical solution. In such cases, one must often resort to numerical integration methods. A number of the complex kinetic examples we present will require numerical integration and will involve the assumptip that the various systems are cloked. Some of the properties of open systems (e.g., steady-state concentrations, etc.) have recently been reviewed by Hoffman ( 2 9 ) , and we will discuss our work in the context of open systems in Part 2 of this series.
Hydrolysis of tripolyphosphate By virtue of their ability to complex calcium and magnesium ions, the metal species which impart hardness to water, condensed phosphates are extensively used as detergent builders. This use accounts for a major source of phosphorus to certain surface waters. In water, phosphates will hydrolyze to orthophosphate, the form most readily available to plants and organisms. With tripolyphosphate, the most effective condensed phosphate detergent builder, hydrolysis takes place in two steps. The kinetic studies of this reaction have not distinguished between the several possible species of tri-, pyro-, and orthophosphate (e.g., degree of protonation). This is denoted by the use of parentheses in the following reactions: (P3OIo5-)
+ H 2 0 ki
A
(Pod3-)
+
(P2074-)
+ 2H+
(1 1)
...
l . . . ~ ' ",..,."~~.-__ .,~
.
.-,-.;.....-.....
............
sdecled flrst-order rate constants and their corresponding f,valuecr
f
k FldGdU mlb wmian~
HaGite 0.Wk
(a-'1
(S)
10'0
-
....MnSO,(aq) ....F%t+ 10'
10 lff
ra-
101 1V
-
....CF+
HP
Mn*+
FeOW+
HO CrOH'+
....Fa!++ .. ..H$
+ So,. .................
+ H+ ................... 8
+ H+ ................... + H+. .............
Fe(0H);c
--
+ HS-. ........................
H+
....Zn(NTA),"
7
ZnNTA-
+ NTA-
0 8
10
............. 11
....NiHP,O,oS N12+ + HPsOqo"............... 12 ....HtCO, CO. + H.0 ...................... 19.14 , = ....Ni, I NP+ + (COO)2* (oxalaleion) ...... 15 mc ....Niglyt Nlgly+ + gly- (gly=glyclne). ....... 16 ....ZnNTAZnt+ + NTA- ................... 11 .GO, H,o. ... HIGO, ............................ 17 ....Nlgly+
-
+ glr
....HCO;
-
+ O H.......................
NI'+
GO,
(gly~glycine)......... 16
17
....Pso,o- H,o. P*O,+ + PO,* ................. 18 HO
10
... ..PzO,o,e9 2P0,s .........................
,LI.dlr
....CH,GHNH,COOH*.**.urn)
....(Alanlrm),
iU.Mnli.fk4
CH,CH,NH,
+ GO, ......
(Alanine), .....................
18
10
20
10"
Volume 15. Number IO. October 1981
1157
. . .
_
.
.-......_I
......
FIGURE 2
Selected second-order rateconstants and their wnesponding I, values plotted as a function of the initialconcentration c. The value of C is assumed to beequalfor both reactantspecies
-
io-'
rs
o-'
1v--
1v--
lW
1v--
1w--
lW
lo---
lo-l--
....C W + + H+ ....Mn2+ + SO,% ....Pbs+ + HNTA-
lC-
1w--
lo.--
....NP+ + HPsO,o'
lo+
lo-'--
10'
1v --
1w
--
IO'
--
I(*
--
1(*
--
IW
1w --
I 0'
1w
--
1(* --
....H+ + HS....H+ + FeOH*+
1
10.
ms
---
10-7 1v-
-
lo--
io-' 1v-
lo-'
10"-
IOIO--
I
C
H.S
.......................... 10 H.0 + Fet+.. ............21
+
............... o ............... 7 + ........... 22
C++ H,O MnSO, (aq) PbNTAH+
NIHP,O,o%...
--
-- ....NP+ + (COO),.- (oxalate Ion)
-
............ 72
2 . c o
...... 75 ....NI*+ + glyNlgly+ (glyglyclne). ........ 98 23 10' -- 10" -- ....C02 + OH- d HCO; ...................... NQ..
-f
(I
I@--
-lo.
-
1v--
-
....F#+ + 0, -* Fe(lll) (pH 8). ................24 ....Fett + F-
10'
I(*
--
I@
--
I(*
1w --
10"
-- ....F#+ + 0, d
FeF+
.......................
25
rnln
l(r
--
l(P
--
-
l(*
h
lw
IW--
d
1w--
-
10. 1w
rno
1158 EnvironmentalScience 8 Technology
....s (-
11)
-
+ 0,
....F#+ + 0,
.Mnt+ + 0,
................. 24
Fe(lll) (pH 7)
S*O*",
so,-, sop., ....... 29
Fe(ll1) (PH 6).
............... . 24
Mn(lW (pH 0)
........... . 27
kz + H2022(P043-) +
(P2074-)
2H+
(12)
If we let PPP, PP, and P denote the total tri-, pyro-, and orthophosphate concentrations, the corresponding rate laws for the various Dhosohates would . . be:
dppp= - k l p p p
hibiting some zero-order characteristics under certain conditions (30). Conclusions. Of the chemical kinetics topics of interest to the natural water chemist, the hydrolysis of polyphosphates is probably among the most simple. The rate laws are linear and no back reactions are involved. This simplicity is directly responsible for the analytical integrability of equations (13-15).
dt
-= dPP dt
-dp = dt
klPPP klPPP
- k2PP
+ 2k2PP
These coupled, linear differential equations may be integrated analytically ( 5 . 6 ) . The time-dependent concentration of orthophosphate is:
where
Since the hydrolysis of tripolyphosphate obeys a first-order rate law, its time-dependent concentration follows equation (4). Based on equations (4) and (16) and mass balance. then, the concentration of pyrophosphate as a function of time may be determined easily. We carried out such calculations for an initial tripolyphosphate concentration of 1.0 X 10-5 M using the values of kl and k2 determined by Clesceri and Lee ( / a ) under sterile conditions. The results, presented in Figure 3a, illustrate the relatively slow kinetics represented by these values of k l and k2. Indeed, about IO days are required to convert half of the tripolyphosphate to orthophosphate. If both rate constants are increased by a factor of 10, the conversion time is reduced to two days (Figure 3b). Such an effect could be caused by an increase in the temperature. Alternatively, the presence of microbes (viz., enzymes) can often alter the kinetics of certain systems rather dramatically. In fact, studies of reactions (11) and (12) indicate that the kinetics change substantially under septic conditions, with the rate law for PPP hydrolysis ex-
Fe(I1) oxidation with ligand present Under anoxic conditions, such as exist in certain groundwaters and in the hypolimnion of eutrophic lakes, iron is generally present in the Fe(I1) state. Acid mine drainage will also contain varying amounts of Fe(ll). When anoxic groundwaters are brought to the surface, when hypolimnetic waters undergo the spring and fall overturns, and when acid mine drainage is discharged to a river, the oxidation by oxygen of Fe(I1) to Fe(II1) takes place. This conversion is accompanied by a reduction in pH and dissolved oxygen concentration, as well as the precipitation of y-FeOOH (lepidocrocite), which then ages to form a - F e 0 0 H (goethite). Changes in the pH and dissolved oxygen levels have obviously important implications for natural water chemistry, and ferric oxides and hydroxides are known to play an important role in determining trace metal ion chemistry through adsorption. In the well-known work by Stumm and Lee on Fe(II) oxidation ( 2 4 ) .the reaction rate was proportional to the product of [Fe(lI)],poz, and [OH-I2.
Recent work by Sung and Morgan (31) indicates that there are certain conditions under which the reaction is autocatalyzed by the surface of the precipitated Fe(1ll) product, but we will not consider this here. In the absence of a ligand and at constant pH andpo,, the time dependence of the ferrous iron concentration is the same as that for a simple first-order reaction. For this discussion, we will assume that a complexed form, FeL2+, is stable with respect to oxidation. Such ligand stabilization has been suggested as an explanation for the apparent presence of Fe(1l) in oxic solutions that contain relatively high levels of organic matter ( 3 2 ) . Indeed, Theis and Singer have found that ligands such as tannic acid and glutamine retard the oxidation of Fe(1l) at pH 6.3 and a poZof 0.5 atm ( 3 3 ) .(On the other hand, other ligands such as citric acid have been found by these same researchers to accelerate the oxidation rate. Because of the strong interest in the rate-decelerating effects of humiclike ligands, we will consider only the stabilization case here.) We may abbreviate this sequence of events with the following set of reactions: ki
FeL2+ eFez+ k-i
+L
( 17)
Reaction ( I 8) is assumed to be essentially irreversible. Since the formation constant for the first Fe(I1) hydroxo complex is only about 3.0 X IO4 M-I (34).even at pH 8.5, the concentration
FIGURE 3
The hydrolysis of tripolyphosphate (PPP) flml to pyrophoophata(PP) and orlhophosphate (P)
k. = 1.0 x
[email protected]' 2.3
X
10-E.-*
k, = 1.0
X
10-6 s-1
(b) k,
PPP
Time (d)
Time (d)
Volume 15. Number 10, October 1981
1159
of FeOH+ is relatively minor, and so we will assume that Fe2+ and FeL2+ are the only important Fe(I1) species. This leaves a total of four unknowns for any given point in time and set of initial conditions. With two massbalance equations, one for total Fe and one for total ligand, we require two additional equations: the differential equations for the system species. A differential equation may be written for each of the four species, but only sets of two will be independent of the two mass-balance equations. For the Fe2+ and FeL2+ pair, the two independent, coupled, nonlinear differential equations are:
d [ FeL2+] = -kl[FeL2*] dt k- [ Fe2+][ L]
+
(20) We will express mol/L of Fe(II1) produced as [Fe(III)], (It is important to note, however, that at neutral pH the [Fe(III)] may include precipitated ferric hydroxide in addition to dissolved ferric species.) The functionality of the oxidation term in (19) was studied by Stumm and Lee ( 2 4 ) .These authors report a value of 1.3 X 10l2 M-2 atm-l s-l for k2. The mass conservation equation for Fe is:
+
+
FeT = [Fe2+], [FeL2+], [Fe(III)lo = [Fe2+] [FeL2+] [Fe(III)] (21) Since Fe(II1) is not involved in any reactions as a reactant species, we may take [Fe(III)lo = 0 without loss of generality. Since FeT is a constant, by taking the derivative of (21) with respect to time, we see that: d [Fe(III)] =---d [Fe2+] dt dt d [ FeL2+] (22) dt The differential equation for [Fe(III)] may be obtained from (22) by substitution using (19) and (20). The second important mass-balance relationship for this system is:
+
LT = [L],
+
+ [FeL2+], = [L] + [FeL2+]
(23)
The differential equation for L may be derived using the derivative of (23) and (20). As with (22), it is not independent of equations (19, 20, 21, 23). Equation (23) allows us to eliminate [L] in (19) and (20), and we obtain: 1160 EnvironmentalScience & Technology
k-1[Fe2+](L~- [FeL2+]) and
k-1 [Fe2+] (LT - [FeL2+]) (25) Equations (24) and (25) are coupled, nonlinear differential equations. The nonlinearity is due to the mass balance on L and is embodied in the [Fe2+][FeL2+] cross-terms. Equations such as (24) and (25) are extremely difficult, if not impossible, to solve analytically in the general case-that is, with no simplifying assumptionsby presently available mathematical methods. It is far easier to integrate them numerically. We carried out such numerical integrations for a variety of initial conditions and rate constant values using a Runge-Kutta integration method (35). Provision was made for a continuous updating of the integration step size in response to the changing stiffness of the coupled differential equations. In all cases, equilibrium between Fe2+, FeL2+, and L was assumed at time zero. A constant pozof 0.2 atm was also assumed. The latter assumption is justified since this po2 corresponds to approximately 3.0 X M 0 2 and the highest concentration of Fe(I1) we will consider is 1.O X l 0-5 M . In order to exemplify the effects of varying the parameters kl, LT, and [OH-], the kinetic plots will be presented in sequences wherein one such parameter is varied and the others are held constant. Effect of pH. Because of the [OH-I2 term in equation (24), perhaps the most important parameter affecting the Fe(I1) oxidation rate is pH. This is evident in Figures 4a-d. For these calculations, pH ranged between 7 and 8.5 with LT = 5.0 X M ; FeT = M ; kl = 1.0 s-l; and k-l = 1.0 X lo6 M - k l . As seen in Figures 1 and 2, the kl and k-1 values are order-of-magnitude estimates for the dissociation and complexation of divalent metal ions with monodentate ligands. With simple complexation/ dissociation kinetics such as these, the stability constant K will then be equal to k-l/kl = IO6 M - l . In the case of more complex kinetics, however, especially those involving multidentate ligands, there is no simple equation relating the forward rate constants and the backward rate constants to the overall equilibrium constant. In Figure 4d, at a pH of 8.5, the
Fe2+ and FeL2+ drop precipitously, and the conversion to Fe(II1) is essentially complete within 10 s. At time zero, [Fe2+] is greater than [FeL2+]. This condition is quickly reversed since the free L grows rapidly as the total ferrous iron concentration diminishes. The [FeL2+]/[Fe2+] ratio stabilizes once [L] approaches LT. In Figure 4c, the reduction of the pH to 8 has a dramatic effect on the overall kinetics. Nevertheless, the oxidation is still complete within 1 min. As in Figure 4d, the rapid buildup of L leads to a greater proportion of the Fe(I1) being bound up as FeL2+. If the pH is reduced as low as 7 (Figure 4a), the oxidation of Fe(I1) becomes considerably slower and approximately 1 h is required to achieve a 90% conversion to Fe( 111). The presence of the ligand in this system leads to a reduction in the rate of Fe(I1) oxidation through the simple attenuation of the free Fe2+ concentration. Its presence can also hinder the oxidation as a consequence of a ratelimiting release of Fe2+ from FeL2+. Whether or not the second mechanism plays a role will be determined by the relative magnitudes of the individual rates at which Fe2+ is produced and consumed. The initial rate of production from FeL2+iskl[FeL2+], (-4.0 X M s-' in Figure 4). The initial rate of consumption by 0 2 is k2[Fe2+],[OH-]2p~, ( E 1.5 X 1O6[OH-I2 M s-l in Figure 4). (The Fe2+ complexation rate is not as important here since the net movement of Fe through the system is from FeL2+ to Fe(III).) At both pH 7 and 7.5 in Figure 4, the production rate is well above the oxidation consumption rate, not only at time zero, but also throughout the course of the reaction. This implies that the release of Fe2+ from FeL2+ will be able to keep pace with the Fe2+ oxidation rate, and therefore a state of very near equilibrium will be maintained between FeL2+, Fe2+, and L during the reactions depicted in Figures 4a and 4b. The role of Q. To demonstrate this phenomenon, we have also plotted the concentration quotient:
as a function of time. The value of Q remains essentially identical to K throughout the course of the reactions depicted in both Figures 4a and 4b. For pH values of 8 and 8.5, however, the initial values of the k2 term in equation (19) are 1.5 X M s-I and 1.5 X M s-l, respectively. These values are of the same order of magnitude as the initial release rate of
when it drops from its maximum value to (by the quadratic equation):
10-5
z .o -c
J(ki- a - k-ILT)2 + 4 k i k - i L ~ ~ ~ I L T
10-4
(30)
9
c m 0
5
10-1
1.0
1.0 10-7
0
4.0 X M s-l; therefore, we can anticipate that the FeL2+/Fe2+/L equilibrium will be disrupted once the Fe2+ oxidation is initiated. As is evident from the plots of Q in'Figures 4c and 4d,this is indeed the case, with the severity of the disruption being greater at pH 8.5 than at pH 8. In both Figures 4c and 4d. the degree of deviation of Q from K goes through a maximum and then stabilizes at a constant value above that of K. This is a most interesting observation. At time zero, the FeL2+/Fe2+/L system is a t equilibrium. Once oxidation begins, however, the balance between the k l and k-l terms is removed, the [Fe2+] is lowered below the equilibrium level; and Q begins to rise. As the [Fez+] falls, the net conversion rate of FeL2+ to Fe2+ begins to rise; [ FeL2+] decreases; and the rate of increase in Q begins to fall off. In this manner, a measure of balance is returned to the FeL2+/Fe2+/L system, and Q itself decreases, then stabilizes. The fact that Q does not decrease further and assume the value of K indicates that the oxidation of Fe2+ is continuing: Equilibrium cannot be attained as long as the total Fe(ll) is changing. This conclusion is a result of the fact that there is no final equi-
C X
0
(b) pH = 7.5
librium state as long as reaction (18) continues to be irreversible. The stacan then be bilization of Q at viewed as the attainment of a stable "kinetic state" wherein thecontinued supply of Fez+ from FeL2+ is due to the constant thermodynamic driving force:
AG
= -RT In Qstab -
(27) K The value of can be predicted by examining the derivative of Q with respect to time. By making use of the time derivatives of [Fe2+] and [FeL2+], as well as thedefinition of Q, we obtain:
a = -klQ + k-1 - k1Q2([L] + dt [Fe2+])
+ k-tQ([L] + [Fe2+])+
kz[OH-12~02Q (28) Once the oxidation has proceeded to the extent such that thevalue of [Fez+] has dropped substantially below that of [L], and [L] is approximatelyequal to LT. equation (28) reduces to:
a = -klQ + k-l dr
- klQ2LT +
k--IQLT + k2 [OH-I2po2Q (29) Q reaches a stable value (dQ/dr = 0)
where a = k2 [OH-I2 pop Only one root will be meaningful. Under the conditions extant in Figure 4c, equavalue of tion (30) predicts a 1.04 X Io6 M-I. After approximately 1 min, Q does indeed reach this value and hold. For the conditions used to calculate the results presented in Figure 4d, Qstab is 1.46 X 10' M-1. In view of the greater disruption caused by the value of the k2 term at pH 8.5 over that at 8, it is not surprising that (&tab is correspondingly larger at pH 8.5. We have discussed this concept of Q S a b largely within the context of Figures 4c and 4d. The same phenomenon would arise for Figures 4a and 4b if the calculations had been carried out to a point such that LT = [L] >> [Fe2+]. For Figures4a and4b. by equation (30). (&tab will assume valuesof 1.0004 X IO6 M-I and 1.004 X IO6 M-I, respectively. Small deviations from K such as these reflect the fact that only a small overabundance of FeL2+ is required to keep pace with the Fe2+ oxidation; as was concluded earlier, equilibrium may be assumed to be essentially maintained in these two cases. Effect of h.While the pH clearly plays a very important role in determining the kinetics of this system, other variables, such as LT. kl, and k-1, can also affect the kinetics markedly. The concentration profiles presented in Figure 5, calculated for pH 8 and three different values of LT. demonstrate the anticipated result that increasing levels of L stabilize the Fe(ll) (Le., retard the oxidation) by attenuating the [Fe2+]. Interestingly, this stabilization also.extends to the FeL2+/Fe2+/L equilibrium, because increasing the level of LT: increases the proportion of Fe(I1) bound as FeL2+, which therefore increases the Fe2+ release rate (h [FeL2+1) decreases the proportion of Fe(I1) present as Fez+ which decreases the oxidative consumption rate (k2[Fe2+1[OH-I2pot). These observations are corroborated by the behavior of the Q versus I plots in Figures Sa-c. Again, the asymptotically approached values of Qstab evident in these figures may be predicted Volume 15. Number 10. October 1981
1161
using equation (30). They are 1.14 X lo6, 1.04 X lo6, and 1.02 X 1O6M-I, respectively. Effect of kl. If all other parameters are held constant, increasing kl will accelerate the overall rate at which the conversion of Fe(I1) to Fe(II1) takes place. This is primarily a result of the fact that higher levels of Fe2+ are available for oxidation since increasing the magnitude of k l reduces the value of the FeL2+ stability constant. Three cases were analyzed, using pH 8 and kl values of 0.1, 1.0, and 10 s-l. The results of the calculations are presented in Figures 6a-c, respectively. As expected, the longevity of the Fe(I1) decreases as kl increases. We also note that the degree of equilibrium maintenance increases with increasing kl. On first analysis, this does not seem surprising. Indeed, for a specific [ FeL2+] value, increasing kl would be expected to promote equilibrium. However, increasing kl with constant k-1 and LT tends to decrease the value of [FeLZ+] and increase [Fe2+], and our discussion of the implications of varying LT concluded that such effects tend to hamper the attainment of equilibrium. We can improve our understanding of these counterbalancing effects by examining the initial conditions in Figures 6a-c. Since the quantity kl[FeL2+lo / k2[Fe2+lop~2[OH-12 takes on the values 0.37,2.85, and 10.8 in Figures 6a-c, respectively, it is clear why the overall effect of increasing kl becomes one of improving the degree of equilibrium: kl increases faster than [FeL2+]/ [ Fe2+] decreases. But why does this occur? First, we note that although perfect equilibrium is never attained, in these cases Q generally remains close to K = k-l/kl. Therefore, for constant k-1 and LT, increasing kl causes a decrease in [FeL2+]/[Fe2+][L] and so the decrease in Q is spread over a decrease in [FeL2+]/[Fe2+]and 1/[L], since [L] must increase as K decreases. The Qstab values predicted using equation (30) are achieved within the 1-min time frame over which the integrations were carried out for both Figures 6b and c. However, for Figure 6a, in which kl was 0.1 s-l, considerably more than 1 min is required to achieve the predicted Qstab value of 1.05 x 107 M-1. We have carried out a similar set of calculations at pH 7 and these results are presented in Figure 7. Because the term k2[OH-I2po2 is so small at pH 7, near-perfect equilibrium is maintained throughout the course of the reactions in Figure 7. Similarly, the Qstabvalues for Figures 7a-c are almost identical 1162
Environmental Science & Technology
to K. As expected, however, varying kl, and therefore K, nevertheless affects the Fe(I1) oxidation kinetics substantially by lowering the free Fe2+ concentration. Conclusions. The results of varying three parameters-pH, LT, and kl-were studied. The Fe(I1) oxidation rate is most sensitive to changes in the [OH-]. Changing the pH from 7 to 8.5 effects a dramatic acceleration in the oxidation rate, even in the presence of a ligand, L, which attenuates the concentration of the free, oxidizable Fe2+.Because the oxidation of the Fe2+drives the FeL2+/Fe2+/L system out of equilibrium, it is natural that the degree of equilibrium (i.e., closeness of Q to K) is also most sensitive to the PH. Since reaction (18) has been assumed to be irreversible (i.e., that there are no back reactions from Fe(III)), as we have described this system, no level of Fe(I1) is stable. In every case, the consumption of Fe(I1) must continue ad infinitum; the value of Q asymptotically approaches that of Qstab; and the thermodynamic driving force for the reaction becomes constant. In the long term, though, we know that a state of.dynamic equilibrium must be reached between the Fe(II), Fe(III), and the ligand L, and the value of Q must reassume that of K. In terms of the analyses presented in this section, however, the equilibrium state for each case will be reached only after considerably more than 1 min. For equilibrium in Figure 4d, for example, the [Fe2+] must be reduced to approximately M . Thus, the abovementioned contradictions will not interfere with the conclusions we have reached in this section, and the assumption of irreversibility remains appropriate for most time scales of interest. In certain cases, then, Q will remain close to Qstabfor relatively long periods of time. If the total Fe(I1) remaining at any given time is known, then Qstab may be used to calculate the relative proportions of Fe2+ and FeL2+. However, since the numerical integration that would provide the Fe(II)T value would also provide the [Fe2+] and [FeL2+] values anyway, Qstab holds no special utility in this regard. In these and other cases, however, the essential arrival of Q at Qstab will ease the difficulty of the integration considerably. Consider the time derivative of lnQstab: dlnQstab - dln[FeL2+l dt dt dln[Fe2+1 --dln[LI dt dt
-0
(31)
Once the [L] value becomes stabilized at LT: dln[L] -dt
-0
and therefore: dln[FeL2+] - dln[Fe2+] (32) dt dt Equation (32) is quite important and implies that the slopes of the log [FeL2+] and log [Fe2+] versus time curves will become parallel once Q attains Qstab. Further, under these conditions:
Consequently, the two slopes become not only equal but constant as well-results which simplify the integration process considerably once Qstab has been attained. (Incidentally, equation (30) may be deduced from equations (32-34).) An examination of the several figures presented in this section will confirm these conclusions. Of course, when very little equilibrium disruption occurs and Q remains essentially equal to K, the same equations apply once [L] approaches LT.
Overall conclusions As the data in Figures 1 and 2 have shown, the velocities at which different aqueous-phase reactions proceed vary over many orders of magnitude. In the case of second- and higher-order reactions, the reaction kinetics can be affected dramatically by varying the concentrations of the species appearing in the rate law. As long as one is concerned with just one irreversible reaction, an analytical solution to a given problem is generally available. When coupled reactions involving opposing steps are of interest, the attendant mathematics quickly become sufficiently complex to require numerical integration methods. Whether the opposing reactions within a given kinetic system will approximate equilibrium as the whole system moves toward the overall equilibrium state specified by the thermodynamics of the particular case is a matter of relative reaction rates. If a reaction that is coupled to a pair of opposing reactions is appreciably
.m. .m.
0 ._
1.5 0 3
C
E
s
D
',lo-'
,.o
10-7
10-7
c
1.0
1
2
-m c m c
0
10~'
10~8
s
10-8
FIGURE6
The oxidation of Fe(ll) in the presenceof ligand at pH 8.0 and varying k, *
Time (s)
FIGURE 7
The oxidation of Fe(1l) in the presence of ligand at pH 7.0 and varying k, * 10-61
F#+
!*;:I
10-6:
15.0
10-8
1
0
(a) k, = 0.1
I-'
1 t
10~0
10
20
30
40
50
60
0
10
20
30
40
50 Time (s)
Volume 15. Number 10, October 1981
1163
The Pesticide Chemist and Modern Toxicology
ACS Symposium Series 160 S. Kris Bandal, Editor 3M Company Gin0 J . Marco. Editor Ciba-Geigy Corporation Leon Golberg. Editor Chemical Industry Institute of Toxicology Marguerite L. Leng, Editor The Dow Chemical Company Basedon a symposiumsponsoredbythe ACS Division of Pesticide chemistry of rhe American Chemical Society.
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faster than the balancing reaction within the pair, the pair must be displaced appreciably from equilibrium. The converse is also true. An examination of the concentration quotient(s) during the course of a given reaction illustrates the degree of the disruption(s). Under certain conditions, which were discussed in detail for one particular reaction system, several assumptions involving Q can occasionally greatly simplify the integration. Two reaction systems were examined in this paper, the hydrolysis of tripolyphosphate and the oxidation of Fe(I1) in the presence of ligand. Part 2 will provide examples from three different reaction systems, the autocatalytic oxidation of Mn(I1) in both the presence and the absence of ligand, and the adsorption of a metal ion on the surface of suspended particles while also in the presence of a ligand. The conclusions drawn from all of these cases will then be used to make a number of general statements concerning the question of when kinetics or equilibrium modeling of natural water chemistry phenomena is more appropriate. References
(18) C l ~ r i . N . L . : L a . G . F . I n l . J . A i r W a t e ~ Pollut. 1965.4: 743-751. (19) King. J. How Chemical Reactions Occur”: Benjamin: Menlo Park, Calif.,
.,”-. ,O&.d
(20) Bader, R. G.. Narraganwtt Marine Lab.. UniversityolRhode IslandOaas. Publ. 1962 1.42. (21) Holmes. L. P.:Cole.D.:Eyring.E.J. Phys. C‘hem. 1968.72,301-304. (22) Korytd. J. Z. Elecktrochem. 1960, 64. 196199~ ... .. . (23) Goodridge. F.; Taylor. M. D. Trans. ’ Faradav Soc. 1963 59.2868-2874. (24) Slumm, W.: Lee, G.F. Ind. Eng. Chem. 1961.53. 143-146. (25) Pouli. F.;Smith. W. MacF.Can. J.Chem. 1960.38.567-575, (26) Mills. G . A.: Urey. H.C. J. Am. Chem. Sac. 1940,62.1019-1026. (27) Morgan. J. J. Ph.D. Thais. Harvard. Cambridge. Mass.. 1964. (28) Cline. J. D.; Richards. F.A. Enuimn.Sci. Techno!. 1969.3.838.843. (29) Hoflman. M. R. Emiron. Sci. Technol. 1981. 15.,~ 345-353. (30) Shannon. J. E.; Lee, G . F. Air Water Polirt. Int. J. 1966, 10.735-756. (31) Sung. W.: Morgan, J. J. Environ. Sci. Terhnol. 1980. 14,561-568. (32) Theis. T. L.: Singer. P. C. I n “Trace Metals and Metal-Organic Interactions in Natural Waters”: P. Singer. Ed.; Ann Arbor Sci. Publ.: Ann Arbor. Mich., 1973. (33) Theis. T. L.: Singer, P. C. Emiron. Sci. Terhnol. 19’14.8.569-573. (34) Smith. R. M.: Martell. A. E. “Criiical Stability Constants”: Plenum Press: New York. 1976: VoI. 4. (35) Ralslon. A,; Will. H. S. “Mathematical Methods for Digital Computers”: WileyInterscience:New York. 1960; Vol. 1. ~
~~~~
e.
( I ) Sill€n. L. C. (a) I n “Oceanography”; M. Sears, Ed.; Am. As~oc. Advance. si.:
Washington, D.C.. 1961. (b) In “Equilibrium Concepts in Natural Water Systems”; R. F. Could.. Ed.;Adv. Chem. Series No. 67. Am. Chem.Soc.: Washington. D.C., 1967. (2) Morel. F.; Morgan. J. J. Emiron. Sci. Terhnol. IW2,6,58-67. (3) Westall, J. C.: Zachary. J. L.: Morel. F. “MINEQL A Computer Program for the Calculation of the Chemical Equilibrium Campition of Aquatic Systems”; Mass. Inst Technol. Dept. of Civil Engineering.: Tech. Note No. 18. Acquisition No, T-76-6, 1976. (4) Frost, A. A,: Pearson. R. G . “Kinetics and Mechanism: A Study of Homogeneous Chemical Reactions,” 2nd ed.: WiIey: New York. 1961. ( 5 ) Benson, S. W. ‘The Foundations of Chemical Kinetics”: McGraw-Hill: New York, 1960. (6) Brezonik, P. L. I n “Waste and Water Pollution Handbmk”; L. L.Ciaccio, Ed.:Marcell Dekker New York. 1974: Vol. 3. (7) Hayes. R. G.; Myers. R. J. J. Chem. Phys. 1964.40.877.882. (8) Hemma, P.. et al. 1 .Phys. Chcm. 1971.75, 929-932 (9) kich,-L. D.: Cole, D. L.: Eyring. E. M. J. Phy.s. Chem. 1969.73.713-716, (10) Eigen. M.: Kustin. K. J. Am. Chem. Sm. 1961.82.5952-5953. ( 1 1 ) Rabenstein. D.; Kula. R. 1.Am. Chem. Soc. 1969.91.2492-2503. (12) Hammes. G. C.; Morell. G. C. 1 .Am. Chrm. Sac. 1964.86, 1497-1502. (13) Roughton, F. J. W. J. Am. Chem. Sm. 1941.63.2930-2934. (14) Scheurer. P. 1.: Brownwell. R.; LaValle, J . J. Phys. Chem. 1958,62,809-812. ( I S ) Teggins. J. E.; Milburn. R. M. Inorg. Chem. 1964.3.364-368, (16) Hammes, J. C.: Stcinfeld. J. 1. J. Am. Chem. Sor. 196284,4639-4643. (17) Welch, M. 1.; Lilton, J. F.; Seck, J. A. J. Phys. Chem. 1969.73,3351-3356.
Dr. James F. Pankow (left) is anassistant professor of enoronmentai science at the Oregon Graduate Center. His research interests include rhe equilibrium and k i netic aspcrs of natural wafer chemisrry as well as the transport of organic pollutants rhrough the environment. He is especially inrerested i n rhe mechanisms by which precipitarion scawnges organic pollutants, the rransporr of organic ollutanrs across rhe airlsea interface. a n i t he development of new techniquesfor the analysis ofrrace organic compounds. He received his B.A. i n chemistry in 1973 and his Ph.D. i n environmental engineering sciencefrom the California Inslirute of Technology i n 1978.
Dr.James J. Morgan (right) is professor of environmenlal engineerin science and vice president for student a j a i r s ai CalTech. H i s work is i n rhefield of aquatic chemistry and surface chemistry i n wafer Ireatmenr. His major research interesrs involve solurion and interfacial processes governing the rransport and removal of aquaricpollulanrs. He was thefirsr ediror of ES&T.
