Thermodynamic, Kinetic, and Extrathermodynamic Considerations in the Development of Equilibrium Models for Aquatic Systems Michael R. Hoffmann"? Environmental Engineering Program, Department of Civil and Mineral Engineering, University of Minnesota, Minneapolis, Minnesota 55455
Thermodynamic and kinetic criteria are examined for the general application of equilibrium models to dynamic natural waters. The hypolimnion of a dimictic lake is used as a specific example. The majority of chemical reactions given consideration in equilibrium models are shown to be kinetically rapid in comparison to the chemical residence times in hypolimnetic waters. Extrathermodynamic relationships are explored as a method for extrapolating kinetic and thermodynamic constants from a limited set of experimental data.
applicability of the law of microscopic reversibility has been demonstrated experimentally (8). From eq 1, it is apparent that the equilibrium constant determines the ratio of rate constants, but it does not give any information about their absolute values. This relationship can be used in various forms to establish appropriate conditions for the application of equilibrium models for the description of chemical phenomena in dynamic systems. O p e n S y s t e m s and Kinetic Considerations
Introduction
Conceptually, development of computer-generated equilibrium models appears to be relatively straightforward (1). However, as Morgan ( 2 )points out, a closed-system equilibrium model may be a poor approximation for a specific natural water system because of flows of matter and energy in and out of the system, In addition, application of steady-state approximations may inadequately reflect the state of the thermodynamic equilibrium, although, for some well-defined systems such as sediment-water interfaces, the concepts of local equilibrium or steady state may be valid. Morgan (2) and Jenne ( 3 )have cited previously the limitations of equilibrium models to adequately characterize the chemistry of natural waters. The major objections to the application of an equilibrium model to a real system are, first, that the equilibrium approximation of a dynamic system is basically a poor approximation and, second, that a system which is thermodynamically favorable may be kinetically hindered (i.e., the relationship between thermodynamics and kinetics is thought commonly to be tenuous and unpredictable). In this paper, thermodynamic and kinetic criteria will be explored for the application of equilibrium models to dynamic natural waters with specific reference to a system defined by the hypolimnion of a dimictic lake.
For a natural system such as a dimictic, eutrophic lake which is open to its environment, energy and matter continually flow in and out. In addition to chemical transformation, transport, mixing, and biological processes are often superimposed. As a first approximation, a lake or a well-defined portion of the lake such as the hypolimnion can be treated as a continuous-flow stirred-tank reactor (CFSTR) in which biological processes are neglected. Under these conditions, for a simple first-order chemical transformation
Closed S y s t e m s and Thermodynamic Considerations
Solution of a chemical equilibrium problem can be readily achieved by solving a system of mass-action equations ( 4 5 ) defined by appropriate stoichiometric coefficients and equilibrium constants (6). An alternative to the mass-action notion of the equilibrium condition is to consider the rate of approach to equilibrium. By definition, a t equilibrium the rate of the forward reaction is equal to the rate of the reverse reaction with no observable change in concentration. For an idealized case in which the stoichiometric coefficients in a chemical reaction correspond to the kinetic reaction orders, the reaction rate equality a t equilibrium gives the law of microscopic reversibility (7,8) K = kf/k, (1) where kf and k, are the rate constants for the forward and reverse reactions, and K is the equilibrium constant. The
+ Current address: W. M. Keck Laboratories, Environment Engineering Science, California Institute of Technology, Pasadena, CA 91125.
0013-936X/81/0915-0345$01.25/0 @ 1981 American Chemical Society
kf
A e B
(2)
kr
taking place in a completely mixed volume, V ,with a volume rate of flow, Q,the steady-state solution to the mass-balance equation for the [A] in the system and the effluent is given in eq 3 (2,9). [A]/[B] = 1/K
+ l/(ln 27,/71/2)
(3)
Equation 3 shows that, for simple first-order processes in a well-mixed volume, if the hydraulic residence time, T,, is sufficiently large with respect to the reaction half-life, 7112, the steady-state condition approaches chemical equilibrium or, in other words, that the ratio of [B] to [A] is given by K. Unfortunately, very few chemical reactions of interest in natural systems follow simple first-order kinetics, although under certain conditions a pseudo-first-order approximation for many higher-order reactions is valid (10). However, reactions such as aquation or hydrolysis of transition-metal complexes frequently follow first-order rate laws. ML5X
+ H20 3ML50H2 + X
(4)
kr
Empirically observed rate laws for aquation as written in eq 4 are of the following form: -d[MLbX]/dt = kf[MLbX] (5) where L is either a substituted ligand or H2O of the primary coordination sphere for an octahedral complex and X is a ligand which is replaced by water in the coordination sphere. When both L and X are HzO, the process is known as water exchange in the primary coordination sphere. Listed in Table I are a few representative aquation reactions and other firstorder processes and their corresponding rate constants. The rate constants range from to 1O1O s-l with corresponding half-lives from lo4 to s. Extended tables of data are available as supplementary material. (See paragraph a t end of text regarding supplementary material.) For development of an equilibrium model of the sediment-water interface in the hypolimnion of a dimictic, euVolume 15, Number 3, March 1981
345
trophic lake during the period of summer stratification, the chemical residence time, 7r, can be defined as the difference in time from the onset of thermal stratification after spring turnover to the end of stratification at fall turnover. From Delfino's ( 1 1 ) temperature profile data for Lake Mendota during 1967, the time difference between the beginning and the end of stratification gives a 7, value of 1.2 X IO7s. The ratio of T r / 7 1 / 2 for the reactions listed in Table I range from l o 3 to These ratios are sufficiently large to cause the second term in eq 3 to become negligible with respect to the first term for most reactions. Anoxic dissolution of FeS, which is potentially a dominant phase in Lake Mendota sediments, has a pseudo-first-order rate constant of sui (12a) and a (12b). These values give corresponding K,, of 7.9 X values of 1/K = loi8 and 71/2/ln 27, = which clearly show that, for the dissolution of FeS under the time-invariant condition for Lake Mendota, the ratio of products to reactants can be approximated satisfactorily by the equilibrium constant. However, the majority of aquatic chemical processes that are amenable to equilibrium modeling, such as complexation, autoxidation, redox electron transfer, and acid-base proton transfer, are second-order kinetic reactions. For example, the commonly accepted mechanism ( 1 3 ) for ligand substitution reactions (also known as either anation or complexation) is the following two-step mechanism M(H~O)P
+~
substitution into eq 8 gives d[ML]ldt = kzKo[M][L] (10) where k2 >> k-2 and k2' = k2Ko, Assuming that complexation can be represented mathematically as a one-step process according to eq 11
M(H~o)+ ~ ~~ +2 & - M ( H ~ o ) ~ L+ H ~ O k-1
one can write the mass balance rate expressions for a continuous-flow system with a completely mixed volume, V, and a volume rate of flow, Q, as d[M]/dt = (Q/V)[M]o - (Q/V)[MI - hi[M][L] + h-i[ML] (12) d[L]/dt = (Q/V)[L]o - (Q/V)[LI - ki[M][LI + h-l[ML] (13) d[ML]/dt = (Q/V)[ML]o- (Q/V)[ML] + ki[MI[Ll - k-i[ML]
3( M ( H ~ o ) ~ ~ + : L ~ - ) (6)
kz k-2
(14)
for a system in which M, L, and ML are introduced at concentration [M]o, [Llo, and [ML]o and for which M, L, and ML are continuously withdrawn at concentrations of [MI, [L], and [ML].The steady-state condition for M, L, and ML is defined by d[M]ldt = d[L]/dt = d[ML]/dt = 0
2 -
1M(Hz0)s2+:L2-).M(Hz0)5L
(11)
(15)
Setting eq 14 equal to zero and rearranging gives
+ HzO
(7)
in which eq 6 is actually subdivided into two steps. The first part of the process represented by eq 6 involves a diffusioncontrolled formation of a weak ion pair with at least two intervening water molecules; the second part of the first process is the slightly slower formation of a stronger ion pair with only one intervening water molecule. Equation 7 represents the formation of the final complex in which expulsion of a coor: dinated water molecule is rate determining. The experimental rate law for this process is first order in each reactant. A similar bimolecular rate law can be derived from the mechanism presented in eq 6 and 7 , starting with the rate-determining step in the mechanism. d [ML]/dt = k 2 [{M:L)]
(8)
and subsequent successive division of each side of eq 16 by [ML] results in eq 17.
When one uses the law of microscopic reversibility, K = kl/k-l, and the assumption that the influent [ML]o = 0, eq 17 can be written as
(18) which can be reduced further to a form similar to eq 3 when [MJo = [LJoand 7, = (V/&).
If the concentration for the ion pair is given by
-[MLI (9)
[(M:LII = KoWI [LI
where K O is an equilibrium constant for ion pairing, then
[MI[LI
7 3
Tr
+ K~I/Z[M]O
(19)
For a second-order reaction the half-life expression, 71/2 =
Table 1. First-Order Reaction Rate Constants, k , , for Aquation or Hydrolysis Reactions with a General M(H20) 4- L ( k l ) and Rate Law Y = kl[M] Stoichiometry M 4-H 2 0 kl, 8-'
4.4 x 8.8X 1.1 3.1 x 3.2 X 3.0 x 8.0 x 3.0x 1.3 6.6 X 5.3 x 1.0 x
Kh5* a
IO-'
5.0 X
107
1.o
IO6
1 .o 1 .o 1 .o 1 .o 1.0 x 2.6X 1.0x 9.0 x
104
109 103
10-2 103
Environmental Science & Technology
1, "C
a
lo-" lo-'*
0.4 1 .o 0.1 0.1 0.1 0.1 0.1 0.5 0.5
10-8 10-2
1.o
Kh is the equilibrium constant for aquation/hydrolysis as written from left to right.
346
Ir, M
10-4
14.5 8.4 25 25 25 25 25 25 25 25 20
pn
re1
9-8 1.5 1.7
51 52 53 54 54 54 53 55 56 56 57 58
1
1 1 1 1 6-8 7-8 7 5
Table II. Second-Order Rate Constants, k2, for Redox Reactions/Electron Transfers That Obey Rate Laws of the Form Y = k2[OX][RED] reactlon
+
Fe(phen)s2+ C102Fe3+ penicillamine Fe(phen)s2+-I-HOC1 Fe3+ ascorbate Fe3+ I-
+
+
+
Fe(phen)s3+ SOs2Cu2+ 4- ascorbate Mn3+ thiourea CU(DMP)~+ H 2 0 2
+ + Fe2++ benzoquinone
k z V aM-’ 8-1
3.0 x 105 4.5 x lo-’ 2.2 x 10-2 30 16 4.2 3.1x 104 2.6x 104 1.2 x 106 9.1 x 10-4
K:’
(AGO = - n F E o )
8.0X
loi6
6.8X 10” 6.0 x 107 6.2 x 1032 6.9 x 1064 2.1 x 1054 4.1x 10-3
Ir, M
T, O C
PH
ref
1 .o 0.1 1 .o 1 .o 0.2 0.5 0.1
20 20 25 0 25 25 30 25 25 25
9 9 7.2 1 1 1 4-6 0 6-9 0
60 61 62 63 64 65 66 67 68 69
0.1 0.5
[OX] and [RED] refer to the concentrations of the oxidant and reductant, respectively. K, is the equilibrium constant obtained from standard redox poten-
tials.
Table 111. Second-Order Reaction Rate Constants, k2, for Complexation/Ligand Substitution with the General 4-H 2 0 ( k 2 ) , and Observed Rate Law v = k2[ML(H20)6”+] Stoichiometry M( H20)6n+ L m- + ML( H20)5n[L “-1
+
reactlon
kp, M-’
s-’
+ oxalate
1.4x 6.4 x 1.2x 1.4x 1.0 x 7.9 x 5.0 X
103 103
Fe3+ Fe3+ Fe3+ Fe3+ Fez+ 4Fez+
+ sulfate + sulphosalicylate + salicylate sulfate + phen
Mn2+ 4- NTA3Mn2+ 4- HNTA2Mn2+ sulfate Mn2+ chloride CU” H2ATP2-
+ + + + + +
serinate co2+ a-alanine Co2+ histidine
cu2+
a
2.0
104
104 106 105
lo*
x 105
4.0X 1.6x 8.8 x 1.8x 1.3 X 1.3x
lo6 107 108 109
loB 107
Kf*’ a ( p = 0 )
P, M
T, o c
PH
ref
5.5 x 107 1.6 x 104 2.6 x 1014 2.8 x 1017 2.0 x 102 7.0 x 105 3.7 x 108 1.7X lo-* 1.8 x lo2 1.1 ( p = 1.0)
0.5 0.5 1.0 1 .o 0.1 1 .o
25 25 25 25 20 25 25 25 20 20 25 25 20 20
0.5 1.3 2 2 1 1
71 72 73 73 74 75 76 76 73 77 78 79 80 80
8.9x 108 5.3 x 104 7.9 x 106
0.1 0.1 0.1 0.1 0.1 0.1
1 1 1-5 1-5 9 9
Kfis the overall formation constant for the complex.
l/(k1[M]o), used above is only valid for the condition of initial equal concentrations of reactants M and L. Equation 19 clearly shows that, if the residence time is large with respect to the product of 7112 times K times the initial reactant concentration, then the steady-state condition for a well-mixed volume approaches equilibrium and concentrations of components can be determined from equilibrium computations. Representative kinetic data presented in Tables 11-IV and VI can be used to test the validity of the steady-state approximation for natural water systems with eq 19. Unlike thermodynamic constants, kinetic constants for reactions of interest in natural waters are not readily available in convenient compendia. Moreover, many of the reactions involving transition metals have been studied in highly acidic medium (Le., 1M HClOI) to avoid problems encountered with metal hydrolysis and subsequent identification of the principal reactive species in solution. Other reactions have been studied over very narrow ranges of pH so that the exact pH dependence of a particular reaction may not be ascertained. Furthermore, extrapolation of data to pH ranges of interest in natural water systems involves a high degree of uncertainty, although some trends in reaction rates are predictable. For example, many of the second-order rate constants given in Table I11 for complexation reactions have been determined at low pH to avoid interference from hydrolysis and subsequent polymerization of the metal ion; but, in general, as a
metal undergoes hydrolysis to give hydroxy complexes, its primary coordination sphere of HzO is labilized (Le., as measured by the water exchange rate) and subsequent ligand substitution occurs more rapidly. This increased rate of complexation is consistent with the mechanism given in eq 6 and 7 in which the rate-determining step is the expulsion of a coordinated water molecule. Other reactions such as acidbase proton transfers are uniformly rapid across a wide range of pH, as can be seen in Table V. Kinetic data over a broad range of conditions have been compiled and presented in Tables I-VI1 for reactions of interest in the development of thermodynamic models for aquatic systems. The principal source of kinetic data is the primary literature, although frequent review articles and books such as ref 14 are useful starting points. In development of any equilibrium model, which will be used to describe chemical distributions in a natural water, it is necessary to determine that the reactions under consideration occur within a reasonable time frame compared to the residence time of a particular body of water, especially when the formalism of “local equilibrium” is not readily applicable. The study of reaction rates applicable to natural waters is further complicated by the fact that many reactions of interest are sensitive to catalytic effects such as trace-metal, specific acid and base, surface, general acid and base, nucleophilic, and photolytic catalysis. In addition, in natural waters microbiological processes play important roles in the catalysis of nuVolume 15,Number 3,March 1981 347
merous reactions. For example, the autoxidation of sulfide (Table VII) has been studied by a number of investigators under slightly different experimental conditions in artificial and natural waters. From comparison of the reported halflives as measured by different investigators, it is obvious that the reaction kinetics are complicated and that the reaction is sensitive to a variety of cata1y;tic influences such as trace-metal catalysis (15, 16), general base catalysis (15), microbial catalysis (171,and surface catalysis (18).The reported half-lives vary from days to minutes under similar concentration conditions, and reported kinetic orders vary from one investigator to another. Nevertheless, given the uncertainty involved in applying laboratory kinetic data to environmental situations, data such as that included in Tables I-VI1 serve as a useful starting point to verify the applicability of equilibrium models for studying complex chemical phenomena in dynamic natural waters. In the specific case of hypolimnetic waters, where a typical chemical residence time has been estimated to be lo7 s ( 1 1 ) , most of the important complexation reactions and protontransfer reactions listed in Tables I11 and IV have secondorder rate constants in the general range of 103-1010 M-l s-l. For metals such as Cu and Ni with typical freshwater concentrations in the range of 10-7-10-5 M (19) and with second-order formation rate constants in the range of 103-109 M-l s-l, reaction half-lives will vary from an upper limit of lo4 s to a lower limit of s, assuming that there is a suitable complexing ligand available in a similar concentration range. From these half-lives it is clear that, for most complexation ] o that the reactions under consideration, T , >> T ~ / ~ K [ Mand steady-state approximation for these reactions is valid. Likewise, proton-transfer processes, which are important in acid-base equilibria, will have half-lives in the range of 10-7-10-3 s for acid-base concentrations in the range of ]o M. In these cases T , is significantly greater than T ~ / ~ K [ Mfor proton transfer. Therefore, it is safe to assume on this basis that the steady-state approximation is valid for acid-base
equilibria in hypolimnetic waters. Problems arise in the application of the steady-state condition for reactions that are known to be relatively slow in the absence of catalytic influences. Typical reactions (20) that are "slow" on a relevant time scale for aquatic environments are certain metal-ion oxidations (e.g., Cr(II1) to Cr(VI), sulfide oxidations, sulfate reduction, precipitation of metal silicates and carbonates (e.g., dolomites), and the dissolution of some sulfide minerals (e.g., CuFeS2). The electron-transfer and redox reactions listed in Table IV have second-order rate constants which range from 10-4 to lo7 M-l s-l with a mean value of ca. lo2.For redox reactions at the lower end of the range with reactant concentrations of M, 71/2 = 1X lo9 s. In these cases T , < T&[M]o and the steady-state approximation would be invalid. On the other hand, redox reactions listed in Table VI11 with small AGO values (Le. no. 9) and moderate hf values will meet the conditions T , >> T~/&[M]o.In the case of redox reactions, care must be exerted in selection of the appropriate reactions for inclusion in an equilibrium model. Reactions that are known to be slow or borderline should be deleted, a t least during initial development. For example, the autoxidation of Mn(II), which is an autocatalytic reaction for which the reaction rate is proportional to the square of the hydroxide ion concentration, has been shown to be extremely slow a t the p H and the temperature of seawater (20,211. These results suggest that the Mn(I1)-Mn(1V) redox couple may be unimportant in natural waters with lower p H conditions (i.e., p H 18). Although the data presented in Tables I-VI1 are far from complete, they represent a first-step in accumulation of kinetic data that may have significant implications for understanding a little-known aspect of aquatic chemistry. Many reactions of interest for natural systems have not been included and many other reactions have not been studied under conditions representative of natural waters. Until that time in the future when most reactions of interest have been studied kinetically, it may be useful to explore the applicability of extrathermo-
+
Table IV. Second-Order Reaction Rate Constants, k2, for Proton Transfer, HA B + HB 4-A, Where HA Is a Suitable Proton Donor and B a Suitable Proton Acceptor and the Rate Law Has the General Form v = k2"I[Bl reaction k2, M-I s-l c. M T, PH ref KpS4a ( p = 0 ) O C
+ OH+ OHC6HsOH + OHH30+ + HSH30+ + NH3 + HC03H3O+ + CHaCO2-
1.4 X 3.4 x 1.3 X 7.5 x 4.0 X 5.0 X 3.2 X 5.0 X
H30+
"4'
~ 3 0 +
+ HCO2-
CH3C02H a
9.9 x 1013 5.6x 104 1.0 x 104
io1' 1010
loio
7.9 x 1.7 x 2.3 X 5.7 x 9.7 x
1010 loio IOio
loio lo6
1073
109
lo6 104
10-2
0
25
0
20
0 0 0 0 0 1
25 25 25 25 25 20
15.75 9.25 4.20 7.24 9.25 3.77 4.20 3.75
82 a2 82 82 82 82 a2 83
Kp refers to the equilibrium association constants for the reactions as written.
Table V. Observed Second-Order Reaction Rate Constants, k2, for Conditions of Constant p and pH for Autoxidatlons having the General Stoichiometry RED O2 -* OX H20 and Rate Laws v = k2[RED][O2]
+
reaction
+
+
k2. M-'
+
H+ Fez+ '/402 -,Fe3+ %H20 Mn2+ H20 1/20z Mn02 2H+ Co(hist)22- O2
+ +
+
Cu(dipy)Z+ 4- 0 Fe(cyst) O2 HSa
+
+
+
+
2
0 2
s-l
9.6 X 10-1 9.1 x 10-2 3.5 x 103 6.5 x 103 5.0 x 103
1.0x 10-1
KO is the equilibrium constant obtained from standard electrode potentials.
348
Environmental Science & Technology
+
Koa
(AO"
-nFE0)
1.3x 1037 2.7
x
1030
-1.0 x 1066 -1 .o x 1042 3.5 x 10'6 Pseudo-second-order constants.
T , OC
20 25 25 25 25 25
PH
7 9
8-1 1 5
re1
a5 86 87
aa
6
89
12
90
Table VI. Second-Order Rate Constants, k2, for Complexation/Ligand Substitution Reactions of Ni( H20)62+ with the Stoichiometry Ni(ll) 4- L + Ni(II)L -k H20 and the Rate Law v = k2[Ni(ll)][L] kz,
”;’
M
no.
llgand
S-
log KfB4a
P7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
histidine phenanthroline
2.2 x 103 4.1 103 1.6 103 5.3 x 103 4.7 x 103 3.4 104 2.0 x 104 7.6 x 103 2.0 x 105 2.6 x 105 2.6 x 104 2.7 104 1.6 x 104 7.9 x 104 3.2 x 103 4.6 x 103 5 . o x 103 -1 10-3 -1 x 10-3 1.2 x 10-1 8.5 X 10’
8.7 8.6 7.0 7.0 6.4 5.9 5.4 3.2 3.2 2.2 1.6 1.5 1.o 52 3.0 1.9 1.2 5.2
0.1 0.1
a
x x
dipyridyl
salicylicate sulfosalylicate proline a-alanine ethyl malonate
x
HEDTA3-
phthalate lactate acetate sulfate oxalate imidazole pyridine thiocyanate methionine glutathione rn-tyrosine cysteine
x
x
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0 0 0.1 0.1
0.1 0.15 0.15 0.15 0.15
5.1 9.8
T, OC
PH
ref
24 25
5-7 6-8
25 20 25 25 25 25 25 25 25 24 25 25 24 24 24 24
6-8 9 3-5
92 93 94 95 95 96 97 98 99 98 100 101 102 103 104 105 106 107 107 107 107
4-5 5.8 3-7 7 6 7 7 6-7 6-7 6-7 6-7
K, is the corresponding equilibrium formation constant.
Table VII. Comparison of Kinetic Results Observed for the Autoxidation of Aqueous Sulfide under Roughly Similar Conditionse no.
1 2
3 4 5 6 7 8 9 10 11
reactlon
HS-+02 HS-+02 HS-+02 HS-4-02 HS-+02 HS-+02 HS-+02 HS-4-02 HS-4-02 HS-+ 02 HS- 4- 0 2
+ CoTSPd
water
PH
t112,’ min
sea water distilled sea water sea water fresh distilled fresh fresh sea water distilled distilled
8.0 7.9 8.2 7.8 7.6 11.0 8.6 8.0 7.7 8.7 8.3
280 3000 24 175 880 570 2200 114 600 2220 7
T,
O C
23.0 25.0 25.0 9.8 25.0 25.0 25.0 25.0 9.0 25.0 25.0
[HS-] n
[021
1 1.34 1 1 1 1 1 1
0.56 1 1 1 1
1 1
1 1
a Calculated from reported data. EDTA added. Simulated. CoTSP = Co(ll) 4,4‘,4”,4”’-tetrasulfophthalocyanine, [CoTSP] = 2 X been summarized in terms of the calculated reaction half-lives.
dynamic relationships in the form of linear free-energy relationships (LFERs) for predicting rates of reactions that may not have been studied under an appropriate set of conditions. Extrathermodynamic Relationships
Empirical correlations between rate and equilibrium constants in the form of linear free-energy relationships (8,22-24) have been used successfully in the study of reaction mechanisms to determine the extent of bond formation and bond breaking in the transition state, the sensitivity of some reactions to electronic effects, the type of bonding in the activated complex and in identification of intermediate species. The earliest examples of LFER were given by Bronsted (25) and Hammett (261, who correlated rate constants with corresponding acid-base ionization constants. Both Bronsted and Hammett used pK, values for proton acids as equilibrium constants in a standard reaction series.
pH range
8-8.5 6-12
4-10 12 7-8.6 6-10 7-8 5-12 5-12 M.
e
ref
108 109 110 111 112 113 114 115 116 117 118
Rate data have
In the former case, rates of reactions sensitive to general acid catalysis were related to the abilities of proton acids to form hydronium ions and in the latter case to explain the impact of electronic effects of ring substituents on the rates of reactions of aromatic compounds. The molecular basis for LFER can be understood in terms of reaction coordinate diagrams shown in Figure 1for a reaction where bond formation is important. For this reaction M
+ Li
---+
MLi
(20)
the ground-state energy for the reactants is given by E1 and the ground-htate energy for the products is given by Ez. The energy diagram represents the most probable path of the reaction system which is represented ideally by a line on a three-dimensional potential-energy surface which follows the lowest possible energy contour between initial and final states (27). On a three-dimensional surface the transition state is the region near the saddle point and the corresponding region in Volume 15, Number 3, March 1981 349
Table VIII. Second-Order Rate Constants, k2, for Fe(II) Oxidations/Electron-Transfer Reactions Which Follow the Rate Law a Y = k2[Fe(Il)][oxidant] no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
oxldant
kp,a M-’
Fe3+ Fe(EDTA)FeCI2+ Fe(ter~y)~~+ Fe(di~y)~~+ Fe(5-n i t r ~ p h e n ) ~ ~ + Fe(5-~hlorophen)~~+ Fe(2,5-dirnethylphen)33f
Fe(tetrarneth~lphen)~~+ benzosemiquinone toluserniquinone duroserniquinone 2,6-dichloroserniquinone
4.0 4.0x 3.73 x 8.5 x 2.7x 1.1 x 2.1 x 7.8 x 2.3X 7.19 X 2.98 X 2.66 x 3.58 6.77 X
FeS04+ A@ AGt in kJ/mol, AGt = -!?Tin
g-1
AGt
69.7 92.5 64.1 44.9 47.7 38.5 42.7 50.8 59.5 73.9 76.1 93.6 70.0 56.9
10-4 10’ 104 104 105 105 103
lo2 lo-’ lo-’ 10-4
lo2
AGO
P, M
T, ‘ C
ref
0
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
25 25 25 25 25 25 25 25 25 25 25 25 25 25
119 120 119 121 121 121 121 121 121 122 122 122 122 123
61.3 -2.5 -18.3 -22.2 -49.2 -36.7 -22.2 -6.7 52.3 62.3 85.8 38.9 6.5
k, t RTIn (kTlb). AG+is the free energy of activation and A @ is the partial molar freeenergy of the reaction. Calculated from Kand kt (Marcus, R. A. J. Cbem. Phys. 1957, 26, 872). Data and reaction numbers apply to Figure 3. a
[H+] = 1.0 M.
the two-dimensional diagram of Figure 1A is given by the distance E,, the activation energy. The fundamental principle of LFER (22) is that changes in activation energies are proportional to changes in the ground-state energy of the products for a related series of reactions, as illustrated in Figure 1A where ML1 is the least stable product and ML3 is the most stable product. The energy diagram can be made more specific, as shown in Figure l B , in which the potential-energy curves represent the relative energies of reactants and products as a function of internuclear distance. The vertical distance between the minima of reactants and products is equal to the energy change of the reaction, and the vertical distance from the reactant minima to the intersection of the two curves represents the activation energy. Assuming, first, that potential-energy curves for a related series of compounds are parallel, second, that free energies are directly related to the heats of reaction, and, third, that the entropy changes for the series are constant, one can show, as illustrated in Figure l B , that changes in activation energy, AE,, are proportional to changes in ground-state energy, AEO, for products of related reactions. This relationship is expressed simply as
AE, = PAEO
(21)
where ,dis a constant, which is obtained from the slope of the curves a t the points of intersection (22). p i s characteristic of a particular reaction series with values ranging from 0 to 1 (8). From the Arrhenius equation (23) which relates the rate constant empirically to the activation energy and from transition-state theory (23) which relates the rate constant to the free energy of activation, it can be shown that
A[AGt]= PA[AGo]
(22)
Equation 22 relates the free energy of activation to the free energy of the bond being formed. It follows from the law of microscopic reversibility that for a symmetrical activated complex the free energy of activation is related to the absolute ground-state energy of the bond that is being broken for the reaction in the reverse direction (8).Equation 21 can be rewritten as In ( k l l k 2 ) =
P In ( K d K z )
(23)
which is the general form for a LFER (35).Linear free-energy relationships have been used widely by kineticists to subLeffler and stantiate postulated mechanisms. Edwards (8), 350
Environmental Science & Technology
@ ,
Transit ion
I REACTION COORDINATE
0 ,
ML, ML. -‘ 1
E
I
DISTANCE
Figure 1. Hypothetical reaction coordinate diagram for complexation of a metal, M, with a related series of ligands, L,. (A) A two-dimensional representation of a four-dimensional potential-energy surface. (B) Morse curves for reactants and products as a function of internuclear distance. Grunwald (27), Chapman and Shorter (28a, b ) , and Shorter (28c) cite numerous examples of the applicability of LFER. Application of the principles of LFER to chemical transformations in aquatic systems may prove to be useful (29). For example, for a related series of reactions such as complexation of a specific transition-metal ion with a variety of ligands, a In k vs. In K plot may be constructed for known reactions. From a LFER plot of this type, rate constants may be estimated from the equilibrium constants for reactions which have not been studied kinetically. In this way, supporting evidence for the application of equilibrium calculations may be obtained. More important, rate constants extrapolated from LFER may be useful for predicting the fate of various chemical contaminants in aquatic systems in the absence of specific laboratory data (29).
The application of the LFER approach will be illustrated with the data given in Table VI for the complexation of Ni(I1) with a variety of ligands. At first glance it appears that there is not a significant variation in the forward rate constants even though there is a wide variation in the overall stability constants. The exchange rate of water appears to be controlling the overall rate of reaction. This is consistent with the mechanism for ligand substitution presented in eq 6 and 7. Equation 7 represents the formation of the final complex in which expulsion of a coordinated water molecule is rate determining. In the terminology of Langford and Gray (30),this type of mechanism is known as dissociative interchange. In a purely dissociative mechanism, the exchange rate of water represents an upper limit for ligand substitution with no apparent dependence on the nature of the complexing ligand. However, if ion pair formation is likely and the substitution rates with different ligands are not all equal to the rate of water exchange for the metal, as in the case of Ni(II), then dissociative interchange is likely. The transition state in a dissociative interchange reaction is characterized more by bond breaking of the leaving group, H20, than bond formation to the incoming group, L. If bond breaking predominates in the transition state, the reverse rate constant, h-2, for a series of related ligand substitutions should be inversely proportional to the thermodynamic stability of the complexes (13). From the law of microscopic reversibility, it is assumed that the activated complex in the transition state is the same in either the forward or the reverse direction. If this is true, the activation energy is related also to the absolute value of the energy of the bond that is being broken; and it follows that a LFER should exist between h-2 and the overall instability constant, 1/K, for a series of ligand substitutions on Ni(I1). Macroscopically, this conclusion is evident. If, for most ligand substitutions on the same metal ion, k f is approximately constant and K = h&,, then -log K and log h , are linearly related. Figure 2 shows a LFER for a series of reactions listed in Table VI. The slope of this plot, which is close to 1,indicates that bond breaking is important in the transition state. Since the transition state is the same in both directions, activation of the forward process, h2, is not assisted by bond formation to the ligand, L, but more likely the transition state probably lies a t a point along the reaction coordinate corresponding to very weak bonding to the leaving group, H20. For ligand substitutions on Ni(I1) that have not been studied kinetically but for which a stability constant is known,
a reliable estimation of the forward rate constant can be obtained from a plot of this type. This should be particularly useful in the development of an equilibrium model when a reaction such as complexation controls the speciation of a particular metal thermodynamically, but little is known about its kinetic feasibility. Generally, redox reactions are thought to be kinetically slow in the absence of catalytic influences, especially when a significant amount of bond deformation or alteration of a permanent nature may occur (e.g., sulfide autoxidation). However, simple electron-transfer reactions, which do not involve changes in chemical bonds, are amenable to theoretical treatment from either a semiclassical (31)or quantum-mechanical (32) foundation. Marcus (31)has derived a general equation relating the rate of electron-transfer reactions to their corresponding free-energy changes. A LFER for a series of related outer-sphere electrontransfer processes is presented in Figure 3. Kinetic and equilibrium data taken from references listed in Table VI11 have been used to calculate a AGZ and AGO. The LFER for these reactions is reasonable with a slope of 0.4. The Marcus equation (31)predicts that a slope of 0.5 should be observed for reactions with an overall zero AGO and a slope >0.5 for reactions with AGO positive. With current (33)theories and corresponding experimental evidence on electron-transfer processes, the tenuous relationship between thermodynamically and kinetically favorable processes is strengthened. Recently, additional experimental evidence relating the energy differences of bonding in the ground state to those in the transition state has been obtained for the catalytic racemization of cyclic metal dipyridyl complexes (34).Additional examples of potentially useful LFER for characterizing the kinetics of reactions important in aquatic systems include the following: hydrolysis rate constants for a series of substituted ethyl benzoates have been linearly correlated with the corresponding ionization constants of substituted benzoic acids ( 2 8 ~ rate ) ; constants for the oxidation of a variety of reduced sulfur compounds by hydrogen peroxide have been linearly correlated with the corresponding electrode potentials and proton basicities of the sulfur compounds (35,36);rate constants for the formation of a series of silver ammine complexes have been correlated with the basic ionization constants of the amines (37);glycosidase-catalyzed hydrolysis of substituted phenyl P-D-glucosides has been correlated with the Hammett substituent constant (38);rate constants for hydrolysis of
LFER FdE) REDOX LFER NU0 COMPLEXATION
a = 0.4 a4
./ 0 3
I
A -50
-30
-10
10 AGO
30
50
70
90
kJ/rnole
for redox reactions involving Fe(ll)with a variety of oxidants. Data, number assignment,and further explanation are given in Table XIII. Flgure 3. A LFER
Volume 15, Number 3, March 1981 351
various transition-metal complexes have been correlated to their corresponding equilibrium constants (39-45); and, finally, the kinetics of metal-ion-catalyzed decarboxylation of P-keto acids have been correlated linearly with the correspoonding metal-substrate association ( 4 6 , 4 7 )or dissociation constants ( 4 8 ) . LFER analysis of an apparent homologous series of reactions can be fraught with error. Even though certain welldefined systems exhibit an apparent linear correlation of reaction kinetics with ground-state thermodynamics, it must be emphasized that this is not always the case. For example, when entropies of activation show considerable variation for a series of related reactions, the linear free-energy relationship frequently breaks down (23). This phenomenon is a direct result of one of the primary assumptions in the development of LFER (22),that is, that the entropy changes accompanying a reaction are the same for an entire reaction series. The assumption that AS and AS1 are constant for a reaction series is often incorrect. AS can vary significantly among individual reactants in a reaction series. However, occasionally this entropy variation is compensated by a corresponding parallel enthalpy variation with the net result being a linear correlation of free-energies. This phenomenon in terms of the activation energies is known as the compensation effect ( 9 ) . In transition-state theory (23), which itself is essentially an equilibrium theory, the rate constant is expressed as
where (kT/h) is the universal frequency factor and AS1 and A H are the activation entropy and enthalpy, respectively. Boudart ( 9 ) has shown that eq 24 can be rewritten as k f = A exp[(-AHI/R)(l/T - I/@]
(25)
where A is the conventional preexponential factor and 0 is the isokinetic temperature. The isokinetic temperature is the temperature at which rate constants for all reactions in a series have the same value. Leffler ( 4 9 ) cites many examples of reactions exhibiting an isokinetic temperature. An additional problem in the application of LFER occurs when reaction rate constants for a series are not reflected in AG ( 8 ) .Ligand substitution reactions, such as the specific case of N i ( H 2 0 ) ~ ~substituted + with a series of ligands, are good examples of this situation. In this case, the value of kfvirtually does not vary with AGO; however, k, was shown to correlate linearly with the instability constant. Hammond (50) has treated this problem theoretically within the limitations of the law of microscopic reversibility.
Conclusions When the majority of chemical reactions that are normally considered in an equilibrium model are kinetically rapid in comparison to the chemical residence time, an equilibrium model is a valid approximation for determination of the distribution of chemical species in a dynamic system. For the particular case of the hypolimnion of a thermally stratified lake, the time-invariant condition appears to be a reasonable approximation of equilibrium, even though concentration gradients are likely to exist vertically in the lake system as a whole. Following the approaches presented in this paper, the validity of an equilibrium approximation for a variety of chemical problems encountered in aquatic chemistry;lim‘nology, and water and wastewater treatment can be established readily. Construction of complete kinetic models for a natural water system is exceedingly complex and is complicated by the lack of kinetic information (IO).Application of computer-generated equilibrium models should prove to be useful as a first-approximation for establishing reasonable chemical boundary conditions and for predicting the relative 352
Environmental Science & Technology
importance of major chemical species under a certain set of conditions. Semiempirical linear free-energy relationships can be useful in the development of reasonable boundary conditions for the applicability of equilibrium models and also for prediction of the time-dependent fate of chemical constituents in the aquatic environment in the absence of experimental kinetic data.
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Received f o r review August 16,1979. Accepted November 13,1980.
Supplementary Material Available: Extended tables of kinetic data ( I 3 pages) will appear following these pages in the microfilm edition o f this volume o f the journal. Photocopies of the supplementary material from this paper or microfiche (105 % 148 rnm, 24X reduction, negatives) may be obtained from Business Operations, Books and Journals Division, American Chemical Society, 1155 16th St., N.W., Washington, D.C. 20036. Full bibliographic citation (journal, title of article, author) and prepayment, check or money order for $13.00 for photocopy ($14.50foreign)or $4.00 for microfiche ($5.00 foreign), are required.
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