Environ. Sci. Technol. 1996, 30, 1919-1922
Experimental Determination of Molar Absorptivities and Quantum Yields for Individual Complexes of a Labile Metal in Dilute Solution BRUCE C. FAUST* School of the Environment and Department of Chemistry, Environmental Chemistry Laboratory, Levine Science Research Center, Duke University, Durham, North Carolina 27708-0328
Although it is increasingly recognized that photochemical reactions of metal complexes can significantly affect the speciation and geochemical cycling of some metals in natural waters, progress has been slow in determining molar absorptivities and quantum yields of individual complexes of a given metal. Such information is needed in order to differentiate the effects of metal-complex stoichiometry from the effects of ligand structure on the photoreactivity and absorption spectra of metal complexes. Hence, a theoretical foundation is provided here for methods that are developed to determine molar absorptivities and quantum yields of individual complexes of a given metal-ligand system. The procedures quantitatively link measurements of average molar absorptivities and average quantum yields with the equilibrium speciation of the metal (measured or computed). Background. Mechanistic investigations of the photochemical reactions of dissolved metal complexes are receiving more attention, as it is increasingly recognized that such sunlight-initiated photoreactions can have large effects on the speciation and geochemical cycling of some metals in natural waters. Much effort has been expended on characterizing the photochemical reactions of metal complexes (1, 2). But the complexity of metal-ligand systems (M ) metal, L ) ligand) has made it difficult to determine quantum yields and molar absorptivities of individual complexes of a metal (3, 4). Knowledge of the molar absorptivities and quantum yields of individual complexes of a metal is needed in order to differentiate the effects of metal-complex stoichiometry (e.g., ML, ML2, etc.) from the effects of ligand structure on the absorption spectra and photoreactivity of metal complexes. A theoretical foundation is provided here for experimental procedures to determine the molar absorptivities and quantum yields of individual complexes of a given metal-ligand system. As will be shown, molar absorptivities and quantum yields of individual complexes of a given metal are * E-mail address:
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1996 American Chemical Society
determined from measurements of average molar absorptivities and average quantum yields that are quantitatively linked with the equilibrium speciation of the metal (measured or computed) for each experimental solution studied. Definitions of the symbols and nomenclature used here are given in Table 1. The experimental methods presented here are for a collimated beam of monochromatic light, as is commonly used in spectrophotometers and in photochemical studies of solutions with continuous illumination. These procedures assume that the individual quantum yields (of product P) for each complex of the metal (ΦP,i) are independent of the solution composition and ionic strength. Also, it is assumed that the solution composition is controlled and that the extent of photolysis is limited, so that photoproducts have negligible effect on the metal speciation, the absorbance, and the photoreactivity of the system. Equilibrium Metal Speciation. The procedures described below are applicable for metal-ligand systems where the equilibrium metal speciation is determined from measurements (e.g., spectroscopic or chromatographic) or from computations. The metal speciation can be computed using any one of several metal-speciation computer programs (e.g., ref. 5), which are based on well-established methods (6) and use databases of equilibrium constants and thermodynamic information. These procedures are applicable for a labile metal (M) and a ligand (L) whose dissolved complexes (e.g., ML, ML2, etc.) interchange (e.g., undergo ligand exchange) rapidly. Based on the rapid water and ligand exchange rates of many metals (7, 6), ligand exchange does not appear to be a limiting factor for this approach. The spectral and photochemical measurements for a given metal-ligand system of interest should be made under the same conditions (temperature, ionic strength, pressure) for which the equilibrium metal speciation is determined. If the equilibrium metal speciation is computed, then the conditions for the experiments may be determined by the conditions for which the relevant equilibrium constants and thermodynamic information are reported. Often equilibrium constants and thermodynamic information are reported for conditions (temperature, ionic strength) that are within the range of normal variability found in natural and environmental aquatic systems. Molar Absorptivities. The measured total absorbance (base 10) of the metal-ligand solution can be expressed as
A ) {M[M]T + Lfree[L]free + Rother}D
(1)
where the parameters are defined in Table 1. Contributions to the total absorbance from substances in the solution excluding any form of the metal or ligand of interest (e.g., background electrolytes, buffer, etc.), (Rother)(D), can be (a) determined explicitly from measurements with pure solvent as a reference or (b) accounted for by zeroing the spectrophotometer with a solution without added metal and ligand but with otherwise identical composition as the metal-ligand solution. In turn, values of Lfree are determined from absorbance measurements of solutions without the added metal but with otherwise identical composition as the metal-ligand solution. Values of [L]free are determined from equilibrium-speciation measurements or computa-
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TABLE 1
Definitions of Symbols and Explanation of the Nomenclaturea symbol
definition
A Rother
measured total absorbance of a solution due to all substances in the solution (excluding solvent) absorbance of all substances in the solution, excluding absorbance due to the solvent and to any form of the metal and ligand of interest, divided by the optical path length (D), cm-1; this accounts for absorbance by the electrolytes, buffer, etc., but not by any form of the metal or ligand of interest optical path length, cm measured average molar absorptivity of the metal due to all forms of the metal, i.e., based on [M]T measured average molar absorptivity of all forms of the ligand excluding complexes of the ligand with the metal of interest, i.e., based on [L]free molar absorptivity of the ith complex of the metal equilibrium fraction (measured or computed) of the total metal present as the ith complex, ) [M]i /[M]T measured volume-averaged incident actinic flux of monochromatic light, Einstein L-1 s-1 (1 Einstein ) 1 mol of photons) spherically integrated solar irradiance (Einstein cm-2 s-1) for a given wavelength at a given point in a natural water measured average apparent first-order photolysis rate constant for all complexes of the metal, ) [ln(10)](Io)(ΦP)(M)(D) for monochromatic illumination with a collimated beam of light apparent first-order photolysis rate constant for the ith complex of the metal: ) [ln(10)](Io)(ΦP,i)(i)(D) for monochromatic illumination with a collimated beam of light, ) [ln(10)](Iλ′)(ΦP,i)(i) for a single wavelength in sunlight average apparent first-order sunlight photolysis rate constant for all complexes of the metal, ) RP/[M]T total concentration of all forms of the ligand excluding complexes of the ligand with the metal of interest wavelength, (nm) total concentration of the metal concentration of the ith complex of the metal measured average quantum yield based on the total absorbance (A) of the solution quantum yield for the ith complex of the metal measured total rate of product P formation from photolysis of all complexes of the metal
D M Lfree i fi Io
Iλ′ j ji jSUN [L]free λ [M]T [M]i ΦP ΦP,i RP
a Except where noted otherwise, parameters refer to a single wavelength. Absorbances and molar absorptivities are expressed in base 10. Units for the parameters are as follows: concentration (M), molar absorptivity (M-1 cm-1 or cm2 mol-1), rate (M s-1), and apparent first-order photolysis rate constant (s-1). All rates, quantum yields, and apparent first-order photolysis rate constants refer to formation of the product P. Average values refer to mean values. The expression for the photolysis rate constant of the ith complex of the metal (ji) in natural waters is based on that for a single compound (8, 9).
tions, for the conditions of each metal-ligand solution studied. In many cases, contributions of the uncomplexed (“free”) ligand species to the total absorbance will be small. The total absorbance of the solution can also be described by
A){
∑ [M] + i
i
Lfree[L]free
+ Rother}D
(2)
i
By equating eqs 1 and 2, an expression is derived that relates the measured average (mean) molar absorptivity M to the individual molar absorptivities of the metal complexes (i):
M )
∑{ [M] }/[M] i
i
T
(3)
of the metal. In principle, there are a large number of adjustable parameters (i.e., the N individual i values) in this linear regression. In practice, usually only a few complexes of the metal (often as few as only two complexes) play significant roles. This issue is addressed in the section Constraints on Linear Regression Analyses. Quantum Yields. As will be seen, the derivation of quantum yields bears some conceptual similarity to that of the molar absorptivities. The measured total rate of product P formation in the solution (from photolysis of all complexes of the metal) can be expressed as
RP ) (Io)(ΦP){1 - exp[-ln(10)(A)]}
(6)
i
Using the definition fi ≡ [M]i/[M]T, this can also be expressed as
M )
∑{( )(f )} i
i
(4)
i
Thus, eq 4 shows that the experimentally measured molar absorptivity is a weighted average, where the weighting is related to the equilibrium speciation of the metal. By definition
∑(f ) ) 1 i
(5)
i
Use of eq 5 with eq 4 allows one fi variable to be eliminated. For a system of N complexes of the metal, a linear regression of the measured M values against the equilibrium-speciation fi values (eqs 4 and 5) is used to determine the individual molar absorptivities (i) for the N complexes
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The parameters RP, Io, and A are measured for each experiment. For each solution, these measured parameters are used together with eq 6 to determine the average quantum yield (ΦP), which is based on the total absorbance (A) of the solution. However, only light absorbed by complexes of the metal of interest, and not light absorbed by other substances in solution, causes the photoreaction of interest. Thus, with complete generality, the rate of product formation can also be described by
RP ) (Io){1 - exp[-ln(10)(A)]}
∑{(Φ i
P,i)(i)[M]i/(A/D)}
(7)
where the quantity (i)[M]i/(A/D) accounts for the fraction of absorbed light that is absorbed by a given complex of the metal. Using the definition fi ≡ [M]i/[M]T, RP can also be expressed as
RP ) (Io){1 - exp[-ln(10)(A)]}[M]T
∑{(Φ
P,i)(i)(fi)/(A/D)}
i
(8) Equating eqs 6 and 8 and rearranging terms gives a relationship between the quantity (ΦP)(A/D)/[M]T, which is comprised entirely of measured and known parameters, and the corresponding fundamental quantum yields for each individual complex of the metal (ΦP,i):
(ΦP)(A/D)/[M]T )
∑{(Φ
P,i)(i)(fi)}
(9)
i
Equation 9 shows that the quantity (ΦP)(A/D)/[M]T is a weighted average (mean), like the molar absorptivities (eq 4), where the weighting is again related to the speciation of the metal. Use of eq 5 with eq 9 allows one fi variable to be eliminated. For a system of N complexes of a metal, a linear regression of the (ΦP)(A/D)/[M]T values versus the equilibrium-speciation fi values (eqs 9 and 5) can be used to determine the individual values of the quantity (ΦP,i)(i) for the N complexes of the metal. As with the molar absorptivities, there are, in principle, N adjustable parameters [the N individual (ΦP,i)(i) values] in this linear regression. In practice, usually only a few complexes of the metal (often as few as only two complexes) play significant roles (see Constraints on Linear Regression Analyses). Metal-Dominated Absorbance. If the total solution absorbance is dominated by complexes of the metal of interest (i.e., M[M]T . Lfree[L]free + Rother, see eq 1), then eq 9 further simplifies to
(ΦP)(M) =
∑{(Φ
P,i)(i)(fi)}
(10)
i
where the quantity (ΦP)(M) is comprised entirely of measured parameters. Thus, eq 10 demonstrates that the measured quantity (ΦP)(M), like the measured molar absorptivity (eq 4), is a weighted average (mean), where the weighting is again related to the speciation of the metal. Use of eq 5 with eq 10 allows one fi variable to be eliminated. For a system of N complexes of a metal, a linear regression of the measured (ΦP)(M) values versus the equilibriumspeciation fi values (eqs 10 and 5) is used to determine the individual values of (ΦP,i)(i) for the N complexes of the metal. Individual Quantum Yields. For each complex of the metal, individual quantum yields of the product P (ΦP,i) are determined by dividing the individual quantity (ΦP,i)(i) by the corresponding individual value of i. Constraints on the Linear Regression Analyses. In principle, there are a large number of adjustable parameters used in the methods described above [i.e., the N individual i values and (ΦP,i)(i) quantities]. However, in practice, solution conditions can usually be controlled so that only a few complexes of the metal (often as few as only two complexes) dominate: (a) the metal speciation, (b) the absorbance attributable to all forms of the metal, and (c) the photochemistry of the system. In any case, additional factors and independent information are needed to constrain the i values and the (ΦP,i)‚ (i) quantities. (1) Values of i and (ΦP,i)(i) for complexes of the metal not involving the ligand of interest should be determined from measurements of the same system (for
the same conditions) but without the ligand of interest. (2) Values of i and (ΦP,i)(i) must, of course, be nonnegative. (3) Molar absorptivities in the ultraviolet-visible wavelength range seldom excede 200 000 M-1 cm-1. Values of (ΦP,i)(i) will generally be smaller than the corresponding i values (i.e., ΦP,i e 1), except for systems where free radical chain mechanisms cause a large secondary thermal production of the product. Four, values of i and (ΦP,i)(i) for a given metal complex, determined from the multi-parameter linear regressions, should be reported only if some of the measurements are obtained under conditions for which the metal complex of interest (a) represents a significant percentage of the metal in solution, (b) contributes significantly to the absorbance attributable to all forms of the metal, and (c) contributes significantly to the photoreactivity of the system. Finally, for the solutions studied, a plot of the calculated values of the quantity of interest [i.e., M, (ΦP)(A/D)/[M]T, or (ΦP)(M)], as computed from the fi values and best-fit values of i or (ΦP,i)(i), versus the measured values of the quantity of interest will verify the quality of fit of these multiple-parameter linear-regression procedures. Potential Limitations of the Method. As noted above, this procedure assumes that the individual quantum yields (ΦP,i) are independent of the solution composition. This assumption can be tested by applying the method to a wider range of conditions (e.g., pH, total metal concentration, total ligand concentration). This procedure relies on knowledge of the metal speciation (measured or computed). If a suitable technique to measure the metal speciation is unavailable and if the database of equilibrium constants and thermodynamic information is insufficient to confidently compute the speciation, then this approach can not be used. Additionally, if the metal-ligand system only undergoes ligand exchange slowly (e.g., metal complexes of hexadentate chelates such as EDTA), then the metal speciation cannot be determined from equilibrium computations. In some systems, polynuclear complexes could form. Often little information is available about such species. Moreover, they may not necessarily be in equilibrium with mononuclear complexes of the metal. However, the possible involvement of polynuclear species can be tested by again applying the method to a wider range of conditions and especially by varying the total metal and total ligand concentrations. Application to Natural Waters. The apparent first-order sunlight photoformation rate constant for a given complex of the metal of interest can be expressed as
∫
ji ) [ln(10)] {(Iλ′)(ΦP,i)(i)} dλ
(11)
where the wavelength (λ) region of integration corresponds to that of sunlight (λ g 280 nm). The contributions of all complexes of the metal of interest to the product photoformation in sunlight can be accounted for by the average sunlight photoformation rate constant (jSUN):
jSUN )
∑{(j )(f )} i
(12)
i
i
where jSUN is a weighted average of the individual photolysis rate constants for each complex (ji), and the weighting is again related to the speciation of the metal (fi). Equations 11 and 12 show that product photoformation in sunlight
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depends on (a) the speciation of the metal (fi), (b) the rate of sunlight absorption by each complex [as measured by (Iλ′)(i)], and (c) the quantum yield for each complex (ΦP,i). General Comments and Conclusions. Using the procedure described here for a given metal-ligand system, one can determine the effect of metal-complex stoichiometry (e.g., ML, ML2, etc.) on the absorption spectra (molar absorptivities) and quantum yields. In turn, this information, together with similar information for complexes of the same metal with other ligands, allows for comparisons of the effect of ligand structure (for a given metal-complex stoichiometry) on the absorption spectra and quantum yields. The results provided here are general in nature. As more methods become available to directly measure metal speciation (e.g., spectroscopic or chromatographic techniques, etc.), this information can be used instead of the computed metal speciation in the above models.
Acknowledgments This work was supported by the Office of Naval Research.
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Literature Cited (1) Balzani, V.; Carassiti, V. Photochemistry of Coordination Compounds; Academic Press: New York, 1970. (2) Adamson, A. W.; Fleischauer, P. D., Eds. Concepts of Inorganic Photochemistry; Wiley-Interscience: New York, 1975. (3) Vincze, L.; Kraut, B.; Papp, S. Inorg. Chim. Acta 1984, 85, 89-96. (4) Horva´th, O.; Papp, S. J. Chem. Educ. 1988, 65, 1102-1105. (5) Felmy, A. R.; Girvin, D. C.; Jenne, E. A. MINTEQsA Computer Program for Calculating Aqueous Geochemical Equilibria; EPA600/3-84-032; U.S. Environmental Protection Agency: Athens, GA, 1985. (6) Morel, F. M. M.; Hering, J. G. Principles and Applications of Aquatic Chemistry; Wiley-Interscience: New York, 1993; pp 399401. (7) Hoffmann, M. R. Environ. Sci. Technol. 1981, 15, 345-353. (8) Leighton, P. A. Photochemistry of Air Pollution; Academic Press: New York, 1961. (9) Zepp, R. G. Environ. Sci. Technol. 1978, 12, 327-329.
Received for review August 18, 1995. Revised manuscript received January 19, 1996. Accepted January 30, 1996.X ES9506162 X
Abstract published in Advance ACS Abstracts, April 1, 1996.
