Experimental and Modelling Studies on the Reactions of the Sulfate Ion Radical

DE TTK 1949 Experimental and Modelling Studies on the Reactions of the Sulfate Ion Radical Ph.D. thesis Dóka Éva Supervisor: Dr. Gábor Lente UNIVERSITY OF DEBRECEN Chemistry Graduate School Debrecen, 2016 DE TTK 1949 Experimental and Modelling Studies on the Reactions of the Sulfate Ion Radical Ph.D. thesis Dóka Éva Supervisor: Dr. Gábor Lente UNIVERSITY OF DEBRECEN Chemistry Graduate School Debrecen, 2016 II III Ezen értekezést a Debreceni Egyetem Természettudományi Doktori Tanács Kémiai Doktori Iskola K/2 Koordinációs és Analitikai Kémia programja keretében készítettem a Debreceni Egyetem természettudományi doktori (Ph.D.) fokozatának elnyerése céljából. Debrecen, 2016. május 27. Dóka Éva Tanúsítom, hogy Dóka Éva doktorjelölt 2012 - 2015 között a fent megnevezett Doktori Iskola K/2 Koordinációs és Analitikai Kémia programjának keretében irányításommal végezte munkáját. Az értekezésben foglalt eredményekhez és az ezekből született publikációkhoz a jelölt önálló alkotó tevékenységével meghatározóan hozzájárult. Az értekezés elfogadását javasolom. Debrecen, 2016. május 27. Dr. Lente Gábor IV V EXPERIMENTAL AND MODELLING STUDIES ON THE REACTIONS OF THE SULFATE ION RADICAL Értekezés a doktori (Ph.D.) fokozat megszerzése érdekében a Kémia tudományágban Írta: Dóka Éva okleveles vegyész és alapokleveles matematikus Készült a Debreceni Egyetem Kémiai Doktori Iskolája (Koordinációs és analitikai kémiai programja) keretében Témavezető: Dr. Lente Gábor egyetemi tanár A doktori szigorlati bizottság: elnök: Dr. Kövér Katalin egyetemi tanár (DE) tagok: Dr. Tóth Ágota egyetemi tanár (SZTE) Dr. Sóvágó Imre professor emeritus (DE) A doktori szigorlat időpontja: 2016. március 8. Az értekezés bírálói: Dr. ........................................... Dr. ........................................... Dr. ........................................... A bírálóbizottság: elnök: Dr. ........................................... tagok: Dr. ........................................... Dr. ........................................... Dr. ........................................... Dr. ........................................... Az értekezés védésének időpontja: 2016. .............................................. VI VII "In the end, everything will be OK. If it's not OK, it's not yet the end." /Fernando Sabino/ XIV Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 1 1. Introduction Sulfur (S) is a nonmetallic element in the 16th group of the periodic table, a member of the oxygen family or chalcogens. The latter name is a hybrid of the Greek word  (khalkos, meaning ore) and the Latinized suffix ‘-gen’, meaning born or produced. This name implies that the first two elements of the group, oxygen and sulfur are highly frequent components of ores and minerals, oxygen being the most abundant element in the Earth’s crust.1 The chalcophile group within the Goldschmidt classification comprises the elements which are likely to form compounds with sulfur, mainly p and d block metals of soft character.2,3 Sulfur is ubiquitous in Nature, it occurs mostly in the following forms:  elemental sulfur in the cap rocks of salt domes or in volcanic eruptions  H2S in natural gas and sulfurorganic substances in petroleum  sulfide ores of metals (PbS, FeS2, CuS, ZnS, HgS etc.).1 Owing to the high abundance of the element, sulfur has been known since the ancient times. It is mentioned several times in the Bible as ‘fire and brimstone’, * in the context of hellfire and eternal misery. Maybe it is the malodorous smell and toxic nature of many sulfur compounds that earned the element such a negative reputation (although pure sulfur does not smell at all). Fire and brimstone also appears in the Greek classics Iliad and Odyssey, although Homer might have mistakenly refer to the smell of sulfur instead of the unpleasant odor of lightning-generated ozone.4,5 The odor of skunk spray is also due to low molecular weight thiol compounds or mercaptans. The origin of the name sulfur is obscure, the earliest appearances date back to early Latin culture of the first centuries BC. It is known not to be a Greek loan word, since Greek authors called the element θεῖον (theion, ancestor of the prefix thiofor sulfur containing compounds). Conceivably, the word developed from sulpur to sulphur and finally to sulfur. The spelling ‘sulphur’ persisted, especially in British linguistic environment, even though IUPAC standardized the ‘f’ spelling a few decades ago, without respect to geographical differences.6,7 Owing to its valence electron composition and catenation property, sulfur features versatile redox behavior with significant environmental aspects, biological relevance as well as industrial applications. * The Lord tests the righteous and the wicked, And the one who loves violence His soul hates. Upon the wicked He will rain snares; Fire and brimstone and burning wind will be the portion of their cup. /Psalms 11:5-6/ Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 2 Sulfur dioxide (SO2) is one of the so-called criteria air pollutants. Detrimental effects of SO2 gas necessitated the admission of upper limits of daily exposure, set by EPA (U. S. Environmental Protection Agency), to avoid potential human health and property damage.8 SO2 is a gaseous compound that accumulates in the atmosphere by volcanic activity and from anthropogenic sources such as combustion of fossil energy carriers (due to sulfur contaminations in coal, petroleum and natural gas) and metallurgy. Upon direct exposure, higher levels of sulfur dioxide cause respiratory irritation, induce bronchoconstriction9 (shortness of breath due to the constriction of pulmonary airways) and increase the symptoms of asthmatic patients.10 A slow-releasing SO2 donor molecule, benzothiazole-sulfinate has recently been discovered, which facilitates the examination of the biological effect of inhaled and endogenously generated SO2.11 On a global level, the major consequence of atmospheric SO2 is the formation of acid rain or acid deposition.12-17 Acid rain is rainwater with a pH below the natural value of 5.6, which is a result of the dissolution of atmospheric CO2. Acidification of rainfall is a relatively modern phenomenon, the consequence of heavy industrialization of the last two centuries. Greenland ice layers from the 1800s have nearly neutral pH values. The pH decrease is aggravated by the formation of sulfuric acid from SO2 according to eq. (1), the autoxidation of hydrated SO2. Autoxidation is a generic term for the reactions where the reductant is oxidized by elementary oxygen from air. The prefix ‘auto’ refers to the fact that the oxidant is seldom added deliberately in these processes but is simply taken up from the environment. 2 H2O·SO2 + O2 = 2 HSO4− + 2 H+ (1) The lowest pH value measured in rainwater was 1.5, detected in Wheeling, West Virginia, in 1979.18 Heavy acid rains lead to extended deforestation, destruction of limestone mountains and man-made objects and buildings, such as the Taj Mahal in India, built entirely from marble. A very interesting and scientifically challenging aspect of the stepwise formation of H2SO4 from sulfur and oxygen is the restoration of the Swedish warship Vasa, which sank on her maiden journey in 1628 and was conserved underwater until 1961, when the wreckage was salvaged from Stockholm harbor. Today the remains of the ship are exhibited in the Vasa Museet in Stockholm. The water which Vasa sank in was rich in sulfate reducing bacteria. Elementary sulfur, as well as sulfur compounds of intermediate oxidation states accumulated in the oak beams of the ship during the 333 years she spent submerged. Ever since the Vasa was brought to the surface, scientists struggle with the massive acidification of the skeleton because the iron in the bolts catalyzes the overall oxidation of sulfur to sulfuric acid.19 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 3 An initiative emerged in the field of geoengineering to counteract global warming based on the global dimming effect of stratospheric sulfate aerosols by the reflection of solar radiation. The concept would endorse the application of missiles and aircraft to deliver sulfur dioxide and other sulfur gases in the stratosphere, to act as sulfate precursors and thus increase the albedo (reflectivity) of the planet. However, the reception of such technological measures has been controversial because they do not replace consistent political and social engagement for the reduction of total greenhouse gas emission. In addition, the long-term harmful consequences of the extra sulfur burden are unpredictable.20-22 Atmospheric autoxidation of SO2 drew considerable attention from inorganic chemists, and detailed mechanistic studies revealed that the intermediates of the autoxidation process are of radical nature (mainly SO3, SO4, SO5). In general, sulfate ion radical (SO4) is one of the most broadly studied free radical in the literature, along with hydroxyl radical (OH).23-25 SO4 is a highly reactive transient species with strong oxidizing power Eº(SO4/SO42V Recently, the role of sulfate ion radical in atmospheric aqueous phase chemistry has been investigated by Herrmann and co-workers.23,26-29 SO4 is considered as an oxidant in so-called advanced oxidation processes (AOPs) in water and wastewater treatment, in order to eliminate organic and inorganic contaminants.30-35 Simultaneously, advanced reduction processes (ARPs) involve UV or ultrasound produced sulfite ion radical (SO3) for the reductive degradation of harmful oxidized pollutants.36 Sulfate ion radical derived oxidation also has biochemical applications. An efficient, tunable footprinting method was developed for monitoring global protein oxidation status. Original FPOP (Fast Photochemical Oxidation of Proteins) utilized OH as oxidant. The introduction of SO4 as a footprinting agent is an improvement due to the slightly lower reactivity and higher target specificity compared to hydroxyl radical.37 The present thesis is dedicated to shed further light on the role of sulfate ion radical in the transition metal catalyzed and the iodide ion catalyzed, photoinitiated autoxidation of hydrated SO2, and more generally, the autoxidation of S(IV). Traditional methods of aqueous solution kinetics are employed, as well as laser flash photolysis for the direct observation of SO4. Computational simulations are carried out to study the potential effect of diffusion and spatial inhomogeneities during laser flash photolysis experiments. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 4 2. Literature overview 2.1. Redox chemistry of sulfur Sulfur has a valence electron configuration of 3s23p4. Therefore, it can occur in compounds in (2)  (+6) oxidation states (even numbers are preferred). Due to the wide range of available oxidation numbers, sulfur exhibits versatile redox chemistry. This section will focus on the redox characteristics of sulfur oxyacids and anions along with the derived sulfuroxy radicals. Table 1. Sulfur oxyacids, conjugate oxyanions and the related free radicals. Formula Name of oxyacid Oxidation state(s) Conjugate anion Related radical H2SO4 sulfuric acid VI sulfate, SO42 hydrogen sulfate, HSO4 sulfate ion radical,a SO4 H2S2O7 disulfuric acid VI disulfate, S2O72 H2SO5 peroxomonosulfuric acid VI peroxomonosulfate, SO52 peroxomonosulfate ion radical, SO5 H2S2O8 peroxodisulfuric acid VI peroxodisulfate, S2O82 H2S2O6 dithionic acid V dithionate, S2O62 H2Sn+2O6 polythionic acid V, 0 polythionate, Sn+2O62 H2S2O3 thiosulfuric acid IV, 0, (or II, II) thiosulfate, S2O32 (H2SO3) H2O∙SO2 (sulfurous acid) hydrated sulfur-dioxide IV sulfite, SO32 hydrogensulfite, HSO3 sulfite ion radical, SO3 H2S2O5 disulfurous acid or pyrosulfurous acid V, III Disulfite; commonly known as metabisulfite or pyrosulfite, S2O52 H2S2O4 dithionous acid III dithionite, S2O42 aalternative names: sulfate radical anion, sulfate radical Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 5 Table 1 gives a brief summary of these kinds of species in the order of decreasing oxidation number.1 The stabilities of these acids and salts vary on a broad scale, e.g. free acid forms of dithionic, ‘sulfurous’ and dithionous acids are virtually nonexistent, they are mostly found in solutions and in salts as the conjugate oxyanions. H2O∙SO2 and SO4 are of utmost importance in the present thesis, their spectral and chemical features will be discussed in further detail in Sections 2.2 and 2.3. Pyrosulfate salts will be presented therein as a source of S(IV) species and other sulfuroxy radicals (SO3, SO5) as congeners of sulfate ion radical. The sulfuroxy species presented in Table 1 participate in various redox equilibria. Standard electrode potentials of a few redox pairs are collected in Table 2. Table 2. Standard electrode potentials of sulfuroxy ions and radicals Redox couplea E° (V) Redox coupleb,c E° (V) S/S2 -0.476 SO2/SO2 -0.17 S/H2Saq 0.142 -0.262 S2O62/S2O3 0.564 -0.288 S2O82/SO42 2.010 -0.31 S2O82/HSO4 2.123 SO3/SO32 0.63 S4O62/S2O32 0.080 0.72 H2SO3/HS2O4 -0.056 0.76 H2SO3/S 0.449 0.89 SO32/S2O42 -1.120 0.73±0.01b SO32/S2O32 -0.571 SO3/HSO3 0.84 SO42H2SO3 0.172 SO4/SO42 2.43 SO42/S2O62 -0.220 2.52-3.08 SO42SO32 -0.930 2.6 SO5/HSO5 1.1 SO5/SO52 0.81±0.02b S2O62/SO3SO32 -0.49 S2O82/SO4SO42 1.39 S4O63/2S2O32 1.07±0.03b a: ref.38; b: ref.39 ; c: ref.40 and references therein Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 6 SO42 and HSO4 are the most frequent sulfur oxyanions in Nature. These are both very stable species, they are formed as oxidation products of sulfur compounds and during the decomposition of peroxodisulfate ions. Sulfate salts take part in the composition of particulate matter in the atmosphere. However, sulfuric acid, especially in its concentrated form, is a highly reactive, strong oxidizing agent, which is able to dissolve electropositive metals. Disulfuric or pyrosulfuric acid is found in fuming sulfuric acid (also known as oleum, oil of vitriol, or spirit of vitriol), it is formed by the dissolution of SO3 in H2SO4. Anhydrous peroxomonosulfate ion (H2SO5) is also called Caro’s acid, it is one of the strongest oxidants known (E°(HSO5/HSO4) = 1.84 V)40. Its conjugate anion, SO52 is available as the composite salt Oxone® (registered trademark of DuPont), which has been recognized as a ‘green’ oxidizing agent as its reduction product is sulfate ion and with the possible by-product elementary oxygen.41,42 Peroxodisulfate ion (S2O82) is generally used in the form of its water-soluble ammonium or potassium salts. Peroxodisulfate salts are utilized as oxidizing and bleaching agents and to initiate radical polymerization processes. Ammonium persulfate (APS) is the most common initiator in polyacrylamide gel electrophoresis. S2O82 has very high electrode potentials (see Table 2), thus it is expected to be a strong oxidizing agent, but its redox reactions are quite sluggish due to a kinetic barrier. Silver(I) ion has been proved to be a powerful catalyst of these reactions. They usually take place as chain reactions and their rate determining step is eq. (2), which is a one-electron transfer that produces highly reactive sulfate ion radical.43-45 Ag+ + S2O82 = Ag2+ + SO42 + SO4 (2) Dithionate ion (S2O62) exhibits quite poor redox chemistry, it is highly inert under common circumstances. It cannot be protonated in the usual pH range, and it reacts even with strong oxidizing agents only at elevated temperatures. If it participates in a redox reaction, the first step is disproportionation in most cases, where sulfur(VI) and sulfur(IV) are the direct products and the latter can be oxidized rapidly by the oxidizing agent.46 Polythionates (O3S-(S)n-SO32 or Sn+2O62) are relatively stable sulfur oxyanions, containing a chain of zero-valent sulfur atoms as a linker between two SO3 groups. Polythionates are frequently found in crater lakes, and they were suggested to serve as efficient markers of forthcoming volcanic eruptions.47 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 7 Thiosulfuric acid has not been isolated so far, it is only known to exist in thiosulfate salts. S2O32ion is a medium strong reducing agent. It is extensively used in iodometric measurements in the laboratory practice and as dechlorinating agent in the textile industry.48 Dithionous acid is unstable both in its pure form and in aqueous solution, dithionites (S2O42) can be isolated as anhydrous salts. In the presence of water, they disproportionate into sulfite and thiosulfate anions (especially in acidic medium). Sodium dithionite is used as a reducing agent in industrial dyeing procedures and for laboratory purposes as well.49,50 2.2. Autoxidation studies of sulfur(IV) Table 1 and Table 2 in the previous section show that +4 is an intermediate oxidation state of sulfur. SO2 is generally considered to be a moderate reducing agent, although depending on the reaction partner, it can be an oxidizing agent as well. A well-known example of the latter is the reaction between SO2 and hydrogen sulfide, which is an important step of the Claus process (patented in 1883), used simultaneously for the desulfurization of crude oils and the production of elementary sulfur.51 The major occurrences of S(IV) atoms are sulfur dioxide (SO2), hydrated sulfur dioxide, H2O∙SO2 and its deprotonated forms: HSO3 and SO32 ions. S(IV) is also found in sulfurorganic molecules, such as sulfonic acids, sulfonates and sulfite esters.52,53 The formula H2OSO2 is generally used to denote dissolved sulfur dioxide because there is no experimental evidence of the existence of the fully protonated H2SO3 molecule. The acid dissociation constant values for the subsequent deprotonation steps of H2O∙SO2 are pKa,1 = 1.86 and pKa,2 = 6.34. The dimerization of sulfite ions into pyrosulfite or metabisulfite ions (S2O52) at higher concentrations is a well-known phenomenon.54 In aqueous medium, the dimeric anion is in equilibrium with the hydrogen sulfite ion, the S2O52 form only occurs in concentrated solutions ( 0.1 M). Therefore, it is convenient to prepare solutions of sulfite ions by dissolving pyrosulfite salts, e.g. Na2S2O5.55 In the subsequent sections of the thesis, the different deprotonated forms of H2O∙SO2 will be collectively referred to as ‘sulfite ion’ or simply S(IV). The atmospheric autoxidation of S(IV), depicted in reaction (1) is the major source of acid rain formation. SO2 can easily accumulate in rain droplets, as its water solubility is much higher compared to other atmospheric gases (Table 3).38 Mole fractions are compared instead of the corresponding Henry’s constants because the concentration of SO2 passes through the limit of validity of Henry’s law.56 The solubility of sulfur dioxide is lower in acidic medium, which provides the basis of the identification of sulfite ions within the Fresenius system.57 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 8 Table 3. Mole fractions (X) of atmospheric gases in aqueous solutions. p = 1 atm, T = 25 °C38 X SO2 2.9102 CO2 7.1104 O2 2.5105 N2 1.3105 The kinetics and mechanism of the process represented by the overall equation shown in eq. (1) drew considerable attention from inorganic chemists, a significant amount of experimental findings have been published on this subject. Catalytic aspects, photochemical phenomena or the role of free radicals in the system have been discussed. 2.2.1. Transition metal ion catalysis in the autoxidation of sulfur(IV) Numerous studies have revealed the role of transition metal ion catalysis in the oxidation of sulfur(IV) species. Kraft and van Eldik investigated the iron(III) catalyzed autoxidation of sulfur(IV) oxides and considered the possible role of metal-sulfito complexes in the mechanism.58-61 A detailed kinetic and mechanistic analysis of the Fe(III)S(IV)O2 system was published later on and proved the catalytic effect of iron(III).62 Brandt and van Eldik examined the influence of pH, the medium and aging in independent experiments.63 These authors also presented a comprehensive overview of the subject, focusing on atmosphericrelevant processes and mechanisms.40 According to this summary, the majority of the published reaction mechanisms for the homogeneous transition metal catalyzed autoxidation of sulfur(IV) oxides suggest radical mechanisms that involve steps from the scheme given by Bäckström, who was the first to publish mechanistic observations on reaction (1).55 He proposed a radical chain mechanism involving sulfoxy radical intermediates, which was supported by the fact that common radical scavengers inhibit the reaction. Fábián and Csordás reviewed the role of metal ions in autoxidation processes from kinetic and mechanistic points of view.64 Elding and co-workers examined the catalytic effect of manganese, chromium and vanadium ions, and they observed the emergence of iron-manganese synergism. They even considered gold(III) as a catalyst in S(IV) autoxidation based on a reduction study of Au(III) complexes by sulfite and hydrogen sulfite ions.65-68 Alexander et al. reported that 9-17% of total sulfate production on Earth can be assigned to oxidation of S(IV) by O2 catalyzed by Fe(III) and Mn(II).69 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 9 Coichev and co-workers studied the radical intermediates that are generated in the autoxidation process catalyzed by Co(II) and Cu(II) complexes, and their potential detrimental effects on DNA chains.70,71 The catalytic effect of Cu(III) tetraglycine complexes on the autoxidation was found and studied by Anast and Margerum.72 DNA damage examinations were also carried out by Burrows’s team with Ni(II) and Mn(II) complexes.73-75 Pioneering work on free radical induced DNA damage was presented by Clemens von Sonntag.76 2.2.2. Photochemical phenomena in the autoxidation of sulfur(IV) Photoinitiated autoxidation in the presence of iron(II) The autoxidation processes of sulfur(IV) in acidic aqueous solution have been extensively studied by Kerezsi et al.55,77-79 They observed that the uncatalyzed or dark reaction is very slow. In the presence of Fe(II), the rate of the reaction increases, but it acts as an auxiliary reducing agent rather than a catalyst. The experiments have been carried out in a diode array spectrophotometer, which is a convenient tool for the simultaneous initiation and detection of the reaction. According to their results, the uncatalyzed autoxidation takes place through excited H2OSO2 and HSO5 intermediates and follows the stoichiometry indicated in eq. (1).78 The reaction rate is independent of the concentration of dissolved oxygen in the pH-range 0.0  1.67, but shows a well-defined dependence on light intensity and sulfur(IV) concentration. In the presence of iron(II), the formation of iron(III) ions was detected, which may be interpreted by the simultaneous presence of two additional pathways. Both pathways contain the oxidation of iron(II) into iron(III) by one of the reaction intermediates. Figure 2.1 presents the suggested scheme of the reaction. Figure 2.1. Photoinitiated autoxidation of sulfur(IV) in the absence and presence of iron(II) ions in acidic aqueous phase. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 16 A detailed kinetic and spectroscopic analysis of sulfite anion radical and its decay mechanisms was given by Waygood and McElroy.135 Laser flash photolysis of sodium dithionate (S2O62)solutions at 193 nm led to the formation of SO3 and SO32. They reported that the decay of sulfite ion radical is second order with a limit rate constant of (4.0 ± 0.3)  l08 M1 s1 and takes place by the simultaneous recombination into dithionate ion and self-reaction into sulfite ion and SO3. The relative rate constant of reaction () compared to (7) was found to be 0.8 ± 0.2. SO3 + SO3S2O62  SO3 + SO3SO32 + SO3 (7) Warneck and colleagues studied the steady state photolysis of SO32and HSO3at 254 nm and analyzed the possible free radical reactions, rate constants and product distribution in the presence and absence of dissolved oxygen.136,137 The literature of peroxomonosulfate radical (SO5) is limited compared to the previously discussed congeners, although due to rapid interconversion reactions, the separate investigation of these radicals is barely feasible. Practically all free radical studies connected to S(IV) autoxidation mentioned earlier deal with the parallel occurrence of SO3, SO4 and SO5radicals, some of them with SO2 as well. The radical is most likely generated by the oxygen addition of sulfite ion radical and decays by several radical-radical and radical-stable species reactions. A paper from T. N. Das focuses mainly on SO5radical, re-evaluating its role in sulfite autoxidation chains in liquid hydrometeors.138 SO5radical shows a weak absorption band between 260-265 nm with a molar absorption coefficient of 1065 ± 80 M1 cm1. 2.4. Effect of inhomogeneities in fast reaction kinetics As mentioned in the previous section of this thesis, the inhomogeneous distribution of the transient absorbing species in the sample is a typical characteristic of laser flash photolysis experiments. This fact should be taken into account during the acquisition and processing of transient absorption data. Sample inhomogeneities can lead to distorted conclusions, especially when the kinetic trace detected is not an exponential curve. Whenever possible, setting pseudofirst order conditions is desirable in order to avoid the usage of actual transient concentrations. Only a handful of papers deal with the experimental and/or mathematical treatment of potential inhomogeneity issues and the resulting errors in kinetic conclusions. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 17 At the dawn of the flash photolysis technique, J. W. Boag analyzed the effect of non-uniform distribution of initial transient concentration along and across the analyzing light beam.139 He considered first and second order reactions and made an effort to give analytical expressions for the difference between the actual and the experimentally determined rate constants. He assumed certain shapes of the initial distribution (see Figure 2.4) and used practically conceivable parameters, such as total absorbance between 0 and 2 (typically 0.4-0.6), h > 0.5, experimental time length at least 4-5 half-life of the observed species. He concluded that first order reactions are less influenced by initial distributions, and with the above parameter values, the difference between 𝑘  (as he assigned the observed rate constant) and k (the actual value) is less than 1%. It is worth noting that he used the linearized versions of the integrated rate laws (which is a source a statistical distortions itself) in his deductions and ignored diffusive motion of the particles within the experimental time. A) B) Figure 2.4. Initial distribution of transient concentration along the analyzing light beam in flash photolysis experiments. A) Extreme cases. a = (sine)2, b = triangular, c = sine distribution; I0 = incident light intensity; I(t) = exit light intensity at time t. B) Expected practical distributions. (a) triangular cap; (b) sine cap; (c) parabolic cap. The parameter h is the ratio between the minimum and maximum of initial concentration (0  h  1). Figures adapted from Boag, Trans. Faraday Soc. 1968, Figs. 1 and 4.139 Bazin and Ebbesen studied the error originating from a poor overlap between the laser beam and the analyzing (or probe) beam in given experimental arrangements of laser flash photolysis. They introduced two kinds of correction factors for such bad overlaps in the laser and probe directions, and provided technical advice on how to detect them.140 Instead of considering actual kinetic measurements, they focused on the deviation of the measured absorbance (ODexp) from the real value (ODtrue). Figure 2.5 shows the possible sources of insufficient overlap between the two light beams. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 18 Figure 2.5. Examples of bad overlap situations in laser flash photolysis experiments. The solid line represents the analyzed space, the dashed line delineates the laser beam. The hatched area shows the overlap. (a) and (b): perpendicular, (c) pseudo front face, (d) collinear arrangement. (the reader might notice that the panels are labelled clockwise instead of row-continuously). Figure adapted from Bazin and Ebbesen Photochem. Photobiol. 1983, Fig. 1.140 The authors assigned x and y for the fractions not covered by the complementary beams for cases (a) and (b) in Figure 2.5. With the notations of the figure, l 'll y  and s 'ss x  ; 0  x, y < 1, and the relation between ODtrue and ODexp is given in eq. (8). ODtrue = ODexp  SF(x,y)  DF(x) (8) SF values are the scale factors 1/(1  x) and 1/(1  y), the proportionality factors for bad overlap across and along the analyzing direction, respectively. An important outcome of their calculations is that the ratio ODtrue/ODexp increases with ODexp, and they defined a distortion factor (DF) as the ratio of the relative errors at high and low experimental value. DF = ( exp true OD OD )/( exp true OD OD )ODexp  0 (9) DF is only a function of x and not y, in the case of bad overlap along the light path, ODtrue is always linearly dependent on ODexp. They suggest that DF tends to cause more trouble in data processing, which confirms that it is preferable to work with low absorbance values, as long as the signal-to-noise ratio is acceptable. Cassidy and Long created a mathematical model to test the validity of experimental rate constants determined under putative pseudo-first order conditions.141 The underlying chemical Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 19 systems were laser flash photolysis studies of CO photodissociation reactions from transition metal carbonyl complexes and the subsequent coupling of the ML6 and ML5 forms. CO(CO)M(CO)M 56 hν (10) 11256 (CO)M(CO)M(CO)M  hν (11) Their model implies that M(CO)5 is generated by a laser beam of circular cross section and monitored by a second beam in collinear arrangement, in a cylindrical volume. M(CO)5 concentration decays reaching zero at the far face of the cuvette, and the absorbance can be calculated from Beer’s law. They also considered the diffusion of M(CO)5 molecules outside of the monitored volume in the case where reaction (11) is sufficiently slow. Mathematically, this system could be described by two-dimensional diffusion equations coupled to a second order reaction. All diffusion coefficients were supposed to be equal. It was assumed that M(CO)6 is in high excess compared to M(CO)5 and the second order rate constant of eq. (11) was calculated by the traditional pseudo-first order method (plotting kobs as a function of [M(CO)6], the obtained slope is k). The percentage differences were given between log(kcalc) and log(kinput) values, where k were used as an input parameter in the model. Upon considering different kinds of concentration inhomogeneities and parameter sets, the following conclusions were drawn from their results: i. when the relative concentration of M(CO)5 was high (violating pseudo-first order conditions), the diffusion outside the cylinder started to interfere below kinput = 107 M1 s1, causing up to 14% decrease ii. inhomogeneous distribution along the light path (due to Beer’s law) did not cause any distortion, as long as the high excess of M(CO)6 was maintained iii. when a Gaussian beam profile was used for the excitation pulse, diffusion caused errors below 107 M1s1, at low relative concentration of M(CO)6; iv. as an interesting complementary information, the pseudo-first order plot for the determination of kcalc was always perfectly linear, even if high errors were found between log(kcalc) and log(kinput). Therefore, the correlation coefficient in this case was not a good indicator for testing pseudo-first order behavior. The results of the above described model calculations were not compared to experimental data because LFP measurements were carried out with perpendicular arrangement. Sample inhomogeneity issues can be assessed by experimental approaches as well. Bonneau and co-workers published a detailed manual for the collection and analysis of transient Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 20 absorption data and emphasized the importance of correct instrument geometry and inhomogeneous transient distributions. They suggested that a mirror should be placed at the side of the sample opposite to the laser excitation spot in order to reflect the pumping light back and thus increase the absorbed fraction of light and offset inhomogeneities at the same time.142 Goez et al. realized the idea above, although they used a solid corner-tube retroreflector instead of a mirror, in order to reduce Beer inhomogeneity (caused by absorption along the light path of the laser).143 Retroreflectors are less sensitive to alignment imperfections than mirrors and make it possible to carry out measurements at different wavelengths without changing the mirror. The incident and the reflected beams were mixed with the aid of the retroreflector which had double beneficial effect: the total absorbed intensity increased and the Beer inhomogeneity decreased (the former by a factor of 1.45-1.65, the latter 2.6-4.8-fold). They found a very simple relation between these two effects. According to their calculations and measurements, in the presence of the retroreflector, the absorbed intensity increased by a factor of 𝜃, then the total decrease of inhomogeneity equals 2/𝜃-1. Distortions arising from the Gaussian beam profile also improved a little by this method, but the authors pointed out that beam shapers or easier experimental settings could correct for the non-uniformity of the beam. Solution inhomogeneities tend to cause distortions in stopped flow (SF) measurements as well, although the source if the non-uniform distribution of concentrations is conceptually different than the ones connected to flash photolysis experiments. In the case of the stopped flow technique, the inhomogeneity arises from the fact that the mixture of the reacting components needs a certain time to fill the observation cell (filling time). For very rapid reactions, whose half-life is comparable to the filling time of the instrument, a significant difference can be formed between the front and the far face of the observation cell (see Figure 2.6). Rorabacher and colleagues developed mathematical models to resolve second order rate constants of such rapid reactions, taking into account the concentration gradient under appropriate conditions. First, they treated irreversible second order reactions, choosing an arbitrary ‘zero time’ of the measurements. They managed to determine a rate constant of 7  106 M1 s1 this way.144 Their model was later improved to remove the reversibility and starting time restrictions and the rate constant limit shifted up by an order of magnitude.145 Finally, they completed their results by another gradient-corrected approach, where they used the steady state absorbance forming in the flow cell before the cessation of the flow.146 The resolved rate constants of both methods, as well as a non-corrected second order treatment for the sake of comparison, were carefully analyzed using fast reactions of transition metal complexes as an experimental model. The rate constants of the latter were predicted from the Marcus theory.147 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 21 As a conclusion, second order rate constants of magnitude up to 108 M1 s1 became available from stopped flow measurements for reactions whose half-lives are less than one half of the instrument dead time (the time required for the solution to travel the distance x + l in Figure 2.6). Figure 2.6. Scheme of sample concentration gradient in a stopped flow experiment. Figure adapted from Dunn et al. J. Phys. Chem. 1996, Fig. 1.146 Considering the sub-nanosecond to millisecond time resolution of the laser flash photolysis technique, it has important overlap with the stopped flow method from the point of view of the order of magnitude of the accessible rate constants. However, the major differences between these techniques still hold, i.e. stopped flow deals with ground state species (even if very short lived), whereas LFP operates with transient species generated by the high energy laser pulse. The laser generation method eliminates the need for manual or instrumental mixing. On the other hand, the ‘batch’ characteristic of LFP measurements is a technical limitation. The point is that all ground state molecules are located in the cuvette at the moment of the laser pulse. Therefore, the effect of the laser on all components has to be tested individually before observing a radical reaction and sometimes it is impossible to isolate a certain component to absorb the total amount of laser energy. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 22 3. Research objectives The present thesis is dedicated to shed further light on the role of sulfate ion radical in the transition metal catalyzed autoxidation of hydrated SO2, more generally, autoxidation of S(IV). The catalytic effect of silver(I) ions in the presence of peroxodisulfate (S2O82) ions is investigated. Traditional methods of aqueous solution kinetics are employed, as well as laser flash photolysis technique for the direct observation of SO4. Computational simulations are carried out to study the potential effect of diffusion and spatial inhomogeneities during laser flash photolysis experiments. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 23 4. Experimental methods 4.1. Materials All chemicals used in this study were of analytical reagent grade and were purchased from commercial sources. Sodium sulfite stock solutions were prepared freshly from Na2S2O5 (Reanal) every day. Whenever necessary, pyrosulfite solutions were deaerated by bubbling Ar (purity > 99.95%) for at least 15 min in order to increase the reproducibility of measurements under anaerobic conditions. Potassium peroxodisulfate stock solutions were prepared from K2S2O8 (Reanal). This solution was oxygenated during the O2-dependent experiments. Silver(I) catalyst solutions were prepared by dissolving a weighed amount of AgNO3 (Reanal) to a known final volume. Doubly deionized and ultrafiltered water from a Millipore Q system was used in the entire work. Most of the experiments were carried out at high and constant acid concentration (0.10 or 0.33 M sulfuric acid). Therefore, additional salt was not used to adjust the ionic strength. 4.2. Instrumentation and softwares 4.2.1. UV-vis spectrophotometric experiments related to S(IV) autoxidation UV-vis spectra were recorded on a Perkin Elmer Lambda 2S or a Perkin Elmer Lambda 25 scanning spectrophotometer. Kinetic experiments were carried out in standard quartz cuvettes (optical path length = 1.000 cm). The overall sample volume was 3.00 cm3 in each case. Constant temperature (25.0  0.1 C) was maintained with an external thermostat and circulating thermal bath. Samples were prepared by the following method. Required aliquots of pyrosulfite, peroxodisulfate and sulfuric acid solutions were mixed with water and thermostated for 3-4 min. The process was started by adding the catalyst immediately prior to commencing the detection in the spectrophotometer. Reaction rates were determined by linear fitting of data curves. One can calculate the actual rate from the slopes of absorbance versus time functions by using the molar absorption coefficient of sulfur(IV), known from independent experiments. From a mathematical point of view, the rate determination requires numerical differentiation. Under the usual conditions, the accuracy of reaction rates are 10%, indicated by the error bars in Figure 5.3-Figure 5.7Figure 5.9. In the silver(I) catalysis study, another method was applied to compute the reaction rates (which were fairly constant within a single experiment) and compare them with the results of numerical derivation. The slopes of the sections preceding the break point were calculated (see Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 24 Figure 5.2) The absorbance change can be transferred into concentration change by dividing it with the (apparent) molar absorption coefficient valid at the pH of the reaction system. The main advantage of the latter method is that it avoids the possible problems arising from imperfect thermostating, which may occur at the beginning of experiments. Reaction rates determined with the two different methods were in very good agreement, and we used the results of the method of initial rates in the final calculations. 4.2.2. Laser flash photolysis measurements Laser flash photolysis (hereafter referred to as LFP) experiments have been carried out in an LKS.60 nanosecond transient absorption spectrometer, shown in Figure 4.1. Figure 4.1. Applied Photophysics LKS.60 nanosecond laser flash photolysis instrument The instrument is equipped with a Quantel Brilliant Nd:YAG laser along with its second, third, fourth and fifth harmonic generators (referred as SHG, THG, FoHG, FiHG). Nd:YAG (Neodymium doped yttrium aluminum garnet) is a common solid-state laser type, where the lasing medium is a Y3Al5O12 crystal doped with 0.1-1% Nd3+. The primary wavelength of an Nd:YAG laser light is 1064 nm, its harmonics emit at 532 (2nd), 355 (3rd), 266 (4th) and 213 nm (5th). Harmonics are generated by specific crystals that exhibit non-linear optical effects. Common non-linear crystals are KDP, DKDP, LaTiO3, BaTiO3 etc. Table 4. Specification of the fourth harmonic of Quantel Brillant Nd:YAG laser Wavelength 266 nm Pulse length (full width at half maximum) 6 ns Repetition rate 10 Hz Energy per pulse (max.) 40 mJ Beam diameter 6 mm Q-switch ON Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 25 In the present thesis, the fourth harmonic of the Nd:YAG laser was used to generate sulfate ion radicals and other transient species. Table 4 contains the characteristic features of the laser beam. Unfortunately, the fifth harmonic generator reduced the energy of the laser pulse too much and it could not be employed for any practical purposes in this study. Q-switching is a frequent technique applied to increase laser power by shortening the laser pulse. Originally, the Q-factor (or quality factor) is a term of electronics, it describes the goodness of an RLC circuit. Q-switching is realized by a Pockels-cell in the LKS.60 instrument. The analyzing light source was a 150 W ozone free xenon arc lamp (OSRAM 150W/CR OFR) and its light beam entered a 1.00  1.00 cm fluorescence quartz cuvette. We have implied cross-beam excitation, the probe beam and the laser beam were introduced into the sample in a perpendicular arrangement. The cuvette was placed in an adjustable sample holder, which was positioned horizontally so that the analyzing light beam hit close to the front face of the cuvette where laser beam entered. This is an important setting as the concentration of transient species generated by the laser excitation typically decreases rapidly with the distance from the front window (see also Section 5.3.7.). A programmable f/3.4 grating monochromator with a symmetrical Czerny-Turner optical configuration was combined with a R928 photomultiplier for the transient signal detection at different wavelengths. Data points were collected by an Agilent Infiniium digital storage oscilloscope (model number DSO8064A, maximum sampling speed 4 GSa/s (0.25 ns between data points), bandwidth 600 MHz, output impedance 50 Ω). The excitation of tris(bipyridine)ruthenium(II) chloride (Ru(bpy)3Cl2) by the third harmonic of the laser (355 nm) was used as a test reaction (Figure 4.2).148 Figure 4.2. A) Transient absorption spectra of tris(bipyridine)ruthenium(II) chloride after 355 nm laser pulse. The signals were recorded at the indicated time points after the laser pulse. B) First order decay of triplet Ru(bpy)32+. The solid line represents the non-linear least square fit to the observed data points. [Ru(bpy)32+] = 31 μM,  ex = 355 nm Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 32 Figure 5.6. Reaction rates as a function of the peroxodisulfate ion concentration in the autoxidation of sulfur(IV). The solid line represents the best fit to the proposed mechanism shown in Scheme 1. [S(IV)] = 3.0 mM; [H2SO4] = 0.103 M; [Ag+] = 0.167 mM; path length = 1.000 cm; [O2] = 0.130 mM; V = 3.00 cm3; T = 25.0 C. Dependence of the reaction rate on pH Finally, the effect of pH on the rate of the zeroth order process is shown in Figure 5.7. The pH was calculated from the concentration of added sulfuric acid considering the fact that hydrogen sulfate ion is not a strong acid (pKa2 = 1.06 at 25 °C) Figure 5.7. Reaction rates as a function of the pH in the autoxidation of sulfur(IV). [S(IV)] = 3.0 mM; [S2O82] = 0.0284 M; [Ag+] = 0.167 mM; [O2] = 0.130 mM; path length = 1.000 cm; V = 3.00 cm3; T = 25.0 C. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 33 5.1.3. Suggested mechanism Our kinetic observations, in accordance with earlier results of investigations77,79 about the autoxidation of S(IV), confirm that a chain reaction takes place in the studied system. The initiation step is very likely to be the well-known reaction between silver(I) and peroxodisulfate ions (2), which produces two reactive chain carriers, silver(II) and sulfate ion radicals. Another initiation step will be mentioned later, as the reverse direction of a chain termination step.     2 44 2 18 2 82 SOSOAgOSAg k (18) Among the propagation steps, the reaction of Ag(I) and SO4 producing silver(II) and HSO4 certainly occurs (19).     2 4 2 19 4SOAgSOAg k (19) Previous studies revealed that the possible direct reaction between sulfate ion radical and sulfur(IV) does not play a role in the autoxidation process.62,77,79 Therefore, it is reasonable to assume that silver(II) reacts with sulfur(IV) in the chain reaction. Ag(II) is a strong oxidizing species, which can easily be reduced by S(IV) in a one-electron step (20).    3 20 2SOAg(IV)SAg k (20) It is also known from the literature that sulfite ion radical reacts very quickly with dissolved oxygen124 and it is likely that the product of this reaction, peroxomonosulfate ion radical (SO5), oxidizes sulfur(IV) (21, 22).   5 21 23 SOOSO k (21)   4 2 4 22 5SOSO(IV)SSO k (22) At this point, one can see that steps (19 – 22) compose a chain in which one cycle leads to the formation of two sulfate ions by the reaction of one oxygen and two sulfur(IV). With regard to chain termination, one can consider the disproportion of silver(II) into silver(I) and silver(III) which is also a previously published assumption.43,45 (III)AgAgAg223 2  k (23) Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 34   2 24 Ag2(III)AgAg k (24) Eqs. (23) and (24) describe a reversible process in which the reverse step acts as initiation, as it produces a chain carrier. This should be taken into account in the derivation of the rate law. Silver(III) is a very reactive species and it react with sulfur(IV) in a two-electron process (25), where silver(I) is reproduced, ready to join the catalytic cycle again.   2 4 52 SOAg(IV)S(III)Ag k (25) Silver(III) is supposed to occur as the AgO+ oxocation in aqueous solution, which means that reaction (25) possibly occurs as oxygen atom transfer.45 The recombination of sulfate ion radicals to peroxodisulfate ion (26) has always appeared in the chain mechanisms of the autoxidation processes of sulfur(IV) examined before.77,79 For this reason, it seems appropriate to consider it in the scheme as a possible termination step.    2 82 26 4OSSO2k (26) Scheme 1. The suggested mechanism of the autoxidation of sulfur(IV) in the presence of silver(I) and peroxodisulfate ions.     2 44 2 18 2 82 SOSOAgOSAg k v1=k18[Ag+][S2O82−] (R1)     2 4 2 19 4SOAgSOAg k v2=k19[Ag+][SO4•−] (R2)     3 obs 20 2SOAg(IV)SAg k v3= 𝑘20 obs[Ag2+][S(IV)] (R3)   5 21 23 SOOSO k v4=k21[SO3•−][O2] (R4)    4 2 4 obs 22 5SOSO(IV)SSO k v5= 𝑘22 obs[SO3•−][S(IV)] (R5) (III)AgAgAg223 2  k v6=k23[Ag2+]2 (R6)   2 24 Ag2(III)AgAg k v7=k24[Ag+][Ag(III)] (R7)    2 4 obs 25 SOAg(IV)S(III)Ag k v8= 𝑘25 obs[Ag(III)][S(IV)] (R8)    2 82 26 4OSSO2k v9=k26[SO4•−]2 (R9) Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 35 Figure 5.8. The suggested mechanism of the autoxidation of sulfur(IV) in the presence of silver(I) and peroxodisulfate ions. 5.1.4. Derivation of the rate law The regular mathematical treatment of chain reactions is the long-chain approach.91 Essentially, it is built on two main pillars:  the steady-state approximation can be applied to every reactive intermediates (Ag(II), Ag(III), SO3, SO4, SO5 in the present study)  the rate of chain initiation or termination is lower than those of the propagation steps. Two generally applied conclusions can be drawn from the assumptions above: i. Each propagation step has the same rate as the rate of the net reaction. ii. The rate of initiation and termination steps are equal. If more than one initiation or propagation steps occur, then equality refers to the sum of the rates of a certain type:   kkT, jjI, vv Here j is the number of initiation steps (I), k is the number of termination steps (T). The rate of reaction (1) may be described by eq. (27).     dt Od dt SOOHd 2 1222   v (27) Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 36 The rate is equal to the rate of any propagation steps, so it can be derived from eqs. (19)- (22). It is convenient to use eq. (20) for this purpose, in order to simplify subsequent calculations. v = v3 = 𝑘20 obs[Ag2+]ss[S(IV)] (28) here subscript ss indicate steady-state concentrations All reactions discussed are supposed to be second order, and these are elementary steps except for reactions (20), (22) and (25). In the latter cases, the deprotonation of H2OSO2 should be taken into account in the pH-range of the study. H2OSO2 and HSO3 forms are related by a fast pre-equilibrium. Consequently, the rate of step (20) is given in eq. (29) v = [Ag2+]ss(k20[H2O∙SO2] + k20′[HSO3]) (29) Considering the equilibrium of the two S(IV) forms, the formula takes the following form: )][H( ][H [S(IV)]][Ag a2020 a ss 2K'kk K v     (30) Implying the long-chain assumption (ii), one can obtain eq. (31): 2 ss426 2 ss 2 23ss24 2 8218 ][SO][Ag][Ag(III)][Ag]O][S[Ag     kkkk (31) The steady-state concentration of silver(III) can be expressed from eq. (32) as follows: 2 ss 2 23ss24 a a2525 ss ][Ag][Ag[Ag(III)] ][H ][H [S(IV)][Ag(III)]      kk K K'kk (32) ][Ag ][H ][H [S(IV)] ][Ag [Ag(III)] 24 a a2525 2 ss 2 23 ss         k K K'kk k (33) The steady-state concentration of sulfate ion radicals can be given from the fact that the rates of propagation steps are equal (v2 = v3): a a2020 ss 2 ss419 ][H ][H [S(IV)]][Ag][Ag][SO K K'kk k      (34) )]([H ][H ][Ag [S(IV)]][Ag ][SO a19 a2020ss 2 ss4Kk K'kk         (35) Substituting the steady-state concentrations in eqs. (33) and (35) into eq. (31) yields: Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 37 2 a19 a2020 2 2 2 ss 2 26 2 ss 2 23 24 a a2525 2 ss 2 23 24 2 8218 )]([H ][H ][Ag [S(IV)]][Ag ][Ag ][Ag ][H ][H [S(IV)] ][Ag ][Ag]O][S[Ag                             Kk K'kk kk k K K'kk k kk (36) From eq. (36), the steady-state concentration of silver(II) can be expressed:   ][Ag ][H ][H S(IV) ][Ag )]([H ][H ][Ag [S(IV)] 1 ]O][S[Ag ][Ag 24 a a2525 24 2 a19 a2020 2 2 23 26 2 82 23 18 2 ss 2                           k K K'kk k Kk K'kk k k k k (37) The combination of eqs. (30) and (37) leads to the final rate law:   ][Ag ][H ][H S(IV) ][Ag )]([H ][H ][Ag [S(IV)] 1 ]O][S[Ag ][H )][H[S(IV)]( 24 a a2525 24 2 a19 a2020 2 2 23 26 2 82 23 18 a a2020                                k K K'kk k Kk K'kk k k k k K K'kk v (38) Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 38 The suggested model fits the observed kinetic data very well. Solid lines in Figure 5.3Figure 5.6 represent the best fit to the proposed mechanism. Figure 5.9 shows the correlation between all reaction rates measured at one given pH ([H2SO4] = 0.103 M). The horizontal axis contains the measured rates, while data on the vertical axis are the rates calculated with the parameters from the best fit. The correlation plot also confirms that the model provides excellent quantitative interpretation of the kinetic results. Figure 5.9. Correlation between measured and calculated rates. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 39 5.2. Laser flash photolysis studies on the reactions of sulfate ion radical 5.2.1. Generation and recombination of the sulfate ion radical (SO4) Previously published generation methods and characterization of the sulfate ion radical, SO4, are extensively discussed in Section 2.3. In our experiments, SO4 was generated by the photolysis of an aqueous solution of K2S2O8 (3). The radicals are the product of the homolytic scission of the peroxy bond in S2O82– ion.   4 2 82 SOOS hν (39) Whenever required for the calculations,  (SO4)450 nm = 1600 dm3 mol–1 cm–1 was used, based on the studies of McElroy.118 Following the photolysis of a 0.0967 M K2S2O8 solution at 266 nm, kinetic traces similar to the one in Figure 5.10 were detected. The recorded kinetic traces in the absence of any other reactants were fitted with second order curves, and they all gave a very good fit. Therefore, it is reasonable to assume that the recombination of sulfate ion radicals into the precursor peroxodisulfate ions takes place under these conditions (40), which is the same process as the last chain termination step in Scheme 1. (R9).    2 82 40 4OSSO2k (40) Figure 5.10. Transient absorption signal of the sulfate ion radical. [K2S2O8] = 0.0967 M, T = 25 °C, V = 3.00 cm3. The solid line represents the fitted second order curve (k = 3.91  108 M1 s1). Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 40 The second order rate constant of the reaction, denoted here by k40, was determined from the fitted parameters considering the molar absorption coefficient mentioned above. The value of the calculated second order rate constant was (4.1  0.2)  108 M–1s–1. The recombination process occurs on the millisecond timescale, which requires special treatment, as the basic settings of the instrument are configured for nanosecond measurements. In order to measure on longer timescales, the impedance of the oscilloscope has to be increased from 50 Ω into the kΩ range, up to 1 MΩ, by the installation of a variable resistance box between the signal cable from the photomultiplier tube and proper oscilloscope channel. Furthermore, at longer timescales, the Xe arc lamp has to be operated in continuous mode instead of pulsed mode, otherwise the lamp flashes would be too short for the detection of the transient species. Theoretically, the LKS.60 instrument with its actual equipment is able to follow chemical reactions on the timescales between 1 ns to a few seconds with the convenient instrument settings and data evaluation methods implied. In order to reveal a possible pH-dependence of the rate constant, k40 was measured over the acidic range. Figure 5.11 shows that the rate constant of the recombination process is independent of pH in the implied range. Figure 5.11. The effect of pH on the recombination rate of sulfate ion radicals. [K2S2O8] = 0.0967 M, T = 25 °C, V = 3.00 cm3. 5.2.2. Reaction of sulfate ion radical and silver(I)-ion In the subsequent stage of our kinetic investigation, the main focus was the role of sulfate ion radical in the autoxidation process of sulfur(IV). One can see from Scheme 1 that SO4 is a chain carrier intermediate, it is produced in the initial step (2) and in a propagation step by the reduction of peroxomonosulfate ion radical. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 41 Figure 5.12. Absorption at 450 nm following the photolysis of K2S2O8 (0.100 M) solution, containing AgNO3 (6.12 mM). The solid line represents the fitted single exponential curve (kobs = 3.8  107 s1). The reaction between sulfate ion radical and silver(I) ion is of high importance in chain propagation, because it is the process through which the catalyst enters the cycle. Therefore, we intended to determine the rate constant of reaction (19) by means of laser flash photolysis. An aqueous solution of AgNO3 was photolyzed at 266 nm as a blank experiment in order to measure its contribution to the overall absorption in the reaction. It was found that silver nitrate does not show any transient absorption at 450 nm, thus the absorption can be attributed solely to sulfate ion radical. Figure 5.12 shows a representative transient absorption curve. The reaction takes place on the nanosecond timescale, and can be fitted with a single exponential curve, thus referring to a first order process. Concentrations of the reactant were set in a way that pseudo-first order conditions would be fulfilled to Ag+ ions over sulfate ion radicals. A surprising observation was made following the photolysis of K2S2O8–AgNO3 aqueous solution. After a few laser shots (3-10 depending on the concentration ratio), dark grey cloudy precipitation was seen in the cuvette, first at the spot of laser excitation. Ag2O formation was excluded upon the facts that the color of the precipitate is different from brownish Ag2O, it does not dissolve in sulfuric acid and it appears even at very low pH. The dark grey precipitation is thus assumed to be elementary silver, the source of which is unknown so far. It was thought to be produced in the photodecomposition of AgNO3, but in the absence of K2S2O8, the precipitation did not occur. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 48 Figure 5.21. Ratio of second order rate constant of the decomposition and the molar absorption coefficient of chlorine molecule anion. [K2S2O8] = 0.100 M, () [H2SO4] = 0 M, (▼) [H2SO4] = 0.0049 M, T = 25 °C, V = 3.00 cm3. On the longer time scale, the second order decomposition of chlorine molecule anion was detected (see Figure 5.21). The second order rate constant of the decay of Cl2 is (1.4 ± 0.04)  105 s1 cm   (Cl2−) at either pH = 2.03 or in unbuffered aqueous solution. 5.2.6. Reaction of sulfate ion radical with bromide ion In the reaction of sulfate ion radical with bromide ion, bromide molecule anion is produced in reactions (47) and (48). Kinetic data were collected at 360 nm (see spectra in Figure 5.22).157 Br + SO4 = Br + SO42 (47) Br + Br ⇌ Br2 (48) Figure 5.22. Time resolved transient absorption spectra of the bromine molecule anion. [K2S2O8] = 0.100 M, [KBr] = 0.010 M, V = 3.00 cm3, T = 25 °C, pH = 7 (unbuffered). Spectra were recorded at the indicated time points after the laser pulse. Negative time on panel A (-0.1 μs) refers to the pre-pulse section of the curves (panel B, t < 0). Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 49 Figure 5.23. Pseudo-first order rate constants as a function of bromide ion concentration in the reaction of sulfate ion radical and bromide ion. [K2S2O8] = 0.100 M, () [H2SO4] = 0 M, (▼) [H2SO4] = 0.0049 M, T = 25 °C, V = 3.00 cm3. The reaction was followed on shorter and longer timescales, under pseudo-first order conditions. The second order rate constant was found to be (1.51 ± 0.02)  109 M1 s1 in unbuffered aqueous solution and (1.59 ± 0.03)  109 M1 s1 at pH 2.03 (Figure 5.23). On the longer time scale, the second order decomposition of bromine molecule anion was detected, which was found to be independent from the bromide ion concentration (see Figure 5.24). The second order rate constant of the decay of Br2 is (1.8 ± 0.1)  105 s1 cm   (Br2−) in unbuffered aqueous solution and at pH = 2.03. Figure 5.24. Ratio of second order rate constant and molar absorption coefficient of the decomposition of bromine molecule anion. [K2S2O8] = 0.100 M, [H2SO4] = 0.0049 M, V = 3.00 cm3, T = 25 °C. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 50 5.2.7. Reaction of sulfate ion radical with tryptophan The kinetics and mechanism of the oxidation of tryptophan (Trp) and its derivatives by peroxomonosulfate ions have been recently studied in our laboratory.158 Since tryptophan is an aromatic amino acid with numerous potential oxidation sites (Figure 5.25), the detailed characterization of its oxidation reactions is a rather challenging task that needs to be approached by a combination of experimental techniques. The laser flash photolysis method provides an efficient tool to study the one electron oxidation reactions of Trp that involve short lived reactive intermediates. The one electron oxidation of aromatic amino acids such as tryptophan and tyrosine plays an important role in the formation of protein radicals,159-163 thus participating in electron transport and redox regulation. Figure 5.25. Structure of L-tryptophan and L-tryptophanamide. The photolysis of tryptophan at 266 nm on a microsecond time scale led to kinetic curves containing a positive and a negative peak (see representative curve in Figure 5.26). The positive peak could be assigned to the transient absorption of the intermediate product of the photolysis of Trp, whereas the negative peak presumably corresponds to the laser induced fluorescence of Trp. Negative peaks are completely interpretable in the context of flash photolysis curves, since the measured value is in fact the difference between the absorbance of the ground state species and the excited species. Figure 5.26A shows that the intensity of the negative peak exceeds that of the positive one. The ratio of these values (in favor of the absorption peak) could be optimized by the alteration of the wavelength of the analyzing beam (Figure 5.26B). The emission-to-absorption ratio was found to be optimal at 580 nm, further kinetic experiments were carried out at this wavelength. The 580 nm peak in the transient spectrum of excited tryptophan was previously assigned to be the photoionization product of Trp, a radical cation (Trp+).164 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 51 Figure 5.26. A) Representative kinetic curve of the 266 nm photolysis of aqueous solution of tryptophan. B) Intensity of light absorption and emission in the 266 nm photolysis of aqueous solution of tryptophan, based on kinetic curves as in panel A. The peak intensity corresponds to the absolute value of the maxima of the positive and negative peaks, in absorbance units. Conditions for both panels: [Trp] = 0.5 mM, unbuffered, V = 3.00 cm3, T = 25 °C, l = 1.000 cm. The plots represent A) the average B) the average and standard deviation of 5 parallel measurements, where fresh sample was applied for each laser shot. Figure 5.27 shows the linear dependence of the observed pseudo-first order rate constants on the concentration of tryptophan. 580 nm was not suitable to follow the reaction at pH 7.21 due to poor signal to noise ratio. 520 nm was found to be optimal with regards to emission to absorption ratio. 520 nm is the reported absorption maximum of TrpN radical.162,165-167 Figure 5.27. Pseudo-first order rate constants in the reaction of sulfate ion radical and tryptophan. The data points and error bars represent the average and standard deviation of 5 parallel measurements, where fresh sample was applied for each laser shot. The measured second order rate constants were (7.4 ± 0.1) × 109 M1 s1 in unbuffered solution and (8.7 ± 0.4) 109 M1 s1 close to physiological pH (7.21). [K2S2O8] = 0.100 M, unbuffered (■), pH = 7.21 (20 mM phosphate buffer, ▼), V = 3.00 cm3, T = 25 °C, l = 1.000 cm. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 52 Figure 5.28. Pseudo-first order rate constants in the reaction of sulfate ion radical and Ltryptophanamide. The data points and error bars represent the average and standard deviation of 5 parallel measurements, where a fresh sample was applied for each laser shot. The solid line represents the linear least square fit to the measured points. The slope of the line is (8.8 ± 0.2) 109 M1 s1 and the intercept is (0.8 ± 0.2) 106 s1. [K2S2O8] = 0.100 M, pH = 7.21 (phosphate buffer), V = 3.00 cm3, T = 25°C, l = 1.000 cm,  = 520 nm. L-Tryptophanamide (TrpA) was tested in experiments similar to those with Trp, the aqueous solution of TrpA and K2S2O8 was photolyzed at 266 nm and the absorbance was followed at 520 nm. The observed rate constant depends linearly on the concentration of TrpA, with a positive intercept on the vertical axis (Figure 5.28). Effect of dissolved oxygen concentration in the reaction of sulfate ion radical and tryptophan We altered the concentration of oxygen in the sulfate ion radical–tryptophan radical system in order to reveal any potential influence of dissolved oxygen on the observed rate constants. When argon or nitrogen was vigorously bubbled into the reagent solutions for 15-20 minutes before the excitation, it had no observable effect on the photolysis of tryptophan solution. On the other hand, upon the application of a laser shot on the de-aerated samples, a previously unobserved yellow product appeared in the solutions. The characterization of the nature of this product would require detailed spectroscopic and kinetic studies. From a kinetic point of view, the increase or decrease of oxygen concentration in the solutions caused only slight differences in the measured rate constants. The measured second order rate constants were (9.4 ± 0.5) × 109 M1 s1 in air saturated solution and (9.0 ± 0.3) 109 M1 s1 in de-aerated samples. Data points obtained upon oxygenation of the solutions still show linear dependence, although the function has a positive intercept on the vertical axis. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 53 Figure 5.29. A) Representative kinetic curves in the reaction of sulfate ion radical with Ltryptophan with different oxygen concentrations. Oxygen was removed by vigorously bubbling argon in the reagent solutions for 20 min, whereas oxygenation was achieved by introducing extra oxygen in the samples from a gas bottle for 15-20 min. B) Pseudo-first order rate constants in the reaction of sulfate ion radical with L-tryptophan from kinetic curves as shown in A). The data points and error bars represent the average and standard deviation of 5 parallel measurements, where fresh sample was applied for each laser shot. The solid line represents the linear least square fit to the measured points. The slope of the blue line is (8.4 ± 0.8) 109 M1 s1 and the intercept is (1.4 ± 0.4) 106 s1. [K2S2O8] = 0.100 M, pH = 7.21 (phosphate buffer), V = 3.00 cm3, T = 25 °C, l = 1.000 cm,  = 520 nm. Figure 5.29A shows three representative kinetic curves recorded at 520 nm in air saturated, de-aerated and oxygenated solutions, respectively. Figure 5.29B shows the observed pseudofirst order rate constants as a function of the applied tryptophan concentration with different oxygen saturation in the samples. The measured second order rate constants were (9.4 ± 0.5) × 109 M1 s1 in air saturated solution and (9.0 ± 0.3) 109 M1 s1 in de-aerated samples. Data points obtained upon oxygenation of the solutions still show linear dependence, although the function has a positive intercept on the vertical axis. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 54 Alternative generation of tryptophanyl radical The literature claims that tryptophanyl radical can be generated by the oxidation of aqueous tryptophan by bromine molecule anion (Br2).165,168 The experimental constraint is that we were not able to detect Br2directly upon the photolysis of bromide solution, but only by the oxidation of bromide with sulfate ion radical (see Figure 5.23). It is still feasible to oxidize Trp by bromine molecule anion in the presence of K2S2O8. However, one should take the oxidation of Trp by sulfate ion radical into account as well according to the above mentioned rates. Therefore, in the SO4–Trp–bromide system, we should consider eq. (47) as well. The introduction of Trp increases the observed rate constant of the reaction in such a mixture (Figure 5.30). The reaction was followed at 360 nm, the absorption maximum of bromine molecule anion, on a microsecond time scale. The decay curves did not fit nicely to a single exponential curve. A second order fit was much better curves in agreement with the second order decay of Br2. Figure 5.30. Second order decay of bromine molecule anion in the absence and presence of Ltryptophan. [K2S2O8] = 0.100 M, unbuffered solution, V = 3.00 cm3, T = 25 °C, l = 1.000 cm. The introduction of 0.3 mM Trp led to a 3-fold increase in the observed rate constant. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 55 5.2.8. Reaction of sulfate ion radical with tyrosine Similarly to the study carried out with tryptophan, another aromatic amino acid, tyrosine (Tyr) was also reacted with sulfate ion radical, in order to measure the second order rate constant of reaction (49). Tyr + SO4  Tyr + SO42 (49) The reaction was followed at 450 nm, because in the applied concentration range, the concentration of Tyr radical that is generated is insufficient to produce decent absorption signal. Under pseudo-first order conditions, the observed rate constants showed linear dependence on the concentration of tyrosine, as indicated in Figure 5.31. The data points measured in acidic solutions fit very well to linear functions going through the origin. The obtained second order rate constants were (6.1 ± 0.1)  109 M1 s1 in acidic medium (black line – the pH is not constant in this case, concentration of added sulfuric acid is increasing) and (5.1 ± 0.1)  109 M1 s1 at pH 2.72. Close to neutral pH, the dependence remains linear, although the linear function has a small positive intercept. The second order recombination of the tyrosyl radical leads to the formation of dityrosine, which is known to have an emission peak at 405 nm and forms the basis of several fluorescent assays in the biochemical laboratory practice.169,170 Figure 5.31. Observed rate constants in the reaction of sulfate ion radical and tyrosine as a function of tyrosine concentrations at different pH values. [K2S2O8] = 0.100 M, ■: changing pH ([H2SO4] from 1.5 to 15 mM), ●: pH = 2.72 (measured); ▲: pH = 7.14 (0.1 M phosphate buffer), V = 3.00 cm3, T = 25 °C,  = 450 nm. Solid lines represent the linear fit to the measured data points. The blue line has a slope of (1.4 ± 0.1)  109 M1 s1 and an intercept of (0.4 ± 0.3)  106 s1. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 56 5.2.9. Reaction of sulfate ion radical with S(IV) In the next phase of our studies, we intended to examine the reaction of sulfate ion radical with sulfuroxy anions (SxOyz-), preferably with the ones containing sulfur in the oxidation state of +4. During the autoxidation cycles of S(IV) (see Figure 5.8), SO3 and SO5 are produced as chain carriers along with the sulfate ion radical. We sought independent methods to study the reactions of these species. We postulated the existence of reaction (): SO4 + SO32SO42 + SO3  To check this possible process, we reacted sulfate ion radical with sulfite solutions. The technique was similar to the one applied previously: sulfate ion radical was generated by the photolysis of potassium persulfate. Sulfur(IV) has an absorption peak at 275 nm (see Figure 5.1) in acidic medium. Therefore, K2S2O8 has to be in high excess in order to absorb the majority of light from the incident (laser) pulse. In our systems, K2S2O8 is usually present in 0.1 M concentration in the samples, whereas the reactant species are applied in the millimolar concentration range, K2S2O8 is in 10-100-fold excess. On the other hand, the reactant species has to be in excess over the generated sulfate ion radical, because pseudo-first order approach is applied in the data processing. The pseudo-first order conditions are favorable in free radical reactions because the actual concentrations of the transient species are not required. The third requirement is that the reactant has to be applied in an appropriate amount in order to reach decent signal-to-noise ratio in the observed kinetic curves. Sometimes other factors are to be considered as well, such as the solubility of the compound, possible light sensitivity, thermal reaction with K2S2O8, etc. Because of the above mentioned factors, the optimal concentration range should be the subject of careful consideration. Sodium pyrosulfite or metabisulfite (Na2S2O5) was used as a source of sulfite ions, similarly to the autoxidation studies of S(IV) presented in the Section 5.1. The effect of pH along with dissolved oxygen concentration was examined on the rate constant of () and the reaction was followed at 450 nm. Significant difference was observed when the medium was changed from neutral to strongly acidic (Figure 5.32). Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 57 Figure 5.32. Pseudo-first order rate constants as a function of S(IV) concentration in the reaction of sulfate ion radical and S(IV). [K2S2O8] = 0.100 M, pH = 6.98 (0.1 M phosphate buffer, ■) or 1.16 (0.045 M H2SO4, ●), V = 3.00 cm3, T = 25 °C,  = 450 nm. The solid lines represent the linear fit to the measured data points. The black line has a slope of (1.3 ± 0.01)  109 M1s1 and its intercept with the vertical axis is at (0.8 ± 0.1) s1whereas the slope of the red line, thus the respecting second order rate constant is (0.4 ± 0.01)  109 M1s1. The negative intercept at neutral pH shown in Figure 5.32 implies that the dissolved oxygen present in the samples might oxidize a portion of the S(IV), thus the indicated values are overestimations of the actual concentrations that participate in reaction (). Therefore, the effect of dissolved oxygen was tested by bubbling argon in the solutions and repeating the experiments with de-aerated samples, in neutral and acidic medium as well (Figure 5.33 and Figure 5.34). The negative intercept becomes insignificant when argon is introduced, confirming the contribution of dissolved oxygen to the oxidation process. Figure 5.34 would also suggest that the concentration of dissolved oxygen actually has an influence on the rate constant under acidic conditions, if only the red and the blue data series were taken into account (since the pH values are very close). However, considering the black points as well, measured in air saturated solutions such as the red one, the more probable explanation is that the rate constant is highly sensitive to pH, and dissolved oxygen has little effect compared to the acidity of the medium. The pH dependence can be observed in de-aerated samples, too (Figure 5.35). Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 64 absorption coefficients are more difficult to determine, which leads to a high level of uncertainty in the  values106 (see Section 2.3). Still, we can make a rough estimation of [SO4]0, where t = 0 represents the time point of the laser excitation (note that the excitation also pulse has a non-zero length in time, ~20 ns but for our purposes it can be handled as a point). A second order fit of the recombination curves gives an idea of the initial concentration, using  = 1600 M1 cm1 published by McElroy118 and l = 0.5 cm as an optical path length. Fitting of pure second order recombination curves of sulfate ion radical gave a value of 2 104 M, which is comparable to the applied concentration of iodide ion. This fact raised some questions: why are the detected curves exponential and why does the dependence on the concentration of iodide ion give a reasonably good straight line? Returning to the above mentioned considerations (i-iv), the concentration of SO4 is a local value, it applies for the location of the detection that is limited to a fractional volume in the cuvette. The total volume is 3.00 cm3, so if the average concentration of SO4 is calculated over the total volume, a much lower value is obtained. Additionally, while SO4 goes through a rapid decay, iodide ion can be permanently replenished by diffusion from outside of the reaction space, thus maintaining the high excess of iodide over sulfate ion radical. In the following paragraphs, an attempt will be made to analyze the effect of diffusion in the sulfate ion radical–iodide ion system and to explore the effect of spatial inhomogeneity on the observed rate constants. 5.3.2. Reaction-diffusion equation using cylindrical coordinates A general reaction-diffusion (hereinafter referred to as: RD) equation has a closed form shown in (51): )( tcRcD c   (51) Here, c is the vector of concentrations, D is the matrix of the diffusion coefficients,  is the Laplace operator or Laplacian and R is a complex operator embedding all of the ongoing reactions in the system. Dc is called the diffusion term, R(c) forms the reaction term. Inspired by the seminal work of Alan Turing on morphogenesis (shape formation in biological systems),182 reaction-diffusion systems stand in the main focus of research on chemical and biological pattern formation183-185 as well as other nonlinear dynamic phenomena such as oscillation, chaos or self-organizing systems.186,187 Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 65 Eq. (51) is a second order, semilinear, parabolic partial differential equation (PDE) system, in which each chemical entities is represented by a time-dependent variable.149,188 The solution of eq. (51) is a vector of c(t) functions, where each concentration is explicitly expressed as a function of time. The Laplacian () is a differential operator with a scalar value that gives the divergence of the gradient of a real valued function. For an f(x,y,z) function in the three-dimensional Euclidean space, 2 2 2 2 2 2 2 zyx ff          (52) where  is the gradient of f,                z , y , x (53) Eq. (52) describes the Laplacian in Cartesian coordinates, but it is favorable for our purposes to transform the problem into a cylindrical coordinate system, since the reaction space is a circular cylinder (see Section 5.3.4). Figure 5.38 shows the transformation from Cartesian to cylindrical coordinates, and eq. (54) is the form of Laplacian in the new coordinate system. 2 2 22 2 2 211             rr rr x (54) The angular (fourth) term in eq. (54) can be neglected in our model because of the cylindrical symmetry. Thus, applying eq. (54) in eq. (51) yields: )( 1 t2 2 2 2cR ccc D c                    r rr x (55) Figure 5.38. Transformation from Cartesian to cylindrical coordinates. x = x, y = r ∙ cos(  ) and z = r ∙ sin(  ). Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 66 In the present case, two concentration functions are to be calculated, )( 4 SO x,r,tc  and )( Ix,r,tc  , the objective is to gather an insight of the spatial distribution of the reactants at each particular time point. 5.3.3. Approximation of the partial derivatives Infinitesimal quantities cannot be handled computationally, partial derivatives involved in eq. (55) need to be approximated. Traditionally, derivatives are approximated by finite differences, that is, the slope of the tangent at a certain point on the graph is close to the slope of a secant lying on two nearby graph points (Figure 5.39).189,190 Figure 5.39. Geometric interpretation of the finite differences approximation. Given with formulae:   Δx ΔxxuΔxxu Δx Δxxuxu Δx xuΔxxu xu' iiiiii i2 )()()()()()(       (56) The left side is the derivative of the u(x) function at point xi,  x is a small arbitrary distance on the x axis and the terms are called forward, backward and central differences, respectively (it is easy to see that the central difference is the average of the forward and the backward ones). It is very common in numerical problems (e.g. integration of functions or searching for roots of polynomials) that an appropriate grid (also called mesh) is defined on the domain of the function, values of the function are known at the grid points and further values are calculated using these previously known values. Equidistant grids (where  x is constant) are useful and practical for many cases, but the grid points have to be chosen cautiously at other times to minimize the error of the approximation (the difference between the real and the calculated Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 67 values). Using such a method, complicated differential and integral issues can be discretized and rendered computationally solvable. Higher order derivatives can be similarly estimated by finite differences, e.g. second order central difference is given in eq. (57):   2 )()(2)( Δx ΔxxuxuΔxxu xu" iii i   (57) It is convenient to rewrite eqs. (56) and (57) into a closed form using matrix multiplication. Assuming an equidistant one-dimensional grid with an interval length of h, the central difference in eq. (56) takes the following form:                  1 1 101 2 1 xi xi xi i u u u h xu' (58) Here, uxi denotes the value of u at grid point xi, and xi+1 = xi + h, xi1 = xi  h. The second order difference can be similarly represented with matrix multiplication, by                  1 1 121 1 xi xi xi i u u u h xu" (59) Eqs. (58) and (59) are easily applicable to multivariate functions. Our model requires a two dimensional grid to resolve the x and r extents of the reaction space. According to eq. (55), 2 2c x  , r c and 2 2c r  need to be approximated. Applying eqs. (56) and (57) in eq. (55) gives: 2 11 11 2 11 2 2 2 2 )()(2)( )()( 1 )()(2)( 1 Δr ,rxc,rxc,rxc Δr ,rxc,rxc r Δx ,rxc,rxc,rxc r c r c r x c jijiji jijijijiji i,j                          (60) Here i and j are the indices of the grid representing x and r, axes and c can either be c(SO4) or c(I). The sum of eqs. (61)-(63) yields eq. (60) if eqs. (58) and (59) are used in a twodimensional arrangement. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 68 2 1 1 2 21 )( )( )( 2100 1 121 1 0012 x ,rxc ,rxc ,rxc x c ji ji ji i j,i                                                             (61) rr ,rxc,rxc,rxc r c ri jijiji j,i 11 010 11 0 11 010 )()()( 111                                                          (62) 2 11 2 21 210 11 2 11 012 )()()( r ,rxc,rxc,rxc r ci jijiji j,i                                                           (63) The limitation of this method (i.e. the production of the differences by matrix multiplication using coefficient matrices) arises from the r = 0 case (eq. (62) contains r in the denominator), which is one of the boundary conditions for solving eq. (55). Physically, r = 0 is at the central axis of the cylinder, no diffusion occurs along this axis as a consequence of the cylindrical symmetry. This condition cannot be handled in the frame of the formulation above, an alternative treatment of the diffusion term is given in the following sections. 5.3.4. Model of the reaction space Our description of the reaction space is a modified version of a model provided by Cassidy and Long (see Section 2.4).141 These authors published a two-dimensional model for measuring pseudo-first order rate constants in laser flash photolysis experiments in collinear arrangement, where the laser beam and the analyzing beam are parallel to each other. In our calculations, we assumed crossed or perpendicular arrangement in accordance based on the experimental setup. Figure 5.40 shows the geometric model applied to represent the physical reaction space. j  j  j  Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 69 Figure 5.40. Model geometry for modelling the role of diffusion in laser flash photolysis experiments. See explanation of the labels in the text below. The arrow labelled ‘hν’ represents the laser beam and the blue arrow is the analyzing beam that is used to follow the concentration of the transient species. Adapting the common notations of Beer’s law, I0 is the intensity of the entering light beam and I is the intensity of the leaving beam. x and r are the labels of the spatial axes, L is the path length of the cuvette in the direction of the laser beam. The cross section of the laser beam is considered to be circular with r0 radius, and the origin is fixed to the center of this circle at the entering point of the laser. The shading refers to the decreasing concentration of the transient species along the laser beam. Sulfate ion radical is generated by the laser pulse along the x axis, radially from the origin within r0, thus forming a cylinder-shaped volume. An external radius is defined (rex, Figure 5.41) around this object, diffusion of both sulfate ion radical and iodide ions is allowed from this outer shell to the internal cylinder and vice versa. Figure 5.41. Cross section of the reaction space, approximated by the unity of an inner (r0) and outer (rex) cylinder. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 70 5.3.5. Discrete random walk model of diffusion According to Section 5.3.4., the reaction of sulfate ion radical and iodide ion takes place in a cylindrical space, surrounded by an external tube which is accessible for the diffusive motion of the particles. In order to calculate the spatial distribution of each reactant, the reaction space (including the outer shell) is divided into small volume units that are shaped like cylinder rings (Figure 5.42.). Figure 5.42. Shape of the volume units. This kind of resolution into elementary volumes or cells replaces the interpretation in 5.3.3, where a two dimensional grid was introduced on the x and r axes. This model is conceptually different, in a sense that instead of approximating each partial derivatives in eq. (55) individually, diffusion in each volume unit is treated as a matter exchange with the neighboring cells. The number of particles in a certain cell in a given time interval changes via two ways: species can arrive from the nearby cells and others can leave the unit to enter those adjacent cells. Therefore, the change of concentration in the cell will be the resultant of the outflow and the inflow via diffusion. It is important to note that only the diffusion term is considered at this point, the reaction term is introduced at a later stage. An arbitrary volume unit is in juxtaposition with four other ones, two in axial, two in radial directions. In the x direction, they form a shape like a longer tube, whereas in the radial dimension, the smaller ones are embedded into the bigger ones. The nearby volume units have common ‘walls’, through which diffusion takes place. In accordance with Fick’s 1st law, the flow rate of the components between two chosen units is proportional to the common surface and the concentration gradient.56 Reactants are only allowed to move until they reach the borders of the reaction space (0  r  rex and 0  x  L), there is no flux of matter on the boundaries. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 71 Let’s take a cylinder ring with an inner radius of r, an outer radius of r + dr and a height of dx. For the sake of simplicity, we can assume that dr = dx = 1. The surface of such an object consists of four parts (see Table 7), two circle rings or annulus of equal area, an outer side and an inner side, both of which have the shape of a curved rectangular. Table 7. Surface area of the parts of a cylinder ring with a grid size of dr = dx =1 outer radius = r Surface area Fraction of the total area Upper annulus     πrπrπr121 2 2 4 1 48 12    r r Lower annulus     πrπrπr121 2 2 4 1 48 12    r r Outer surface rπ2 48 2 r r Inner surface πr)12(  48 22   r r Total πr)48(  1 Clearly, the inner surface of this unit is the outer surface of the one inside it and reversely, its outer surface is equal to the inner surface of the next one with increased radius. The volume of the described object is: V = r2   (r1)2  = (2r1)  (64) The volumes of its neighboring cells are (2r1)  (upper and lower; x±1, r), (2r+1)  (outside; x,r+1) and (2r3)  (inside; x, r1). As stated before, the diffusion term can be decomposed into outflow and inflow from and into each particularly volume unit. The outflow term is quite straightforward to demonstrate. It is assumed that all particles leave the cell, at each surface part according to its ratio to the total surface (Table 7, column 3). In a general volume unit, where the actual concentration is c(x,r), the scheme of the diffusion is shown in Figure 5.43. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 72 Figure 5.43. Exchange of matter in a volume unit of general position. The unit shown has an inner radius of r  1 and an outer radius of r, having an index of n in the radial direction (r = 0 is the first one, where n = 1). The ratios given in the figure have to be multiplied by the diffusion coefficient. Blue arrows represent the outflow from the cell, right arrows correspond to the inflow from the neighboring units. Figure 5.43 refers to a cell in general position, i.e. it is located inside the reaction space, it does not reach any of the boundaries (r  0; r  rex; x  0; x  L). When the cell has one or two border surfaces, then it is assumed that the fraction of the particles corresponding to that surface area do not leave the cell. The inflow part is further divided into the inflow from the x and r neighbors. In the x direction, cells have the same volume (since r is constant). Therefore, the number of residing particles has the same ratio as their concentrations. Eq. (65) gives the x inflow into the c(x,r) unit.                    r,x r,x r,x c c c ,rxc,rxc 1 1 4 1 0 4 1 )1( 4 1 )1( 4 1 (65) In the case of the radial inflow, not only the common surface, but the ratio of the volumes is considered in the following multiplication: radial inflow = n n n,tot n,out n n n,tot n,in V V A A x,rc V V A A x,rc 1 1 1 1 1 1)1()1(       (66) Substituting the known volumes and surfaces into eq. (66) gives: 12 32 128 22 )1( 12 12 48 2 )1(         n n n n x,rc n n n n x,rc (67) Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 73 Upon simplification, the following equation is obtained:                   )( 0 )( 48 22 )1( 48 2 )1( 11 ru rv ccc n n x,rc n n x,rc r,xr,xr,x (68) Here v(r) and u(r) are the coefficients describing the inflow from the inner and the outer cells, respectively. Separate matrices were defined for coefficients containing the x inflow (DX), the r inflow (DR) and the outflow as well (DE). Using these matrices, the right hand side of eq. (55), still without the reaction term, can be reconstituted as eq. (69) D∙(DXc + cDR + DE.c) (69) Here, ∙ is the regular multiplication sign,  is the matrix multiplication and . is the elementwise multiplication of matrices. Matrix multiplication is sensitive to the order of the factors, and the matrix dimensions have to match for a valid operation. 5.3.6. Numerical integration According to the Newton-Leibniz formula, one of the fundamental theorems of calculus, the definite integral of an u(x) univariate function between xi and xi+1 point is given as   1 1)()()( i i ii x xdssu'xuxu (70) Eq. (70) is in the background in the numerical solutions of kinetic differential equations, where the independent variable is time and the dependent variables are concentrations. Our previous efforts were targeted to give a reasonable approximation of the right hand side of (55), i.e. the derivative of concentrations with respect to time. The next step is to define how to obtain the c(t + h) concentrations in the possession of the c(t) values. Definite integration is another operation that computers cannot handle directly (although symbolic softwares such as Mathematica contain numerous primitive functions). A plethora of algorithms exist for numerical integration, the approximation of definite integrals between two time points (or whatever is the independent variable).191 The most widely used algorithm in chemical kinetics is the fourth-order Runge-Kutta method (RK4),91 because it is relatively easy to code and it has high accuracy, given that time step length (h) is chosen correctly. Adaptive time stepping is practical in most cases, where the step size gradually changes as the iteration proceeds. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 80 of the rapid nature of the reaction of silver(I) and sulfate ion radical became available by the application of the laser flash photolysis technique. A newly acquired Applied Photophysics laser flash photolysis (LFP) instrument of nanosecond time resolution allowed us to study the reactions of the sulfate ion radical in a direct manner. SO4 was generated by the photolysis of K2S2O8 by a 266 nm pulse and followed by its absorption at 450 nm. The second order rate constant of its reaction with silver(I) ion was determined under pseudo-first order conditions, the measured value is (7.7 ± 0.5)  109 M1 s1 in strongly acidic medium. This value approaches the diffusion controlled limit, affirming the role of reaction (19) in chain propagation. Similarly to the case of silver(I), the second order reactions of sulfate ion radical with previously recognized catalysts of the autoxidation of S(IV) were studied. Rate constants for Ce(III) and iodide ions were determined, as well as further rate constants with halide ions, a few biomolecules and S(IV). In the sulfate ion radicaliodide ion system, interesting kinetic behavior was observed. The recorded kinetic curves fitted to double exponential kinetics, and both fitted parameters showed linear dependence on the concentration of iodide ion. One of the two dependence functions went through the origin. Therefore, the respective rate constant was assigned to reaction (42). The other line had a significant intercept with the vertical axis, it was thus concluded to belong to the reversible reaction of iodide ion and iodine atom, producing iodide molecule anion (I2). This was suggested as a novel method for generation iodine atoms and the equilibrium constant of (43) was determined. Pseudo-first order behavior was observed in the sulfate ion radicaliodide ion system, even though the prerequisite for pseudo-first order kinetics, the high excess of iodide ions, was not reached locally at the site of observation. The potential replenishment of iodide ions by diffusion from outside the reaction space was examined by a numerical model. The reaction space was divided into unit volumes and an outer radius was defined inside which diffusion of particles was allowed. The random walk model of diffusion was applied to simplify the solution of the operative partial differential equation (PDE) system. Using a chemically conceivable parameter set, it was shown that for a rate constant in the 109 M1 s1 order of magnitude, diffusion does not influence the determination of the rate constant within the time frame of the reaction. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 81 7. Összefoglalás Ez az értekezés közvetlen folytatása és kiegészítése a kutatócsoportunkban korábban lezajlott vizsgálatoknak a kén(IV) autooxidációjának kinetikájával és mechanizmusával kapcsolatosan. Ezek a korábbi munkák feltárták a Ce(III) és I ionok katalitikus szerepét valamint a Fe(II) ionok lehetséges részvételét a fotoiniciált autooxidációs mechanizmusban. A kísérletek során diódasoros spektrofotométert alkalmaztak a reakciók indítására és követésére egyaránt.54,75-77 Az autooxidációs folyamatok mechanizmusa gyökös láncreakció, amelyben szulfit-, szulfátés peroxomonoszulfát-iongyökök (rendre SO3, SO4 és SO5 a közös láncvivők. Az eredmények azt mutatják, hogy a szulfátiongyök (SO4) kiemelt szerepet tölt be a katalízisben, kétféle okból. A szulfátiongyök reakciója a katalizátorral fontos láncvivő lépés valamint a peroxodiszulfát ionokat termelő másodrendű rekombinációja többnyire az egyetlen jelentős lánclezáró lépés. Az autooxidáció mechanizmusának mélyebb megértése érdekében célul tűztük ki a mechanizmust felépítő részrendszerek független vizsgálatát. Egy ilyen részrendszer a S(IV) reakciója a lánczáró lépésben képződő peroxodiszulfát-ionnal, amely vélhetően szinproporciós folyamatban szulfátiongyököt termel. Korábbi eredmények alapján az említett reakció savas közegben, katalizátor távollétében rendkívül lassú. Az S2O82 ionok redoxireakciói kinetikai gátlás miatt többnyire igen lassúak a nagy redoxipotenciál ellenére is. Az ezüstionok a peroxodiszulfát ion oxidációs reakcióinak jól ismert katalizátorai, mivel az iniciáló lépés termékei a szulfátion mellett igen reaktív részecskék, Ag(II) és szulfátiongyök.43-45 Vizsgálataink során ezért ezüst-nitrátot adtunk a S(IV)–oxigén–peroxodiszulfát reakcióelegyekhez, erősen savas közegben. Előkísérletek alapján a S(IV) autooxidációja ily módon hatékonyan iniciálható. A fotokémiai mellékreakció kizárása érdekében a diódasoros spektrofotométer alkalmazását kísérleteink során kizártuk, normál spektrofotométer segítségével követtük a reakciót. Ez a dolgozat részletesen ismerteti a S(IV) ezüst ionok által katalizált autooxidációjának kinetikai vizsgálatát, peroxodiszulfát-ion, mint ko-katalizátor jelenlétében. A kísérletekből levont mechanisztikus következtetések ugyancsak a dolgozat tárgyát képezik. Kilenc lépésből álló mechanizmust javasoltunk (Scheme 1), valamint egy némileg összetett, ugyanakkor kinetikailag megalapozott sebességi egyenletet vezettünk le (38) a kinetikai adatok értelmezésére. Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 82 A szulfátiongyök és az ezüstionok közötti másodrendű reakció (Scheme 1; R2) az ezüstionok által katalizált autooxidáció általunk javasolt mechanizmusának egyik láncvivő lépése. A sebességi egyenlet levezetése során a hosszú láncot feltételező megközelítést alkalmaztuk, amely feltételezi, hogy a láncvivő lépések sebessége nagyobb, mint a láncindító és a lánclezáró lépéseké (mivel másképp a láncreakció nem tudna lezajlani), illetve azt is, hogy a láncvivő lépések sebessége megegyezik. Annak igazolását, hogy a szulfátiongyök (SO4) és az Ag+ ionok között igen gyors reakció játszódik le, lézeres villanófény-fotolízis módszer alkalmazása tette lehetővé. A tanszékünkön nemrégiben megvásárolt Applied Photophysics nanoszekundum időfelbontású villanófény-fotolízis készülék segítségével közvetlenül tudtuk tanulmányozni a szulfátiongyök reakcióinak kinetikáját. Az SO4 gyököket K2S2O8 oldat fotolízisével állítottuk elő 266 nm hullámhosszúságú lézerimpulzus segítségével. A reakciókat 450 nm-en, a szulfátiongyök elnyelési maximumán követtük. Az ezüstionnal való reakciót pszeudo-elsőrendű körülmények között vizsgáltuk, és a másodrendű sebességi állandó mért értéke (7.7 ± 0.5)  109 M1s1 volt, erősen savas körülmények között. Ez az érték megközelíti a diffúzió kontrollált reakciók sebességi állandójának maximális értékét, megerősítve a (19) reakció láncvivő lépésként való részvételét a mechanizmusban. Az ezüstionokhoz hasonlóan tanulmányoztuk a szulfátiongyök másodrendű reakcióit az autooxidáció korábban megismert katalizátoraival is. Meghatároztuk a másodrendű sebességi állandók értékeit Ce(III) ionokkal és I ionokkal, valamint egyéb halogenidionokkal, néhány biomolekulával és S(IV) tartalmú részecskékkel. A szulfátiongyök–jodidion rendszerben érdekes kinetikai viselkedést figyelhettünk meg. A reakciót 340 nm-en, a jód molekulaion (I2) elnyelési maximumán követtük és a felvett kinetikai görbéket két exponenciális görbe összegével illesztettük. Mindkét illesztett sebességi állandó típusú paraméter lineáris függést mutatott a jodidionok koncentrációjától. Az egyik lineáris függvény átment az origón, így ezt az állandót hozzárendeltük a (42) reakcióhoz. A másik függvénynek jelentős függőleges tengelymetszete volt, így ebből a függvényből meghatároztuk a (43) reakció odaés visszairányú sebességi állandóját, valamint a folyamatot jellemző egyensúlyi állandót. Ezzel új módszert találtunk a jódatomok előállítására. A szulfátiongyök–jodidion rendszerben pszeudo-elsőrendű viselkedést tapasztaltunk, noha ennek feltétele, a jodidionok nagy feleslege lokálisan nem teljesül a reakcióelegyben, mivel a szulfátiongyök egy kis térfogatrészben képződik a lézersugár haladási iránya által meghatározott módon. Matematikai modellt vezettünk be annak vizsgálatára, hogy van-e mód Éva Dóka: PhD Thesis – Reactions of the sulfate ion radical 83 a jodidionok diffúzió általi pótlására a reakciótéren kívüli térrészből és ezáltal a pszeudoelsőrendű kinetikához szükséges koncentrációarány visszaállítására. A henger alakú reakcióteret kis térfogatelemekre bontottuk és meghatároztunk egy külső sugarat, amelyen belülről megengedett a részecskék diffúziója a reakciótérbe és onnan kifelé. A diffúzió véletlen bolyongás modelljét alkalmaztuk a rendszert leíró parciális differenciál egyenletrendszer megoldásának egyszerűsítésére. 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