Oxidation of hydroxymethanesulfonic acid by Fenton's reagent

culoc6981/89 $3.00+0.@3 Q 1989 Pergamon Press pit

Atmospheric Enuironnmr Vol. 23, No. 3, pp. 563-568, 1989. Printed in Great Britain.

OXIDATION

OF HYDROXYMETHANESULFONIC FENTON’S REAGENT

ACID BY

L. R. MARTIN, M. P. EASTON, J. W. FOSTER and M. W. HILL Aerophysics Laboratory, The Aerospace Corporation, El Segundo, CA 90009, U.S.A. (First received 13 May 1988 and in final

form 1 August 1988)

Abstract-We have studied the oxidation of hydroxymethanesuIfonic acid (HSMA) in Fenton’s reagent (hydrogen peroxide and ferrous ion), and we have used a relative rate method to estimate the absolute rate constant for oxidation of HSMA by OH radicals. We conclude that the atmospheric oxidation of HMSA by OH is slow if Fenton’s-type reactions are the only source of OH, but &et sources of OH may be present. Fenton’s reaction in the absence of other reactants is a weak source of OH radicals at atmospheric concentrations of hydrogen peroxide and iron. On the other hand, the estimate of the absolute reaction rate, based on the rate relative to the oxidation of pinacol by OH, is 1.25kO.25 x lo9 M- ’ s- I. This rate implies that HMSA can be oxidized fairly rapidly by OH in atmospheric aerosols, based on present models for the OH concentration in atmospheric aerosols. We also measured the rate of decomposition of HMSA and the acetaldehyde-bisulfite complex, HESA. The rate of decomposition of HESA at pH 2 is 300 times faster than that of HMSA, and on this basis we believe that HESA would not be very effective at protecting sulfur (IV) from oxidation in the atmosphere.

Key word index: Cloud chemistry, hydroxymethanesulfonic

INTRODUCTlON

Hydroxymet~anesulfonic acid (HMSA) is the complex formed between formaldehyde and S(IV) in solution. It is of interest in atmospheric chemistry, not only because it has been found in atmospheric liquid water, but also because it is highly resistant to oxidation, maintaining S in the (IV) oxidation state. The existence of HMSA in atmospheric liquid water was first inferred by the presence of S(IV) concentrations in excess of those allowed by Henry’s Law for SO, (Klippel and Warneck, 1980; Richards et al., 1983; Munger et al., 1983; Chapman, 1986). It was first positively identified in fog water by Munger et al. (1986), and has also been found in rain and snow (Ang er al., 1987). Recent studies of the kinetics of formation of HMSA (Boyce and Hoffmann, 1984) and of the equilibria (Deister et al., 1986; Dong and Dasgupta, 1986) have appeared in the literature. HMSA has been observed to be resistant to oxidation by HsOs, and 0, (Richards et al., 1983; Hoigne et al., 1985; Kok et al., 1986). Some unpublished work in our own laboratory indicates that it is also resistant to oxidation by O,/Fe ” . Because of the importance of OH radicals in cloudwater, (Chameides and Davis, 1982; Jacob, 1986) the effect of OH on HMSA should be explored. In this paper, we report on laboratory studies of the oxidation of HMSA by Fenton’szeagent, which produces OH radicals in solution, and we estimate the absolute rate of reaction of HMSA with OH radicals by a relative rate method. The possibility that

acid, hydroxyl radical.

Fenton’s-type reactions might occur in atmospheric liquid water was suggested by Graedei et ai. (1985). Fenton observed in 1894 that mixtures of ferrous ion and H,O, have an enhanced ability to oxidize organic compounds (Fenton, 1894). A review of the chemistry of Fenton’s reagent has been given by Walling (1975) and also by Kochi (1973). The traditional reaction scheme for Fenton’s reagent appears in those reviews. We present here a list of reactions and rates that are likely, in our view, to be present in Fenton’s reagent in the absence of organic materials. These reactions are based on the traditional scheme, but are written in terms of OH-complexed iron(III) products. When cast in this form, the reactions are more thermodynamically favorable (although some still have positive Gibbs energy changes). The pH dependences observed for these systems are thought to be due to the acidity of HOz rather than the iron speciation (Barb et al., 1951). HzOz+Fe2+=FeOH2*+OH OH+Fe2+=FeOH2* OH+H,O,=H,O+HO,

k,=76

k2=5x108 k3 = 2.7 x 10’

Hz0,+Fe3+=Fe2++H++H0,

k,=l

x 10e3

H02+Fe3+=Fe2i+02+H+

k, =

3.3 x lo5

HO, + Fe*+ = FeOOH’*

k,=7.2

x lo5

(FeOOH+ + H,O = H,O, + FeOH*+). (All rates are in MY1 s-i. k, is from Walling and Goosen, 1974, all others from Graedel et al., 1986.)

563

L. R.

544

MARTIN d al

If organic materials are present, additional reactions must be included (following Walling, 1975):

OH+RH=H,O+R

DECOMPOSITION PH

OF ti202

' 2.11 Fe++

=

1

x 1O-4

M

t = 25 C.

k7=“107-IOio

R f Fe3 f = Fe2 + + Product R+Fe2i+H,0=FeOH2’

+RH

R+O,=RO,

1000

R0,+FeZ++H10=R0,H+FeOH2+. (Several R’s or reactive sites may be involved. We have again included complexed iron(III) as possible reaction products.)

EXPERIMENTAL

800 600

MEI%K3DS

Four kinds of experiment were performed: (1) decomposition rate of H&I; in the presence of iron catalyst; (2) decomnosition rate of HMSA and HESA in the absence of catalysts; (3) oxidation of HMSA in Fenton’s reagent at various conditions; (4) oxidation of pinacol and HMSA in Fenton’s reagent. The decomposition ofH,U, was measured by trapping the molecular oxygen released from a known reaction vofume in an inverted graduated cylinder and measuring the gas volume as a function of time. The decomposition of HMSA and the Fenton’s oxidation of HMSA were studied by ‘mixing H,O,, catalyst if required, HCI to set the pH, BaCI, and the HMSA. The rate was followed by filtering aliquots separated at the beginning of reaction at various times and weighing the BaSO, farmed. The aliquots were stirred in an ultrasonic cleaner to dislodge precipitate from the walls before fihering. Because the Iass of H202 is sI5w in these ~~~~rnents~ the rate oftoss ofHMSA due to d~rn~s~tju~ or oxidation was roughly first arder. For this reason, most ofthe “survey” runs for experiments of type (3) above involved only one or two weighings and an assumption of first order behavior. The pinacol runs were studied by measuring pinacol loss and acetone production by means of gas chromatography. Aliquots of the reaction mixture were extracted with a fixed volume of tr~cb~oroethy~eneand a known volume of this was injected into a gas chromato~aph. The GC used was a Hewlett Packard Modef HP-%4Q with a flame ionization detector and an HP ~~e~tor~plotter. The column was a 30m LIB-1 fused silica column from J & W Science, He carrier gas, 100” Celsius oven temperature. Most of the reagents were from J. T. Baker. The pinacol was from Aldrich and the ferric perchlorate was from Gallard-Schlessinger. The H,O, was 30% GR tin stabilized from EM Science. The stannate ion in this material would have a concentration of less than 10e7 M in our typical Fenton’s mixtures. Nevertheless, there was concern that the stannate. which sequesters metal ions, might inhibit the Fenton’s reagents. For this reason, we compared rates of Fenton’s reactions of HMSA prepared with this material with similar mixtures preoared with H,O, purified by quiescent distillation. The r&es were th; same. HMSA was prepared by potentiometric titration of bisulfite with formaldehyde, and the Na salt was precipitated with atcohol.

0. 10

Figure 1 shows the First order decomposition of H202 as measured by O2 evolution. This was done with 0.1 M H,Oz, 0.01 M HCl, lo-” M ferrous sulfate.

30

40

TIME.

50

60

?D

80

90

lU0

HlxJRS

Fig. 1. Decomposition rate of hydrogen peroxide, pH = 2.0 (HCI), 1 x lO+ M FeSO,, 2.5”C, as measured by the evolution of oxygen.

The curve is consistent with: -d(HzG,t/dt=k(Fe2*f(H,0,) k=O.O44 M-” s-* at pH 2.0, 25°C. (I) Note that at such higb ratios of H,O, to Fe, this reaction rate should be roughly the same for ferrous or ferric starting material. Thus, our rate may be compared with the rate for ferric ion in perchloric acid reported by Walling and Goosen (1973): k=0.12 M- 1s- 1 at pH 2.0, 30°C. Using the temperature dependence suggested in that paper, the corresponding rate at 25°C would be k =0.054 M- ’ s- * at pII 2.0. Eary (1985) studied this reaction in H,SO,. He quotes a rate in terms of free ferric ion that is equivalent to k=0.024 (ZST) on our basis, which is total iron, for the same pH, Since the iron(II1) in H,SO, solutions is mostly complexed, Eary’s results suggest to us that the form of the iron is not critical; otherwise his rate would have been IO times or more slower than reported. The decomposition of Hz& was considerably slower in the Fenton’s reagent mixtures with organics studied below. Decotnposition

RESULTS AND DISCUSSI0N

20

of HMSA

The d~omposition rate of HMSA is of interest becauseit determines the basehne rate offo~ation of SOi- that we will see in the Fenton’s reactions to follow. This rate will be subtracted from the observed Fentan’s rates in order to establish the actual rate of oxidation by free radicals.

565

Qxidation of hydroxymethanesulfonic acid

Solutions of lO_’ M HMSA, IO-" M BaCI,, and lo- l M H20, were buffered to various pH values and allowed to stand. The progress of the reaction was followed by weighing the BaSO, formed. Because of the rapidity of the reaction of H& witb bisulfite ion, it is strai~~tfo~ard to show that for these high concentrations of H,O, the rate limiting step in the formation of SO:- will be the decomposition of HMSA. (This conclusion is further supported by the fact that we found the rates to be independent of H,O, concentration,) The buffers were KCl/HCl for pH 1,2, and 3; citrate for 4, 5, and rj; tris(hydroxymethyl)aminomethane for 7 and 8. Figure 2 sh5ws the measured first order decomposition rates for HMSA as a function of pH. Rates observed by Kok et al. =(1986)and by Deister et af. (1986) are also shown for comparison in the figure. Both of those papers applied the law of microscopic reversibility to show that the decomposition rates are consistent with measured formation rates and the eq~lib~um constants (Dong and Dasgupta, 1986). The shape of this curve suggests that doubly ionized HMSA decomposes more readily than singly ionized HMSA, although one would not expect the rate to level off until the second ionization is complete, and this should not happen until much higher pH values. We performed a similar experiment for the acetaldehyde-bisulfite complex “HESA”. For this

OfCOMPOSITION

OF

complex, we observed: - d(HESA)/(HESA)dt = 6 x 10- 6 s - ’ pH = 2,25”C, (2) Since HESA should be found mostly in the aqueous phase in the atmosphere, this rate should translate directly into an atmospheric rate, i.e. about 2% h- ’ at pH 2, Thus, the acetaldehyde complex, HESA, should not be as effective as HMSA at protecting S(iV) from oxidatian. Fenton’s reagent studies

As in the previous series, the experiments were very simple: mixtures of BaCl,, HMSA, H,U,, and Fe” were prepared at different pH values. The pH was set with inorganic bu&rs in order to avoid side reactions with Fenton’s reagent. We used KCl/HCl for pH 1,2 and 3, and AlCI, for pH 4. As before, BaSO, was weighed to follow the reaction. Once first order loss of the HMSA was established for typical reaction conditions, additional rates in a series were based on a single weighing. The loss is roughly ftrst order because at these high ratios of H,& to Fe, the fractional loss of H,O, is small during the length ofa run. Note that the appearance of SO:- does not imply that the S(IV) is the initial site of attack of the oxidizing specie on HMSA. If attack on the organic moiety took place first, the freed HSO; ion would immediately be oxidized by H,O, to produce SO:-, and the observations would be the same. Figure 3 shows the first order oxidation rates of FENTON’S

HMSA

OXtgATlON

OF

o

*pHl

lo-*

M HMSA

0 *

.pHZ -pH3

10-Z

M

IO“

1a-’i

HMSA

BaC$

IM vz

0 0

0

10-f ‘b* SEC.1 IO-(

,

0 0

10-8

Yo‘9 ”

1

L

3

d

5

6

7

8

pH

Fig. 2. ~~orn~s~t~o~ of HMSA in the presence of 0.I M hydrogen peroxide as a function ofpI% The buffers are listed in the text. Circles are our data, triangles are data from Kok et al. (1986), square is from Ikister et al. (1986).

Fig. 3. Qxidation of HMSA by Fenton’s reagent as a function of iron concentration and pH. Hydrogen peroxide is fixed at 0.1 M. The lines are plots of Expression (3) in the text. All data points are carrected for the spontaneous decomposition rate given in Fig. 2.

L. R. MARTINet al.

546

10m2 M HMSA in 10-i M H,Or and 10T2 M BaCl, as a function of pH and initial Fe(H) concentration. First order in Fe(H) is indicated, with the slight fall-off at higher iron concentrations probably due to depietion of the H,C& and the consequent departure from first order kinetics. The lines in the figure are based on the final rate Expression (3) below. Since the pH 1.0 data lie systematically below the line, we think that the unionized form of HMSA is less reactive, but this was not investigated quantitatively. Figure 4 shows similar rates for fixed HMSA and Fe concentration as a function of pH and H,O, concentration. These data indicate an approximately threehalves order dependence on the HzOz concentration. The leveling-off of the rate at low pH and high H20, is seen in studies of the ferric ion catalyzed decomposition of H,Oz (Walling and Weil, 1974) and is probably due to a change in mechanism. Figure 5 shows similar rates for fixed HMSA, iron and HzO, concentration as a function of pH. Note that inverse dependence in H+ is seen at low pH, with a maximum near 3.5. This pH dependence is similar to the proportion of Fe(II1) in the form of FeOH’ +. All of the above rates are corrected for the decomposition rates in the absence of catalyst. The data from pH 1 to 3 may be approximated by the empirical rate law:

FENTON’S 0 0 A

I”

OXIDATION

. ptl 1 * pti 2 * ptl3

OF

HMSA

10“

M

HMSA

10-Z

M

BaC12

10-4

M

WI2

*

10-b

10-3

10-2

10“

1

Fig. 4. Oxidation of HMSA by Fenton’s reagent as a function of pH and hydrogen peroxide concentration. The lines are plots of Expression (3) in the text. All data points are corrected for the spontaneous decomposition rate given in Fig. 2.

10-‘-l 1

f 2

3

4

PH

Fig. 5. Oxidation of HMSA by Fenton’s reagent over a wide range of pH. Hydrogen peroxide is 0.1 M and iron is I x 10e4 M. Note maximum near pH 3.5.

The form of this rate expression is similar to the one seen previously for the Fe(II1) catalyzed decomposition of hydrogen peroxide, although the conditions are somewhat different. See the discussion in Walling and Weil (1974). This similarity suggests that the concentration of oxidizing free radicals in Fenton’s reagent is proportional to the rate of decomposition of the H,O,, which is consistent with the presently accepted mechanism. Extra~lation of this kind of rate expression to the environment is perilous, because free radical mechanisms can lead to changes of the rate law when there are large changes in concentrations. Nevertheless, for purposes of discussion, we will estimate the consequences of this rate law in the atmosphere. If we assume that all ofthe HMSA is in the liquid phase, and that 10m5 molar peroxide and lo-” molar Fe are present at pH 3, then the rate of oxidation of HMSA by this process is only 2.3 x 10m4% h- *, an extremely low rate. Even for more extreme conditions such as pH 3, lo-” molar Fe, and 10m4 molar H,O,, the rate would be about 0.1% h- ‘. Thus, this process in isolation seems unlikely to be a significant sink for HMSA, even considering the uncertainties. A brief search for other Fenton’s reagents that might be present in the atmosphere is shown in Fig. 6. This figure compares the rate of oxidation of HMSA in H,O, with Cu, Mn and Pb ions. Note that there is no Fe-G synergism (see below). Since Fe is the most abundant of these metals in the atmosphere, other Fenton’s reagents are unlikely to compete with Fe,

567

Oxidation of kydroxymcthanesulfonic acid FENTON’S

~f~cTloN

ptI 2.0,

HMSA-0.01

M. ~~0~.

0.1 M

lo-

k.sec-'

Pb+'

METAL

ION CONCENTRATION,

M

Fig. 6. Oxidation of HMSA for different types of Fenton’s reagent. Rates in this figure are nat corrected for backgtound rate, which is shown as the dashed line in the figure.

Tabfe 1. Comparison diMSA and pina& oxidation tates We did some studies on the relative oxidation rate of HMSA and pinacol in Fenton’s reagent with the hope of estimating an absolute rate of oxidation of HMSA with OH radicals in solution. Pinacol has the structure:

in Fenton’s reagent (Allsolutions wereO.1 M H,O,, pH =2.0 HCI, 10U4 M FeZ”, 10ez M&XI,, 25°C) (All rates in s-l) (Averages of ten runs) lo-’ M HMSA (alone): 3,0*0,3x lo-& 1O-2 M Pinacoi (atonei: Pinacol Ioss 8.5~3,OX lo-’

Acetone gain 7.9+0.9x W7

lo-’ M HMSA+ IO-’ M Pinacoi HMSA loss 5.5f0.4 x lo-’

This molecule is known to be oxidized to acetone by Fenton’s reagent (Mertz and Walters, 1949), and the acetone product is more slowly attacked by OH than by pinacol, so it may serve as an additional indication of the reaction rate (two acetone molecules are produced for each pinacoi oxidized). The rates of reaction of OH with both pinacol (k=3.2x 10’ M-‘s-r) and acetone (4.3 x IO7 M - ’ s - ‘) are reported in the literature (Anbar and Nuta, 1967). Because of this difference in rate, and because the initial loss rate of acetone is very small compared to the production rate, the correction in converting from acetone gain to pinacol toss is negligible for small conversions. Table 1 presents rates of oxidation of pinacol in our best characterized Fenton’s reagent: Fe* ’ = 10s4 M, H,O,=O.t M, pH=2.0. The pinacol concentration was lo- 2 M. Pinacol loss and acetone appearance were followed by gas chromatography (see Experimental section). The table also presents rates for IOF M each of pinacot and HMSA in the same reagent, and contrasts those with HMSA atone. As before, SOi- formation was followed by BaSU, weighings. Note that the oxidation rates for each molecule are different in the separated and mixed reactions. This is

Pinacol loss 1.4_+0.3x 1o-7

k(BMSA)/k(Pinacol) E 3.9 &o.a

ti, be expected from the presently unde~tood

mechanism for Fenton’s reaction (Walling, 1975) which predicts that the steady state concentration of free radicals will depend on the chemistry of the organic substrates in solution. Thus, a rate comparison is only valid if both reactants are present in the same solution. (The data show, however, that the ratio of the two rate constants is similar in the mixed or unmixed systems.) Our inference of a relative rate constant for OH is further based on the assumption that OH is the primary oxidant in these Fenton’s reagents, and that other radicals such as HO, are not competing significantly for the organic substrates. The only evidence we have that this is true is the absence of an Fe-Cu synergism (see Fig. 6). Copper is a catalyst for the destruction ofHO,, so the absence of an effect tends to support the idea that HO1 is not the oxidant. This conclusion also must be taken with caution, however, because Cu ions may alter the system chemistry in other ways. We conclude, subject to this assumption, that HMSA is more reactive with OH than is pinacol, by a

568

L. R.

MARTIN et al.

factor of 3.9 kO.8. Thus, if we take the absolute rate of reaction of pinacol with OH radicals as 3.2 x lo* M-Is-‘. HMSA + OH = products

This rate constant suggests that HMSA may be consumed fairly rapidly Sn tropospheric clouds, because OH from other sources should be sufficiently abundant to give a large reaction rate. This conclusion is based on models of the OH concentration in clouds. Jacob (1986), for example, has modeled the OH steady state concentration in remote cloud droplets. His prediction is that the steady state concentration in the interior of the droplets is 2.3 x lo-i3 M. This translates into an HMSA loss rate of 100% h- ‘! Of course,

such a perturbation would change the prediction profoundly, but the implication is that this reaction is an effective sink for HMSA. Acknowledgements-This work was supported by the U.S. Environmental Protection Agency Grants Administration Division, Washington, D.C. The assistance id. number was CR-812301-01-1, and the technical administrator was Dr Marcia C. Dodge, U.S.E.P.A., Research Triangle Park, NC. Although the research described in this article has been supported by the United States Environmental Protection Agency, it has not been subjected to Agency review and therefore does not necessarily reflect the views of the Agency and no official endorsement should be inferred. Mention of trade names or commercial products does not constitute endorsement of recommendation for use. We want to thank Dr Daniel Jacob for valuable conversations.

REFERENCES Anbar

M. and Neta P. (1967) A compilation of specific bimolecular rate constants for the reactions of hydrated electrons, hydrogen atoms and hydroxyl radicals with inorganic and organic compounds in aqueous solution. Int. J. appl. Radiat. Isotopes

18, 493-523.

Ang C. C., Lipari F. and Swarm S. J. (1987) Determination of hydroxymethanesulfonate in wet deposition samples. Enuir. Sci. Technol. 21, 102-105. Barb et al. (1951) Reactions of ferrous and ferric ions with hydrogen peroxide. Trans. Faraday Sot. 47, 591-616. Bovce S. D. and Hoffmann M. R. (1984) , , Kinetics and mechanism of the formation of hydroxymethanesulfonic acid at low pH. J. Phys. Chem. 88, 4740-4746. Chameides W. L. and Davis D. D. (1982) The free radical chemistry of cloud droplets and its impact upon the composition of rain. J. Geophys. Res. 87, 48634877.

Chapman E. G. (1986) Evidence for S(IV) compounds other than dissolved SO, in precipitation. Geophys. Rex Letters 13, 1411-1414. Deister U., Neeb R., Helas G. and Warneck P. (1986) Temperature dependence of the equilibrium CH,(OH), + HSO; =CH,(OH)SO; + H,O in aqueous solution. J. Phys. Chem. 90, 3213-3217. Dong S. and Dasgupta P. K. (1986) On the formaldehyde-bisulfite-hydroxymethanesulfonate equilibrium. Atmospheric

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Eary L. E. (1985) Catalytic decomposition of hydrogen peroxide by ferric ion in dilute sulfuric acid solutions. Metall. Trans. B. 16B, 181-186. Fenton H. J. H. (1894) Oxidation of tartaric acid in presence of iron. J. Chem. Sot. 65, 899. Graedel T. E., Weschler C. J. and Mandich M. L. (1985) Influence of transition metal complexes on atmospheric droplet acidity. Nature 317, 240-242. Graedel T. E., Mandich M. L. and Weschler C. J. (1986) Kinetic model studies of atmospheric droplet chemistry. 2. Homogeneous transition metal chemistry in raindrops. J. Geophjs. Res. 91, 5205-5221. Hoiene J. H.. Ader W. R.. Haae W. R. and Staehelin (1985) Rate constants of readtions”of ozone with organic and inorganic compounds in water, III. Water Res. 19, 993-1004.

Jacob D. J. (1986) Chemistry of OH in remote clouds and its role in the production of formic acid and peroxymonosulfate. J. Geoihys. Res. 91, 9801-9826. _ _ Klinoel W. and Wameck P. (1980) The formaldehvde content df’the atmospheric aerosol. Aimospheric Enviionment 14, 809-818.

Kochi J. K. (1973) Oxidation-reduction reactions of free radicals and metal complexes. In Free Radicals (edited by Kochi J. K.), pp. 591-683. John Wiley, New York. Kok G. L., Gitlin S. N. and Lazrus A. L. (1986) Kinetics of the formation and decomposition of hydroxymethanesulfonate. J. Geophys. Res. 91, 2801-2804. Mertz J. H. and Walters W. A. (1949) Some oxidations involving the free hydroxyl radical. J. Chem. Sot. 1949, S15-S25. Munger J. W., Jacob D. J., Waldman J. M. and Hoffmann M. R. (1983) Fogwater chemistry in an urban atmosphere. J. Geophys. Res. 88, 5109-5121. Munger J. W., Tiller C. and Hoffmann M. R. (1986) Identification of hydroxymethanesulfonate in fog water. Science 231, 247-249. Richards L. W., Anderson J. A., Blumenthal D. L., McDonald J. A., Kok G. L. and Lazrus A. L. (1983) Hydrogen peroxide and sulfur(IV) in Los Angeles cloudwater. Atmospheric Environment 17, 911-914.

Walling C. (1975) Fenton’s reagent revisited. Chem. Rev. 8, 125-131. Walling C. and Goosen A. (1973) Mechanism of the ferric ion catalyzed decomposition of hydrogen peroxide. Effect of organic substrates. J. Am. Chem. Sot. %9,2987-2991. Walling C. and Weil T. (1974) The ferric ion catalyzed decomposition of hydrogen peroxide in acid solution. Int. J. Chem. Kinetics. VI, 507-516.