Introduction

Studies of systems involving iodine reactions, such as periodic and chaotic reactions [1,2,3,4], the consequences of a nuclear accident [5,6,7] and the marine boundary layer chemistry [8,9,10] require the understanding of simpler sub-systems. The models of these complicated systems involve a large number of reactions with unknown or poorly known rate constants and the aim of our work is to reduce their number. We present new measurements of the kinetics of I(+ 3) reactions in acidic solutions as well as a new analysis of previous results for inorganic reactions of iodine compounds between them and with H2O2. Adopting the terminology Up and Down reactions proposed by Liebhafsky, we continue to explore the iodine house [11] with 6 floors from I(− 1) to I(+ 5) and an elevator, H2O2.

Table 1 is an update of the model proposed in 2010 [12] taking into account our new study of the kinetics of I(+ 3) reactions. The observed effect of [H+] has consequences for other reactions. A model of iodine reactions must also explain the very unusual kinetics of the oxidation of I2 by H2O2 (reaction O) discussed below. Some kinetic constants were known. Others were adjusted in 2010 to explain this kinetics and we have updated them to take into account the results presented in this work.

Table 1 Units: (mol/l, s)
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The former explanation of the effects of oxygen on the Bray–Liebhafsky (BL) oscillations is also modified. Sharma and Noyes [13] had studied these effects and had even proposed, without success, that they could explain the oscillations. This explanation has been discarded but the importance of oxygen reactions has been confirmed [14,15,16]. To take these effects into account, the 2010 model included the global reaction \({\text{I}}^{ - } + {\text{ H}}^{ + } + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}{\text{O}}_{{2}} \to {\text{HOI}}\) with an empirical rate law. This reaction seems simple but is actually a complicated light-catalyzed radical reaction that deserves further experimental study. In the meantime, other studies suggest that it involves the HOOI intermediate proposed by Ball and Hnatiw [17] to explain the kinetics of the reduction of I(+ 1) by H2O2 in buffered solutions. The existence of this intermediate has been confirmed later [18, 19]. HOOI is also an important intermediate for the explanation of the reduction of iodate by H2O2 [20]. The classical rate law of this reduction is no longer valid when the concentration of H2O2 is larger than about 0.2–0.3 M and a radical reaction path appears. The reaction \({\text{HOI }} + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{I}}^{ - } + {\text{ H}}^{ + } + {\text{ O}}_{{2}} + {\text{ H}}_{{2}} {\text{O}}\) must be split into \({\text{HOI }} + {\text{ H}}_{{2}} {\text{O}}_{{2}} \rightleftharpoons {\text{HOOI }} + {\text{ H}}_{{2}} {\text{O }}\left( {{\text{R5}}} \right)\) and \({\text{HOOI}} \rightleftharpoons {\text{I}}^{ - } + {\text{ H}}^{ + } + {\text{ O}}_{{2}} \left( {{\text{R6}}} \right)\) and a radical reaction path would be initiated by a reaction between HOOI and H2O2 [21, 22]. Reaction R6 could also explain the observation of E. Szabo and P. Ševčik [23]. They measured accurately the rate of O2(g) production during the BL reaction and identified two precursors. One is O2(aq). The concentration of the other precursor increases with [H2O2] and HOOI explains this observation. All these works suggest our new explanation of the effect of oxygen on the BL reaction: R6 is highly reversible and followed by reaction R12. We conclude that HOOI is a member of the iodine house with consequences in many systems.

Some rate constants in Table 1 are calculated using the relation Keq = k+/k- between an equilibrium constant and the kinetic constants in the forward and backward directions. It is sometimes justified by the principle of microscopic reversibility. However, many reactions of Table 1 are not elementary and the use of this principle may be open to criticism. We have offered another proof introducing the concept of quasi-elementary reaction [24]. It is defined as a reaction with a well-defined stoichiometry and rate orders corresponding to this stoichiometry. Take reaction R4 as an example. It is the sum of reactions (4a) and (4b) [25].

$${\text{HOI }} + {\text{ I}}^{ - } \rightleftharpoons {\text{I}}_{{2}} {\text{OH}}^{ - }$$
(4a)
$${\text{I}}_{{2}} {\text{OH}}^{ - } + {\text{ H}}^{ + } \rightleftharpoons {\text{I}}_{{2}} + {\text{ H}}_{{2}} {\text{O}}$$
(4b)

The quasi-stationarity of [I2OH−] gives r4a = rab and

$${\text{r}}_{{4{\text{a}}}} = { }\frac{{{\text{k}}_{{4{\text{a}}}} {\text{k}}_{{4{\text{b}}}} { }\left[ {{\text{HOI}}} \right]\left[ {{\text{I}}^{ - } } \right]\left[ {{\text{H}}^{ + } } \right] - {\text{ k}}_{{ - 4{\text{a}}}} {\text{k}}_{{ - 4{\text{b}}}} \left[ {{\text{I}}_{2} } \right]}}{{{\text{k}}_{{ - 4{\text{a}}}} + {\text{k}}_{{4{\text{b}}}} \left[ {{\text{H}}^{ + } } \right]}}$$

The rate law in Table 1 is observed when \({\text{k}}_{{{\text{4b}}}} \left[ {{\text{H}}^{ + } } \right] \gg {\text{k}}_{{ - {\text{4a}}}}\). At equilibrium k4ak4b [HOI]eq[I−]eq[H+]eq = k-4ak-4b [I2]eq. This kinetic expression must be equivalent to the thermodynamic expression of the equilibrium. It follows that k4ak4b/k−4ak−4b = Keq and, with numbering of Table 1, k4/k−4 = K4. We have applied this principle of equivalence between the kinetic and thermodynamic expressions of the equilibrium to other quasi-elementary reactions in Table 1.

Section “I(+ 3) disproportionation” proposes a new interpretation of our kinetic study of the reaction \({\text{2HOIO}} \to {\text{IO}}_{{3}}^{ - } + {\text{ HOI }} + {\text{ H}}^{ + }\) (R13) published previously [26]. Section “I(+ 3) autocatalytic disproportionation” presents a new kinetic study of the reaction \({\text{HOI }} + {\text{ HOIO}} \to {\text{IO}}_{{3}}^{ - } + {\text{ I}}^{ - } + {\text{ 2H}}^{ + }\) (-R1) and the next section presents a new kinetic study of the oxidation of I(+ 3) by H2O2 (R9). Section “Oxidation of I2 by H2O2 with iodate added initially (Reaction O)” shows that the model in Table 1 with updated values of some kinetic constant explains also our former results. If there is no iodate initially, the I2 + H2O2 reaction begins with a non-reproducible induction period. It is discussed in section “Oxidation of I2 by H2O2 without iodate added initially” on the basis of our recent calculations [27] of the nullclines corresponding to the model in Table 1. Section “I(+ 1) reduction by H2O2” shows that this induction period allows measurements of the kinetic constant of the reaction \({\text{HOI }} + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{HOOI }} + {\text{ H}}_{{2}} {\text{O}}\) (R5). Appendix (in the Supplementary Information) gives details about the calculations of the kinetic constants given in Table 1.

Experiments and calculations

The I(+ 3) solutions are prepared as explained before [28] by the reaction of weighed amounts of I2 and KIO3 in concentrated H2SO4. They contain about 1 to 2% I(+ 1). The calculation of the initial composition of the experimental solutions takes this into account. The other reagents are of the best purity commercially available and are used without further purification. The 18 MΩ water is supplied by a Barnstead Micropore ST model. The initial concentrations [H+] are calculated using the Pitzer model of the H2SO4 solutions. The absorbance measurements are made with an Agilent Cary 60 scanning spectrophotometer. The addition of the samples of I(+ 3) in H2SO4 to the aqueous phase (of composition depending on the kind of experiment) being exothermic, the aqueous phase is cooled before mixing so that the temperature after mixing is close to 25 °C. The spectrophotometer cell is thermostatically controlled at 25 °C and contains a small stirrer.

The sample of I(+ 3) (20 to 30 mg) must be added quickly to the aqueous phase (3 to 5 cm3). Mixing in the reverse order causes a local overheating giving some fast initial reactions and wrong results. Two mixing methods were used. The “syringe” method consists of placing the aqueous phase in a cell with a small circular opening, injecting the sample of I(+ 3) with a fast mixing syringe, closing the cell with a Teflon stopper, inverting the cell to mix well and place it in the spectrophotometer. This cell is tight and allows absorbances measurements up to large conversions, but the first measurements can only be made about 15–20 s after the injection of the I(+ 3) sample. The “paddle” method consists of placing the cooled aqueous phase in a cell with a wide opening (1 cm2) and weighing the I(+ 3) sample on a small paddle. It is introduced quickly into the cell, shaken briefly to mix well, and a Teflon cover is placed over the cell. The cell also contains a small stirrer. This method allows absorbances measurements after less than 5 s and gives accurate values at small conversions, but the cell is not tight and the absorbances are sometimes too small at large conversion. Both method were used for the measurements of the rate of disproportionation of I(+ 3) in the presence of crotonic acid, which is relatively slow, and for the measurements of the rate of autocatalytic disproportionation which begins with an induction period. They gave similar results. Oxidation of I(+ 3) by H2O2 is faster and we used the “paddle” method.

The spectrophotometer results are transferred to a computer and the analytical calculations are done with Excel. The differential equations associated with the model are integrated using the ode15s function of Matlab specially adapted to stiff differential equations. The main program calls the fminsearch function which determines the kinetic constants minimizing the sum of squares of the difference between the absorbances measured and calculated by ode15s. Calculations usually take less than 2 min and the relative error on the iodine mass balance is less than 10–13.

I(+ 3) disproportionation

The I(+ 3) disproportionation is autocatalytic as explained in the next section. To study the kinetics of reaction 2HOIO → IO3− + HOI + H+ (R13), we carried out experiments with crotonic acid (CA) [26]. It reacts very quickly with HOI according to reaction (1) to form an iodohydrin HOICA.

$${\text{HOI }} + {\text{ CA}} \to {\text{HOICA}}$$
(1)

Reaction (R13) becomes the rate determining step giving the kinetic law (2) when the concentration of CA is large enough. We used [CA]0 = 5 × 10–3 to 1.2 × 10–2 M and [I(+ 3)]0 = 5 × 10–4 to 2 × 10–3 M. The kinetic constant of reaction (1), kCA = 4730 M−1 s−1, has been measured previously [29].

$$- \frac{{{\text{d}}\left[ {{\text{I}}\left( { + 3} \right)} \right]}}{{{\text{dt}}}} = { }2{\text{ k}}_{{{\text{exp}},{\text{disp}}}} \left[ {{\text{I}}\left( { + 3} \right)} \right]^{2}$$
(2)

[I(+ 3)] represents the total concentration [I(+ 3)] = [HOIO] + [H2OIO+] = (1 + K19 [H+])[HOIO] so that k13 = (1 + K19 [H+])2 kexp, disp. The integration of the kinetic law (2) gives the equation below which makes it possible to calculate kexp, disp by making as only assumption that the absorbance measured at 275 nm varies linearly with the extent of the reaction [26]. However, it is necessary to estimate the value of A∞ by extrapolation of the A values at long term.

$$\frac{{{\text{A}}_{\infty } - {\text{ A}}_{0} }}{{{\text{A}}_{\infty } - {\text{ A}}_{{\text{t}}} }} = 1 + 2{\text{ k}}_{{{\text{exp}},{\text{disp}}}} \left[ {{\text{I}}\left( { + 3} \right)} \right]_{0} {\text{t}}$$

We also analyzed the experimental results using Matlab to integrate the differential equations associated with the model in Table 1 without the H2O2 reactions but with reaction (1) added. The fminsearch function adjusts the kinetic constant k13 to minimize the sum of squares \(\sum {\left( {{\text{A}}_{{{\text{calc}}}} - {\text{ A}}_{{{\text{exp}}}} } \right)^{{2}} }\) where Acalc = εI(+3) [I(+ 3)] + εI(+5) [I(+ 5)] + εCA [CA] + εHOICA [HOICA]. New measurements gave \(\varepsilon_{{{\text{I}}\left( { + {3}} \right)}} = {121},\quad \varepsilon_{{{\text{I}}\left( { + {5}} \right)}} \, = \,{11}.{4},\quad \varepsilon_{{{\text{CA}}}} \, = \,{6}.{4}0{\text{ and}}\,\varepsilon_{{{\text{HOICA}}}} \, = \,{399}\,{\text{at}}\,{275}\,{\text{nm}}\). The agreement between the rate constants calculated with the order two rate law or with Matlab is excellent.

The rate constant kexp,disp increases very quickly with decreasing acidity [26]. We had explained this effect by reaction (R20) followed by reaction (R21).

$${\text{HOIO}} \rightleftharpoons {\text{H}}^{ + } + {\text{ OIO}}^{ - }$$
(R20)
$${\text{HOIO }} + {\text{ OIO}}^{ - } \to {\text{IO}}_{{3}}^{ - } + {\text{ HOI}}$$
(R21)

The acidity constant of HOIO is unknown but its order of magnitude is K20 = 10–5 to 10–6 M [30] so that [OIO−] = K20 [HOIO]/[H+] is much smaller than [HOIO] under our experimental conditions.

$${\text{r}}_{21} = {\text{ k}}_{21} \left[ {{\text{HOIO}}} \right]\left[ {{\text{OIO}}^{ - } } \right] = {\text{k}}_{21} { }\frac{{{\text{K}}_{20} }}{{\left[ {{\text{H}}^{ + } } \right]}}[{\text{HOIO}}]^{2}$$

and [HOIO] = [I(+ 3)]/(1 + K19 [H+]) give

$${\text{k}}_{{{\text{exp}},{\text{disp}}}} = \frac{{{\text{k}}_{21} {\text{K}}_{20} }}{{\left[ {{\text{H}}^{ + } } \right]{ }\left( {1 + {\text{ K}}_{19} { }\left[ {{\text{H}}^{ + } } \right]} \right)^{2} }}$$

The large effect of [H+] on kexp,disp could be explained by the factor (1 + K19 [H+])2 if K19 ~ 3 M−1. However, the study of the autocatalytic disproportionation rate of I(+ 3) in the next section shows that K19 is much smaller so that this explanation must be discarded and we propose a new explanation of our 2013's results. We assume [H2OIO+] ≪ [HOIO] under our experimental conditions and explain the effect of [H+] on kexp,disp by the formation of the intermediate I2O4H−, similar to the oxide I2O3 known in the gas phase. Its formation will be supported by our new study of the I(+ 3) oxidation by H2O2 discussed below.

$${\text{HOIO }} + {\text{ OIO}}^{ - } \rightleftharpoons {\text{I}}_{{2}} {\text{O}}_{{4}} {\text{H}}^{ - }$$
(R22)
$${\text{I}}_{{2}} {\text{O}}_{{4}} {\text{H}}^{ - } \to {\text{IO}}_{{3}}^{ - } + {\text{ HOI}}$$
(R23)
$${\text{I}}_{{2}} {\text{O}}_{{4}} {\text{H}}^{ - } + {\text{ OH}}^{ - } \to {\text{IO}}_{{3}}^{ - } + {\text{ IO}}^{ - } + {\text{ H}}_{{2}} {\text{O}}$$
(R24)

Reaction R13 is obtained by combining these reactions with the acid–base quasi-equilibria,. Assuming \([{\text{I}}_{{2}} {\text{O}}_{{4}} {\text{H}}^{ - } ] \, = {\text{K}}_{{{22}}} \left[ {{\text{HOIO}}} \right]\left[ {{\text{OIO}}^{ - } } \right]\), we obtain \({\text{r}}_{{{13}}} = \left( {{\text{k}}_{{{23}}} + {\text{k}}_{{{24}}} \left[ {{\text{OH}}^{ - } } \right]} \right){\text{ K}}_{{{22}}} \left[ {{\text{HOIO}}} \right]\left[ {{\text{OIO}}^{ - } } \right]\) or \({\text{r}}_{{{13}}} = \left( {{{{{k^{\prime}}}_{{{13}}} } \mathord{\left/ {\vphantom {{{{k^{\prime}}}_{{{13}}} } {\left[ {{\text{H}}^{ + } } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{H}}^{ + } } \right]}} + {{{{k^{\prime\prime}}}_{{{13}}} } \mathord{\left/ {\vphantom {{{{k^{\prime\prime}}}_{{{13}}} } {\left[ {{\text{H}}^{ + } } \right]^{{2}} }}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{H}}^{ + } } \right]^{{2}} }}} \right) \, \left[ {{\text{HOIO}}} \right]^{{2}}\) where \({{k^{\prime}}}_{{{13}}} = {\text{k}}_{{{23}}} {\text{K}}_{{{2}0}} {\text{K}}_{{{22}}}\) and \({{k^{\prime\prime}}}_{{{13}}} = {\text{k}}_{{{24}}} {\text{K}}_{{\text{w}}} {\text{K}}_{{{2}0}} {\text{K}}_{{{22}}}\). The experiments give kexp,disp = r13/[HOIO]2 and the plot of kexp,disp [H+] as a function of 1/[H+] in Fig. 1 shows that this equation explains our former results. The intercept and slope give the values of \({{k^{\prime}}}_{{{13}}} \,{\text{and}}\,{{k^{\prime\prime}}}_{{{13}}}\).

Fig. 1
figure 1

I(+3) autocatalytic disproportionation

Effect of the acidity on the rate constant of reaction (R13). Experimental values (×) and linear regression line (—) giving \({{k^{\prime}}}_{{{13}}} \, = \,0.0{45}\,{\text{s}}^{{ - {1}}} \,{\text{and}}\,{{k^{\prime\prime}}}_{{{13}}} \, = \,0.0{65}\,{\text{M}}\,{\text{s}}^{{ - {1}}}\) . 

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I(+ 3) autocatalytic disproportionation

The kinetic of the I(+ 3) disproportionation without added CA is not simple. At the beginning, the main reaction is (R-1) followed by the fast reactions (R2) and (R3) giving the autocatalytic reaction \({\text{2HOIO}} + {\text{HOI}} \to {\text{IO}}_{{3}}^{ - } + {\text{H}}^{ + } + {\text{2HOI}}\). The induction period depends on the concentration of I(+ 1) in the preparation of I(+ 3) and overlaps the period required for the solution to be well mixed so that there are no mixing problems. The concentration [HOI] increases and can become as large as 20% of [I(+ 3)]0 at high acidities. Then, it goes through a maximum shown in Fig. 2 and the main reaction becomes \({\text{HOIO}} + {\text{2HOI}} \to {\text{IO}}_{{3}}^{ - } + {\text{H}}^{ + } + {\text{ I}}_{{2}} + {\text{ H}}_{{2}} {\text{O}}\). When [HOIO] becomes small, some HOI remains and the direction of reactions (R2) and (R3) is reversed. [HOI] decreases very slowly according to \({\text{5HOI}} \to {\text{IO}}_{{3}}^{ - } + {\text{ H}}^{ + } + {\text{ 2I}}_{{2}} + {\text{ 2H}}_{{2}} {\text{O}}\). The overall stoichiometry is \({\text{5I}}\left( { + {3}} \right) \to {\text{3IO}}_{{3}}^{ - } + {\text{ 3H}}^{ + } + {\text{ I}}_{{2}} + {\text{ H}}_{{2}} {\text{O}}\) but the experimental values of [I2] at the end of the experiments is always lower than [I(+ 3)]0/5 because the disproportionation of I(+ 1) is very slow under our experimental conditions. Fig. 2 gives examples of evolutions over time.

Fig. 2
figure 2

Left hand scale: Experimental (o) and calculated (—) absorbances at 462 nm at low (a) and high (b) acidities. [H2SO4] = 0.068 M and [I(+ 3)]0 = 4.34 × 10–4 M (a). [H2SO4] = 0.76 M and [I(+ 3)]0 = 5.41 × 10–4 M (—). Right hand scale: Calculated I(+ 1) concentrations (- - -)

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Matlab simulates the experimental curves and the function fminsearch adjusts k−1 and the initial value of [I(+ 1)]. The electronic supplement gives the results. The effect on k−1 of likely modifications of the kinetic constants, including k13, is lower than the experimental inaccuracies. On the other hand, k-1 depends on the values of K18 and K19. The concentrations known experimentally or calculated by mass balances are the total concentrations \(\left[ {{\text{I}}\left( { + {1}} \right)} \right] = \left( {{1} + {\text{K}}_{{{18}}} \left[ {{\text{H}}^{ + } } \right]} \right) \, \left[ {{\text{HOI}}} \right]{\text{ and }}\left[ {{\text{I}}\left( { + {3}} \right)} \right] = \left( {{1} + {\text{ K}}_{{{19}}} \left[ {{\text{H}}^{ + } } \right]} \right) \, \left[ {{\text{HOIO}}} \right]\). Thus, the individual concentrations [HOI] and [HOIO] calculated by Matlab depend on K18 and K19 and, therefore, also the value of k-1. To estimate the values of K18 and K19 we apply the principle of equivalence between the kinetic and thermodynamic expressions of an equilibrium. The kinetic of the reaction (R1) in the forward direction is well known [41]: its rate is proportional to [H+]2. Consequently, k-1 must be independent of [H+]. If K19 > 0.1 M−1, the calculated values of k-1 increase with [H+] as shown in Fig. 3 and, moreover, the fit of the experimental curves becomes bad if K19 > 0.5 M−1. We therefore neglect K19. Fig. 3 also shows that k-1 is nearly independent of [H+] if K18 ~ 0.3 M−1. We had estimated previously K18 ~ 0.5 M−1 [25] and these results suggest to decrease slightly this value. These experiments give k-1 = 210 ± 6 M−1 s−1 (t-Student 95% confidence interval).

Fig. 3
figure 3

Values of k-1 adjusted by Matlab if K18 = 0.5 M−1 and K19 = 0 (O), K18 = 0.3 M−1 and K19 = 0.2 M−1 (+), K18 = 0.3 M−1 and K19 = 0 (×)

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Oxidation of I(+ 3) by H2O2

When a sample of I(+ 3) is added to an acidic solution of H2O2, the main reaction is the oxidation \({\text{HOIO }} + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{IO}}_{{3}}^{ - } + {\text{ H}}^{ + } + {\text{ H}}_{{2}} {\text{O}}\) (R9). The amount of iodine produced is negligible showing that the reduction of HOIO by reaction R7 is much slower. The results show that the rate of R9 is proportional to the square of the concentration [HOIO] and inversely proportional to [H+]. The rate constants k9 in Table 2 were calculated for the rate law \({\text{r}}_{{9}} = {\text{ k}}_{{9}} {{\left[ {{\text{HOIO}}} \right]^{{2}} \left[ {{\text{H}}_{{2}} {\text{O}}_{{2}} } \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{HOIO}}} \right]^{{2}} \left[ {{\text{H}}_{{2}} {\text{O}}_{{2}} } \right]} {\left[ {{\text{H}}^{ + } } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {{\text{H}}^{ + } } \right]}}\) using the Matlab function fminsearch minimizing the sum of the squares of the deviations between the calculated and measured absorbances of HOIO at 275 nm over time. The electronic supplement gives examples. The average value is k9 = (1.7 ± 0.2) × 105 M−1 s−1 (t-Student 95% confidence interval). New measurements of the molar absorption coefficient of HOIO at 275 nm gave ε(HOIO) = 121 ± 2. Other values used in this work are ε(H2O2) = 5.7, ε(IO3−) = 11.4, ε(HOI) = 107 and ε(I2) = 121 at 275 nm.

Table 2 Rate constants of the HOIO oxidation by H2O2 (units: mol/l and s)
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We had shown that H2O2 reduces the HOI monomer and oxidizes its dimer I2O. The rate law of the HOIO disproportionation with an excess of CA discussed above and the kinetic law of HOIO oxidation by H2O2 suggest that the HOIO reactions are similar to those of HOI. HOIO also forms a dimer and H2O2 reduces its monomer and oxidizes its dimer. Also, the effect of [H+] on these reactions suggests that this dimer is I2O4H− introduced above that reacts according to (R24).

$${\text{I}}_{{2}} {\text{O}}_{{4}} {\text{H}}^{ - } + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{IO}}_{{3}}^{ - } + {\text{ HOIO }} + {\text{ H}}_{{2}} {\text{O}}$$
(R25)

The oxidation R25 is much faster than the disproportionation of HOIO from the smallest values of [H2O2] and the quasi-stationarity of [I2O4H−] gives (k-22 + k25 [H2O2]) [I2O4H−] = k22 [HOIO][OIO−] and the rate law

$${\text{r}}_{25} = \frac{{{\text{k}}_{25} {\text{k}}_{22} {\text{K}}_{20} \left[ {{\text{HOIO}}} \right]^{2} \left[ {{\text{H}}_{2} {\text{O}}_{2} } \right]}}{{{ }\left( {{\text{k}}_{ - 22} + {\text{k}}_{25} \left[ {{\text{H}}_{2} {\text{O}}_{2} } \right]} \right)\left[ {{\text{H}}^{ + } } \right]}}$$

This expression explains the rate law of R9 in Table 1 with k9 = k24k22K20/k-22 and α9 = k24/k−22. The term in α9 [H2O2] has no effect on the results in Table 2 because the concentration of H2O2 was small but will be important to simulate the H2O2 effect on the rate of reaction O (Fig. 4).**

Fig. 4
figure 4

Experimental rate constants of the I2 oxidation by H2O2 at 25 °C with about 0.01 mol/l iodate added initially. [HClO4] = 0.04 (open diamond), 0.10 (×), 0.20 (O), 0.40 M (+) and values calculated with the model in Table 1 (lines)

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Oxidation of I2 by H2O2 with iodate added initially (Reaction O)

The oxidation of iodine by H2O2 without iodate added initially begins with a non-reproducible period of induction discussed below. It is suppressed and a fast oxidation of I2 is observed if iodate is added initially. We had carried out more than a thousand experiments in a wide range of concentrations and temperatures and published the main results previously [12, 31]. When the initial iodate concentration is about 0.005 to 0.1 M, the rates are complicated functions of acidity, [H2O2] and [I2] but are independent of the iodate concentration. To analyze these rates, it is convenient to define the function kexp = − d(ln[I2])/dt although the reaction is not exactly of order 1 with respect to [I2]. Fig. 5 shows values of kexp at 25 °C if [I2] = 4 × 10–4 M. The values of kexp do not directly give values of kinetic constants, only relations between them. A modification of one constant requires readjusting the kinetic constants of the reactions R3, R7 and R10. The previously published model [12, 31] explained our results but we had to modify the values of some kinetic constants to take into account our new results. The proposed model with updated rate constants explains not only our new experimental results but also the previous ones. Our model centered on the reactions of HOI and I2O is robust in the sense that slight modifications of it can be compensated by adjustments of some kinetic constants. Fig. 4 shows that the rate constants in Table 1 give excellent simulations of the kexp values. Note the decrease in kexp when the concentration of the reagent H2O2 increases.

Oxidation of I2 by H2O2 without iodate added initially

H2O2 does not react directly with I2. An acidic solution containing only I2 and H2O2 gives a quasi-steady state corresponding to the slow decomposition \({\text{H}}_{{2}} {\text{O}}_{{2}} \to {\text{H}}_{{2}} {\text{O }} + {\text{ O}}_{{2}}\). To obtain the oxidation of I2, the concentration in iodide must be decreased. This can be obtained by addition of iodate as in the previous section, by precipitation of AgI(s) or by formation of HgI+ but can also appear spontaneously after a more or less long time. Fig. 6 of reference [31] gave an example and the electronic supplement gives another one. Since the only concentration that changes significantly during the quasi-steady period is that of produced oxygen, we explained the transition to reaction O by the oxidation of iodide by oxygen. We had shown that the reaction \({\text{I}}^{ - } + {\text{ H}}^{ + } + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}{\text{O}}_{{2}} \to {\text{HOI}}\) with an empirical rate law allows to explain the experimental observations [12, 31]. The direct oxidation of iodide is much too slow but a radical pathway can be fast. Studying the nullclines calculated with the model in Table 1 [27] we concluded that the transition is explained by a saddle-node bifurcation occurring when the concentration of oxygen in solution reaches a critical value. Anything that promotes the transfer of oxygen to the gas phase delays the transition. This explains that it occurs later if the contact surface between the solution and the gas phase is increased and if the solution is stirred [32,33,34].

An essential feature of the proposed model is the competition between the Down reactions R5 followed by R6 and the Up reaction R10. The concentration of the intermediate I2O is very small in aqueous solutions, is quasi-stationary and approximately proportional to [HOI]2. The ratio between the rates r10 = k10 [I2O] [H2O2] and r5 = k5 [HOI] [H2O2] is therefore approximately proportional to [HOI]. If [HOI] is small, H2O2 mainly reacts as a reducing agent. If [HOI] is large, H2O2 mainly reacts as an oxidant. The reaction R4 being a quasi-equilibrium, [I−] must be small enough for H2O2 to act mainly as an oxidant giving reaction O.

It is difficult to quantify the effect of oxygen. The transfer reaction R14 to the gas phase is an over-simplification and the solution can become highly supersaturated in oxygen. Complicated phenomena of transfer at the interface and germination-growth of bubbles must be taken into account. Moreover, the rate of oxidation of iodide is photocatalyzed and depends on complicated radical reactions [36]. Finally, it can be noted that the reaction \({\text{2H}}_{{2}} {\text{O}}_{{2}} \to {\text{2H}}_{{2}} {\text{O }} + {}^{1}{\text{O}}_{{2}}\) is thermodynamically possible [37]. We could produce singlet oxygen which would react much faster with iodide. These complications explain that the induction period of the oxidation of I2 by H2O2 without iodate added initially seems stochastic [34, 35]. A phenomenon seems stochastic when it depends on parameters that we ignore or do not control. A qualitative explanation of the transition between the quasi-steady state and the reaction O when the oxygen concentration reaches a critical value is already a success of the proposed model.

I(+ 1) reduction by H2O2 (reactions R5 and R6)

Furrow [38] measured the absorbances at 354 and 460 nm of solutions of I2 and H2O2 without iodate during the quasi-stationary period. These absorbances allow to calculate the concentrations of I2 and I3−, then that of I− and finally that of HOI corresponding to the quasi-equilibrium R4. The sum of the reactions R5 + R6 + R11 gives the decomposition \({\text{H}}_{{2}} {\text{O}}_{{2}} \to {\text{H}}_{{2}} {\text{O }} + {\text{ O}}_{{2}}\) and Furrow assumed that their rates are equal during this period giving \({\text{k}}_{{5}} \left[ {{\text{HOI}}} \right] = \left( {{{k^{\prime}}}_{{{11}}} + {{ k^{\prime\prime}}}_{{{11}}} \left[ {{\text{H}}^{ + } } \right]} \right) \, \left[ {{\text{I}}^{ - } } \right]\). The values of \({{k^{\prime}}}_{{{11}}} \,{\text{and}}\,{{k^{\prime\prime}}}_{{{11}}}\) being known, k5 can be calculated. Furrow obtained k5 = 3 M−1 s−1 but its values were widely dispersed. We recalculated his results using updated values of the parameters (ε(I2, 354) = 17; ε(I3−, 354) = 26,250; ε(H2O2, 354); K4 = 1011 M−2 when I = 0.10 M) and plotted k11 [I−]/[HOI], where \({\text{k}}_{{{11}}} = {{k^{\prime}}}_{{{11}}} + {{ k^{\prime\prime}}}_{{{11}}} \left[ {{\text{H}}^{ + } } \right]\), as a function of [HOI]. If the above equality was verified, the values of k11 [I−]/[HOI] would be independent of [HOI]. Fig. 5 shows on the contrary a decrease when [HOI] increases. This explains why the values calculated ignoring this dependence were so dispersed.

Fig. 5
figure 5

Analysis of Furrow's results [38] explained in the text

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The model in Table 1 explains this effect of [HOI]. Numerical simulations show that reactions R2 and R10 cannot be neglected. Their sum gives the same global reaction as R11, \({\text{I}}^{ - } + {\text{ H}}^{ + } + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{HOI }} + {\text{ H}}_{{2}} {\text{O}}\). If R3 was an equilibrium and if we could neglect the effect of R12, we would have k5 [HOI][H2O2] = k11 [I−] [H2O2] + k10 [I2O] [H2O2] giving k11 [I−]/[HOI] = k5 – k10 k−3/k3 [HOI]. This approximate expression explains the effect of [HOI] seen in Fig. 5 and suggests k5 ~ 5 M−1 s−1 or a little larger by extrapolation at \(\left[ {{\text{HOI}}} \right] \to 0\). The numerical simulations of these experiments taking the kexp values for reaction O into account give k5 = 6 M−1 s−1. The electronic supplement shows a comparison between the experimental values and the values calculated with the model in Table 1.

Liebhafsky [39] measured the rate of oxygen production due to the reaction \({\text{HOI }} + {\text{ H}}_{{2}} {\text{O}}_{{2}} \to {\text{I}}^{ - } + {\text{ H}}^{ + } + {\text{ O}}_{{2}} + {\text{ H}}_{{2}} {\text{O}}\) (R5 + R6) in the presence of solid iodine and an excess of Tl+ ions. The concentration of iodine in solution was equal to its solubility, the concentration of iodide was fixed by equilibrium \({\text{Tl}}^{ + } + {\text{ I}}^{ - } \rightleftharpoons {\text{TlI}}\left( {\text{s}} \right)\) and that of HOI by reaction (R4). He deduced the values of k5 = (d[O2]/dt)/([HOI] [H2O2] in Fig. 6 giving on average k5 = 37 M−1 s−1. We recalculated his results [12] using more recent values of the equilibrium constants, but keeping Liebhafsky calculations assumptions, and proposed k5 = 23 M−1 s−1, a value significantly greater than k5 = 6 M−1 s−1 adopted above.

The effect of [H+] revealed by Fig. 6 suggests that the dispersion of the k5 values obtained by Liebhafsky is not due to experimental inaccuracies and that the system is more complicated than he thought. This is confirmed by numerical simulations using the model in Table 1 supplemented by reaction (3) and with [I2] fixed by the solubility of iodine. Under these conditions, oxygen is produced not only by reaction R5 followed by R6 but also by reactions R7 and R12. d[O2]/dt > r5 explains that the values of k5 obtained by Liebhafsky are too large. The model also explains a surprising observation reported by Liebhafsky. If he added only Tl+, the iodide concentration became very small, the rate of oxygen production much too high, and iodate was produced [ref. 39, bottom of p. 3504]. To avoid this, he had to add a large amount of solid TlI(s). If k12 ~ 107 M−1 s−1, order of magnitude estimated in the appendix, an important reaction path is R2 + R3 + R4 + R5 + R12 giving overall \({\text{2I}}^{ - } + {\text{ 2H}}^{ + } + {\text{ 3H}}_{{2}} {\text{O}}_{{2}} \to {\text{I}}_{{2}} + {\text{ O}}_{{2}} + {\text{ 4H}}_{{2}} {\text{O}}\) and the observed decrease in [I−]. The addition of solid TlI(s) was necessary to avoid this decrease and to obtain the overall reaction \({\text{2TlI}}\left( {\text{s}} \right) + {\text{2H}}^{ + } + {\text{3H}}_{{2}} {\text{O}}_{{2}} \to {\text{2Tl}}^{ + } + {\text{ I}}_{{2}} + {\text{ O}}_{{2}} + {\text{ 4H}}_{{2}} {\text{O}}\). This reaction greatly contributes to the production of oxygen and varies [H+]. The model explains Liebhafsky observations, shows that the system he studied gives too large values of k5 and is too complicated to allow an exact measurement.

Fig. 6
figure 6

k5 values (M−1 s−1) given by Liebhafsky in Ref. [39], p. 3506, Table 2

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Shin, Lee and von Gunten [40] studied the reaction between H2O2 and I2 in buffered solutions between pH 4 and pH 12. They confirmed that this reaction is catalyzed by the buffers. The large differences between the values of the kinetic constants published in the literature can, in part, be explained by the effect of the nature and the concentration of the buffers. These authors obtained k5 = 29 ± 5 M−1 s−1 but this value was obtained in acetic buffers and the value in unbuffered solutions is certainly smaller. In addition, the extrapolation to pH 1 of results obtained above pH 4 is very approximate. The value k5 = 6 M−1 s−1 that we propose is therefore compatible with the results of Shin, Lee and von Gunten.

Conclusion

The model in Table 1 explains the kinetics of the reactions in acidic solutions of iodine compounds at oxidation states from − 1 to + 5 with each other and with H2O2. The new experiments confirm the existence of the HOOI intermediate compound proposed previously [17,18,19,20,21,22]. The effect of acidity on the disproportionation of HOIO and on its oxidation by H2O2 shows that OIO− is much more reactive than HOIO and suggests that the mechanism of these reactions involves an intermediate compound noted I2O4H− (reaction R22). This model, with updated kinetic constants, also quantitatively explains the kinetics of many reactions previously studied under very different conditions and especially the complicated kinetics of the oxidation of I2 by H2O2 with or without iodate added initially. This model becomes more and more likely that it allows to explain a greater number of experiments of different types. None of these results suggest radical reactions in the dark, except in the mechanism of the oxidation of iodide by oxygen. Light, including that of a spectrophotometer, can initiate radical reactions briefly discussed in the appendix (in the Supplementary Information) which deserve further work.