Equilibrium Constant for the Reaction ClO + ClO ↔ ClOOCl Between

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Equilibrium Constant for the Reaction ClO + ClO ↔ ClOOCl Between 250 and 206 K Kelly L. Hume, Kyle D. Bayes,* and Stanley P. Sander Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109-8099, United States ABSTRACT: The chlorine peroxide molecule, ClOOCl, is an important participant in the chlorine-catalyzed destruction of ozone in the stratosphere. Very few laboratory measurements have been made for the partitioning between monomer ClO and dimer ClOOCl at temperatures lower than 250 K. This paper reports absorption spectra for both ClO and ClOOCl when they are in equilibrium at 1 atm and temperatures down to 206 K. The very low ClO concentrations involved requires measuring and calibrating a differential cross section, ΔσClO, for the 10-0 band of ClO. A third law fit of the new results gives Keq = [(2.01 ± 0.17) 10−27 cm3 molecule−1] e(8554∓21)K/T, where the error limits reflect the uncertainty in the entropy change. The resulting equilibrium constants are slightly lower than currently recommended. The slope of the van’t Hoff plot yields a value for the enthalpy of formation of ClOOCl at 298 K, ΔHof , of 129.8 ± 0.6 kJ mol−1. Uncertainties in the absolute ultraviolet cross sections of ClOOCl and ClO appear to be the limiting factors in these measurements. The new Keq parameters are consistent with the measurements of Santee et al.42 in the stratosphere.

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INTRODUCTION Forty years have passed since Molina and Rowland first published a warning that photolysis of the chlorofluorocarbons in the stratosphere could be a threat to the earth’s ozone layer.1 Subsequent work has shown that there are several reaction cycles involving the ClO radical that consume ozone catalytically. One of these that is especially important in the polar stratosphere involves chlorine peroxide, which can be formed by the three body recombination of ClO molecules, reaction R.1, where M is any collision partner.2−5 The peroxide bond is quite weak, ClO + ClO + M → ClOOCl + M

K abs =

■

(R.1)

Cl + O3 → ClO + O2

[ClOOCl] [ClO]2

(R.2)

(1)

Assuming the Beer−Lambert law for both ClOOCl and ClO, eq 1 can be written as Keq = K abs

(σClO)2 L σClOOCl

(R.3)

Special Issue: Mario Molina Festschrift

(2)

Received: October 6, 2014 Revised: December 9, 2014

where σ refers to cross sections (to base e), L is the optical path length, and the term Kabs involves only measured absorbances. © XXXX American Chemical Society

(3)

For the conditions used, −117 °C and 730 Torr, the ClO radicals primarily dimerized to form ClOOCl, reaction R.1. The flowing gases then entered a Pyrex tube maintained at about −150 °C, which trapped the dimer as a solid. The amount of ClOOCl trapped was in the range of 5−7 mg hour−1. After accumulating the dimer for about 1 h, the laser, Cl2 and O3, were turned off but the flow of nitrogen continued for another hour to remove excess reactants O3 and Cl2. Then when the cold trap was slowly warmed, the solid dimer vaporized and was swept into a cell by the flowing nitrogen, still at 730 Torr. This absorption cell, with a path length of 92.1 cm, was maintained at a constant temperature by circulating cold methanol through the cell jacket; a diagram of this flow system has been

The work described below is an attempt to measure the equilibrium constant for ClO and ClOOCl, Keq, at temperatures approaching those of the earth’s stratosphere. Keq =

(absClO)2

EXPERIMENTS The ClOOCl was generated, trapped, and used as described previously.7 Briefly, a dilute mixture of molecular chlorine and ozone in nitrogen was irradiated by an excimer laser at 351 nm (10 Hz, ∼60 mJ per pulse) to form chlorine atoms, which then generated ClO radicals by reaction R.3. The average ClO concentration during this generation phase was about 1 × 1013 molecules cm−3.

about 70 kJ/mol,6 so the reverse reaction, R.2, can also be important, depending on the temperature. ClOOCl + M → ClO + ClO + M

absClOOCl

A

DOI: 10.1021/jp510100n J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A Table 1. Reactions Used in Modelinga

published.7 The residence time of the dimer in the absorption cell varied from 47 s at 250 K to 57 s at 206 K. Absorption spectra were recorded continuously as the dimer evaporated. Each spectrum consisted of data accumulated during a 15 s time period. The light source was a wellcollimated beam from a 300 W xenon arc. The beam made four passes through the cell, for a total optical path length of 368.4 cm and then was focused on the entrance slit of a 0.25 m spectrometer. A slit width of 0.11 mm gave a resolution (fwhm) of 0.30 nm at the photodiode array detector. The temperatures in the absorption cell were measured by two platinum resistance thermometers immersed in the coolant flow, one at the entrance and the other at the exit of the absorption cell. The temperature difference between the two varied from 0.8 to 2.0 °C, the largest difference at the lowest temperature, with the exit flow being warmer. The average of these two measurements was used for the temperature of the gas phase reactants. Both thermometers were calibrated against a more precise thermometer (Omega HH376, calibration traceable to the National Institute of Standards and Technology, 2013). Since the flow through the absorption cell was slow, about 2 cm per second, any significant decay of the ClOOCl during its transit would complicate the subsequent analysis. Decay rates of the dimer could be measured by stopping the flow through the cell while monitoring the absorbance at 245 nm, the peak of the ClOOCl absorption. While the decay rates at the lower temperatures were slow, those at 250 and 240 K were fast enough (exponential decay rates of 2 to 3 min−1) to require remediation. Modeling showed that these decays were caused by the chlorine atom catalyzed chain decomposition of ClOOCl, reactions R.4 and R.5. The source of the chlorine atoms was the bimolecular reactions R.6 and R.7, both of which generate chlorine atoms. Cl + ClOOCl → Cl 2 + ClOO

(R.4)

ClOO → Cl + O2

(R.5)

ClO + ClO → Cl + OClO

(R.6)

ClO + ClO → Cl + ClOO

(R.7)

(R.1)

ClOOCl → ClO + ClO

(R.2)

Cl + O3 → ClO + O2

(R.3)

Cl + ClOOCl → Cl 2 + ClOO

ClOO → Cl + O2

(R.4)

(R.5)

ClO + ClO → Cl + OClO ClO + ClO → Cl + ClOO

(R.6) (R.7)

Cl + Cl 2O → Cl 2 + ClO Cl + OClO → 2ClO

(R.8) (R.9)

ClO + ClO → Cl 2 + O2

(R.10)

ClO + OClO → Cl 2O3

(R.11)

Cl 2O3 → ClO + OClO

(R.12)

ClO → wall Cl → wall

(R.13) (R.14)

a

Rate constants for reactions R.1 through R.12 were taken from ref 9; rate constants for R.13 and R.14 were estimated.

required. Since the residence time of the gas in the absorption cell was much longer than these relaxation times, the absorption spectra reported in this work are predominately of equilibrium mixtures. While the ClOOCl spectrum is a continuum, the ClO spectrum contains many sharp rotational lines. This means that the effective ClO cross sections will depend on the resolution of the spectrometer. Because of this sensitivity to resolution and the unique way of defining the amplitude of the 10-0 band, described below, it was necessary to measure the absolute cross section of ClO when using this same definition of amplitude. This was done by reacting chlorine atoms with OClO, reaction R.9. This reaction is fast and appears to be stoichiometric.10,11 Cl + OClO → 2ClO

(R.9)

A microwave discharge in argon containing a small amount of molecular chlorine was used as the source of atomic Cl. Approximately 25% of the Cl2 was dissociated by the discharge, as measured by the decrease in Cl2 absorption at 330 nm when the discharge was turned on. A second flow of 10% Cl2/He was passed over solid sodium chlorite (NaClO2) at room temperature to generate OClO and then was added to the chlorine atom flow downstream of the discharge. Sufficient time was allowed for the complete consumption of OClO before the flow entered the absorption cell. The total pressure was varied between 0.6 and 1.6 Torr, which resulted in residence times within the absorption cell of 27 to 60 ms. Most measurements were made at 296 K. Attempts to carry out this calibration at 250 K and below gave inconsistent results, suggesting that the ClO molecules were being lost to the walls.

By adding a small amount of Cl2O continuously just upstream from the absorption cell, reaction R.8 was able to inhibit the chain decomposition of dimer,8 reducing the decay rates by a factor of 10 or more. Cl + ClOCl → Cl 2 + ClO

ClO + ClO → ClOOCl

(R.8)

At lower temperatures, the concentrations of ClO are lower, reducing the number of ClO + ClO collisions, and also the rate constants for both reactions R.6 and R.7 are smaller,9 so the addition of Cl2O was not needed. The slow decay of ClOOCl at low temperatures shows that photolysis by the light beam was not a significant problem. Does the system remain in equilibrium as the gas flows down the absorption tube? Because the dissociation of ClOOCl into two ClO is an endothermic process, the rate constant for reaction R.2 decreases rapidly with decreasing temperature. The concentration of ClO needed for equilibrium also decreases as the temperature decreases. Modeling this system using FACSIMILE and the reactions in Table 1 showed that starting from only ClOOCl in 730 Torr of N2, with no ClO present, only 0.1 s was required for the [ClO] to reach 95% of its equilibrium concentration at 250 K. At 205 K, 1.5 s were

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RESULTS Examples of absorption spectra are shown in Figure 1 for two different temperatures. The strong absorption by ClOOCl at about 245 nm was used to determine the concentration of ClOOCl at equilibrium.12 In the region from 265 to 295 nm, the well-known spectra of ClO can be seen as a series of small peaks in the 250 K spectrum.13 In contrast, the 206 K spectrum in Figure 1 gives no visible B

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The Journal of Physical Chemistry A

Figure 1. Absorption spectra at two temperatures. The absorbance (base e) due to the small amount of Cl2O present in the 250 K spectrum, 0.26 absorbance units at 245 nm, has been subtracted. The solid black line shows the least-squares fit of the 250 K spectrum to the recommended ClOOCl spectrum.9 The ClO bands can be seen as a series of small peaks in the 265 to 295 nm region of the 250 K spectrum but not in the 206 K spectrum. The inset shows a scan of the mercury 253.7 nm line with the same optics as used for all spectra; the green line is a Gaussian fit to that data.

indication of absorption by ClO; however, the same ClO peaks are present but too weak to be evident on this plot. Quantifying the amplitudes of the ClO peaks was done in the following way. An experimental spectrum was least-squares fitted to the known ClOOCl spectrum plus a baseline, even for those cases where ClO bands were clearly present. The resulting fitted function was then subtracted from the experimental spectrum to give a difference spectrum. Two examples of these difference spectra are shown in Figure 2. These show the known vibrational progression of the A2π ← X2π electronic transition of ClO.13 Figure 2 shows only the 9-0, 10-0, and 11-0 vibrational absorption bands. The current study will focus on the 10-0 band, close to 280 nm, as that has the largest difference in amplitude between the peak and the valley to shorter wavelength. The strong peaks correspond to the A2π3/2 ← X 2π3/2 progression. The shoulder to the long wavelength side of the 10-0 peak is the weaker A2π1/2 ← X2π1/2 band of the 9-0 transition. One disadvantage to using this difference method for the ClO bands is that the true baseline for the ClO bands is not determined. Rather than attempting to recover the baseline, the following procedure was used: on a plot of the difference spectrum, a straight line was drawn between the minima of the adjacent valleys of the 10-0 band, at approximately 279.3 and 282.0 nm, as shown in Figure 2. Then the vertical difference in absorbance between the maximum at 279.8 nm and the straight line was used as the absorbance of the 10-0 band, absClO. This method does not establish a real baseline, but it does serve as a reproducible measure of the peak amplitude. Note that a sloping baseline does not affect the vertical amplitude as long as the baseline is reasonably straight over this narrow wavelength range. The peak amplitude of the least-squares fitted ClOOCl curve at 245 nm, minus the fitted baseline, was used for absClOOCl. However, the ClO spectrum does have a true continuum at

Figure 2. Difference spectra derived from the two spectra shown in Figure 1. (a) is for 250 K and (b) is for 206 K. The vertical red arrows, drawn from the peaks to the hand-drawn straight lines connecting the two adjacent valleys, are used as a measure of the absorbance of the 10-0 ClO bands.

wavelengths less than 263 nm.14,15 So when strong ClO bands were present (i.e., at the higher temperatures), a small correction to absClOOCl was made to account for the ClO continuum contribution at 245 nm. This correction was 2% or less for the measurements at 250 K and became progressively smaller at lower temperatures. Using the absClO and the corrected absClOOCl, values for Kabs could be calculated for each spectrum. For most measurements, three to ten spectra having similar amplitudes were averaged before calculating a Kabs. Then the several Kabs measured on a given run were averaged; these average values have been entered in Table 2 along with their sample standard deviations, sabs. Also entered in Table 2 are values of N, the total number of 15 s spectra that were used to calculate each average Kabs. Entries with N of 12 or less were taken during stop flow events when the absorbances were changing fairly rapidly so a Kabs was calculated for each spectrum before averaging. Converting Kabs values to Keq requires absolute cross sections for both ClOOCl and the 10-0 band of ClO. Since the dimer spectrum goes from a bound ground state to a repulsive upper electronic state,12,16,17 its cross section should not be strongly dependent on the spectral resolution or the temperature. So a recent value for the maximum at 245 nm, (7.6) × 10−18 cm2 C

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The Journal of Physical Chemistry A Table 2. Averaged Values of Kabs and Keq and Related Parameters T (K) 250.4 250.4 250.5 250.5 240.6 240.6 240.6 240.6 230.4 220.6 220.6 220.6 220.7 215.9 215.9 215.9 211.0 206.3

Kabs 618 656 606 603 2.36 × 103 2.39 × 103 2.28 × 103 2.24 × 103 1.115 × 104 5.01 × 104 4.72 × 104 4.70 × 104 4.97 × 104 1.111 × 105 1.127 × 105 1.136 × 105 2.404 × 105 5.63 × 105

sabsa 13.7 7.45 3.58 22.9 22.2 82.9 42.4 52.7 507 2147 2260 1210 1210 3.98 × 7.04 × 5.23 × 8.21 × 3.27 ×

Nb

103 103 103 103 104

ΔσClOc(10−18 cm2)

70 5 90 9 50 12 12 20 64 16 52 40 110 50 60 45 55 80

6.78 6.78 6.78 6.78 7.05 7.05 7.05 7.05 7.35 7.64 7.64 7.64 7.64 7.79 7.79 7.79 7.95 8.10

sKeqd (cm3 molecule−1) 3.05 1.66 7.98 5.10 5.35 2.00 1.02 1.27 1.33 6.08 6.39 3.42 3.42 1.17 2.07 1.54 2.52 1.04

× × × × × × × × × × × × × × × × × ×

−14

10 10−14 10−15 10−14 10−13 10−13 10−13 10−13 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−10

Keq (cm3 molecule−1) 1.378 1.461 1.351 1.344 5.68 5.77 5.50 5.39 2.92 1.417 1.334 1.330 1.406 3.27 3.32 3.34 7.37 1.79

× × × × × × × × × × × × × × × × × ×

10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−11 10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−10 10−9

a

The sample standard deviation when averaging Kabs. bThe number of spectra averaged when calculating Kabs. cThe differential cross section of ClO for the 10-0 band (peak-to-line connecting adjacent minima, cf. Figure 2), as determined in this work. dThe calculated standard deviation for Keq.

molecule−1, will be used for σClOOCl for all temperatures in this study.9,12 The ClO bands contain both sharp rotational structure and some continuum, so their cross sections are expected to be functions of both the spectral resolution and temperature. Because of the unique definition of cross section used in this study and the sensitivity to resolution, it was necessary to determine an absolute cross section for the 10-0 band with the same optical system used for the experiments on the equilibrium mixture of ClOOCl and ClO. Reaction R.9 was used to generate ClO in the absence of ClOOCl. An example is given in Figure 3. With the discharge

off, the known bands of OClO are evident at the longer wavelengths. When the discharge was turned on, the OClO peaks disappear completely because there is a surplus of atomic chlorine, and the ClO peaks appear at shorter wavelengths. The peak absorbances of the four strongest OClO bands shown in Figure 3, a(13) through a(16), were used to determine the OClO concentration using the Beer−Lambert law. Then assuming a 2:1 stoichiometric ratio for reaction R.9, it was possible to calculate the concentration of ClO, which ranged from (5 to 11) × 1013 molecules cm−3. Combining the ClO concentration with the observed amplitude of the 10-0 peak, as defined above, resulted in a differential ClO cross section, ΔσClO, that will be used in calculating the equilibrium constant. Calibrations of ΔσClO at 296 K were made at several pressures between 0.6 and 1.7 Torr, resulting in residence times of 79 to 37 ms. Within this range, there was no evident effect of calculated cross section on residence time, which means that the loss of ClO on the halocarbon wax-coated walls was not a significant factor, at least at room temperature. Using the calibration method of converting OClO to ClO, as shown in Figure 3, also requires absolute cross sections for the bands of OClO. There are two sets of recommended peak cross sections for OClO, one by Wahner et al.18 and the other by Kromminga et al.19 These two studies disagree in absolute magnitude of σOClO, with the Kromminga et al. cross sections smaller by 8 to 14% for the a(13) to a(16) bands at room temperature. By comparing the ratios of peak band intensities, a(14)/a(13), a(15)/a(13), and a(16)/a(13), for these two studies, the Wahner et al. (resolution 0.25 nm) appeared to be a better match to the ratios observed in the present work (resolution 0.3 nm), and so the Wahner et al. peak values for σOClO were used in this study. It was also necessary to consider corrections for any loss of ClO within the absorption cell. Losses were modeled using the program FACSIMILE and the reactions shown in Table 1. Using realistic initial concentrations of OClO and Cl (e.g., [OClO]o = 4 × 1013 and [Cl]o = 9 × 1013 molecules cm−3) showed that the conversion of OClO to ClO was more than 95% complete before the gases entered the absorption cell. The

Figure 3. Use of reaction R.9 to generate known concentrations of ClO. When the discharge is turned on, the excess Cl completely converts the OClO to ClO. D

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The Journal of Physical Chemistry A

absorbance between 253.1 and 256.4 nm and using the average cross section20 of 1148 × 10−20 cm2, the ozone concentration was calculated. Assuming a 100% conversion of O3 to ClO, the ClO concentrations together with the absClO for two different runs give the two ΔσClO values shown in Table 3. It should be noted that questions have been raised about the assumption of one ClO molecule being formed for each O3 reacted. Vandersander and Birks21 observed oxygen atoms being formed as well as ClO; the yield of oxygen atoms was less than 0.5% of the ClO. Burkholder et al.22 found significant deviations from a 1:1 ratio but only for a large excess of chlorine atoms. Neither of these effects should impact the present experiments, where the initial concentration of Cl in Figure 4 was approximately double that of O3. The continuum part of the ClO spectrum also can be used to estimate ΔσClO. Because it is a true continuum, the amplitude is not highly dependent on the resolution and there is general agreement on absolute cross sections in this region. Using the observed absorbances in the region from 256 to 263 nm and their recommended cross sections,23 the ClO concentrations were calculated for the two runs with ozone. Combining those with the absClO for the 10-0 band resulted in the last two entries in Table 3. All three methods used to calculate ΔσClO appear to give reasonable values. Without a good reason to favor one over the other, all ten values in Table 3 were averaged to give 5.71 × 10−18 cm2 with a sample standard deviation of 6.0% of the mean. Measuring the temperature-dependence of ΔσClO gave questionable results. As the temperature was decreased, the apparent ClO cross sections initially increased, as expected, but then decreased at temperatures of 215 K and below, suggesting that ClO molecules were being lost on the walls. Varying the flow rate at low temperature confirmed that wall loss was the problem. So the temperature-dependence of the ClO cross sections was estimated in the following way. The program PGOPHER24 together with the molecular parameters25−28 for ClO was used to calculate the expected spectrum by imposing a resolution of 0.30 nm on each rotational line. Calculations were carried out for the 9-0, 10-0, and 11-0 bands of both Cl35O16 and Cl37O16, and then the intensities of these six bands were added together, using the weighting factors of 0.75 for the Cl35O and 0.25 for the Cl37O molecules. Figure 5 shows a fit of two experimental spectra with the PGOPHER curves. These calculations were repeated for temperatures down to 210 K. The resulting spectra were then plotted and treated as if they were experimental spectra. The vertical amplitudes (ΔσClO)T were divided by the amplitude at 296 K; the results were wellrepresented by eq 4.

dominant loss of ClO was formation of ClOOCl followed by its rapid destruction, reaction R.4. The largest correction for this loss, about 10%, was for the lowest pressure runs, due to the longer residence time; these corrections were made before the final values were entered in Table 3. The modeled steady state Table 3. Experimental Values of ΔσClO at 296 K method

pressurea

ΔσClOb

Cl + OClO

0.62 0.62 0.65 0.84 1.02 1.7 1.02 1.02 1.02 1.02

5.93 5.49 5.39 5.49 5.77 5.67 5.27 5.56 6.12 6.37

Cl + O3 continuum a

Units of Torr. bUnits of 10−18 cm2 molecule−1.

concentration of ClOOCl was much too small to interfere with the absorption measurements of the 10-0 band. An average of runs on six different days, all at 296 ± 1 K, gave a mean value of 5.62 × 10−18 cm2 molecule−1 with a sample standard deviation of 3.6% of the mean. An alternate way of measuring ΔσClO involved reacting chlorine atoms with ozone, reaction R.3. The setup was similar to that used for Cl + OClO. Figure 4 shows the strong ozone absorption when the discharge is off and then the ClO spectrum when the discharge is turned on. A surplus of Cl assures that no ozone absorption interferes with the ClO absorbance measurement. Averaging the strong ozone

(ΔσClO)T = 2.629 − [4.616E(− 3)]T (ΔσClO)296 − [1.316E(− 5)]T 2 + [3.433E(− 8)]T 3 (4)

Using the average value of the room temperature ClO cross section determined above, it is then possible to calculate the ΔσClO for other temperatures using eq 4. These cross sections have been added to Table 2 for the temperatures used to measure Kabs. Inserting the values for σClOOCl, ΔσClO, the optical path L (368.4 cm), and Kabs into eq 2 yields values for Keq; these have been added as the last column of Table 2. Also entered in Table 2 are sKeq, estimates for the standard deviations

Figure 4. Use of reaction R.3 to generate known concentrations of ClO. When the discharge is turned on, the excess Cl converts the O3 to ClO + O2. Modeling showed that about 5% of the ClO was lost due to ClOOCl formation. E

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The Journal of Physical Chemistry A Keq (cm 3 molecule−1) = A e(B/T)

(5)

The parameter A can be related to the change of entropy for reaction R.1 at 298 K, ΔSo, by ⎛ eR′T ⎞ ⎛ ΔS o ⎞ ⎟ + ln⎜ ln(A) = ⎜ ⎟ ⎝ R ⎠ ⎝ NAv ⎠

(6)

where ln represents logarithm to the base e, R′ is the gas constant (83.145 cm3 bar mol−1 K−1), and NAv is Avogadro’s constant (6.0221 × 1023 molecules mol−1).35 These conversion factors are needed since the simple relationship between ln(Keq) and the Gibbs free energy change36 requires that the equilibrium constant be in units of pressure, while the values being used above are in concentration units. The corresponding expression for B is

Figure 5. Comparison of the measured 10-0 band of ClO with the spectra calculated using PGOPHER (imposed resolution of 0.3 nm). The 210 K spectrum and curve have been displaced by +0.001 to avoid overlap. The intensity scale comes from the PGOPHER fitting; the intensity of the experimental spectra have been scaled to fit.

⎛ ΔH o + RT ⎞ ⎟ B = −⎜ ⎠ ⎝ R

(7)

where ΔH is the change in enthalpy of reaction R.1 at 298 K and R is now in energy units (8.3145 J mol−1 K−1). Since the molecular parameters for both ClO and ClOOCl are fairly well-established, it is possible to calculate their absolute entropies and thus obtain a value of ΔSo for reaction R.1. Recent calculations show good agreement, with So for ClO (225.0 ± 0.1 J K−1 mol−1)31,37,38 and So for ClOOCl (301.7 ± 0.7 J K−1 mol−1),9,30,31,37,38 both for a standard state of one bar of an ideal gas at 298 K. These values result in a ΔSo of −148.3 ± 0.7 J K−1 mol−1. When this entropy change is used with eq 6, ln (A) = −61.4735 or A = (2.01 × 10−27 cm3 molecule−1). The parameter B was determined by fitting the experimental values of Keq to eq 5 using a nonlinear weighted least-squares calculation employing the Marquardt−Levenberg algorithm. The weighting factors used were the inverse of the expected variance of the mean values of Keq, namely N/(sKeq)2. Using the values of Keq, N, and sKeq in Table 2 and fixing the intercept at (2.01 × 10−27 cm3 molecule−1), as calculated above, gives the equation o

of the Keq, calculated from sabs by the same procedure that converted Kabs to Keq. The experimental values of Keq have been plotted in Figure 6 together with previous values for Keq from

Keq = (2.01 × 10−27cm 3 molecule−1) e(8554K/ T )

(8)

The resulting 95% confidence limits for B are ±3 K. This third law fit is shown as a solid blue line in Figure 6. However, a more realistic estimate of the uncertainties involved comes from including the uncertainty in ΔSo (i.e., using the maximum and minimum limits for ΔSo in eq 6 to give maximum and minimum values for the parameter A and then redoing the third law nonlinear weighted least-squares fitting for each A). This results in the following error limits,

Figure 6. Van’t Hoff plot of the equilibrium constant Keq as a function of the reciprocal of the absolute temperature. The solid blue line is a fit to the present results only, with an imposed intercept of (2.01 × 10−27) cm3 molecule−1.

Keq = [(2.01 ± 0.17) × 10−27cm 3 molecule−1] e(8554 ∓ 21)K / T

(9)

29−34

While van’t Hoff plots such as Figure 6 are the literature. in general not perfectly linear, in the present case, the fit to a straight line is very good because the change in heat capacity, ΔCp, for reaction R.1 is small.37

where a larger value of A requires a smaller value of B, and vice versa. A second law fit of Keq to eq 5, where both A and B are determined by nonlinear weighted least-squares, gives a very similar relationship,

■

DISCUSSION The values for the equilibrium constant will be fitted with the same approach used by the JPL/NASA data evaluation panel,9 namely,

Keq = [(2.13 ± 0.56) × 10−27 cm 3 molecule−1] e(8539 ∓ 64)K / T F

(10) DOI: 10.1021/jp510100n J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A where the ± values now represent 95% confidence limits. Since the second law fit requires a long extrapolation to give A, the close agreement between the second and third law fits is at least partially fortuitous. A best fit expression for Keq should take into account the many previous measurements that have been made using approaches different from those used in this study. Many of these values have been included in Figure 6. For comparison with these higher temperature results, the third law fit of the current data has been extrapolated as a dashed blue line in Figure 6. As can be seen, more of the measurements made at temperatures higher than 250 K fall above the extrapolated blue line than below it. This raises the important task of assigning error limits for the current values of Keq. As can be seen in Table 2, the sample standard deviations for duplicate measurements of Kabs are only a few percent, reaching about 6% for some lower temperatures. Much more important are the uncertainties in cross sections used to convert Kabs to Keq. Reported values for the cross section of ClOOCl at 245 or 248 nm vary from (6.40 to 8.85) × 10−18 cm2, representing a deviation of about ±17% from the value used in this study (7.6 × 10−18 cm2).9,39−41 The uncertainties in the values of ΔσClO in Table 1 come from at least two sources. Since the ΔσClO at low temperatures are scaled from the average room temperature value, they all will share the uncertainty of the room temperature value. Twice the standard deviation for the values in Table 3 would be 12%. However, this value was determined by using the cross sections of OClO reported by Wahner et al.18 An alternate set of OClO cross sections for the a(13) through a(16) bands have been reported by Kromminga et al.19 that are about 10% smaller. Using smaller OClO cross sections in the calibration experiments reported above would result in smaller ClO cross sections. These uncertainties in the ClO cross section have twice the impact as those of ClOOCl because when converting Kabs to Keq, the square of ΔσClO is used but only the first power of σClOOCl. Previously published spectra of ClO also can be used to calculate ΔσClO using the method demonstrated in Figure 2. The spectra from two studies that claim to have comparable resolution to that used here give significantly lower values of ΔσClO for the 10-0 band (4.3 × 10−18 cm2)14 and (4.8 × 10−18 cm2);15 however, neither of these spectra show the distinct dip in intensity at 280.6 nm that can be seen in Figure 2, and that is also evident in the PGOPHER spectra, indicating that their resolution was really lower than claimed. Also, by visually comparing the 9-0 bands in these two studies, it is evident that the spectrum of ref 15 has slightly higher resolution than that of ref 14. Feierabend et al.41 have published a room temperature spectrum of the 10-0 band at much higher resolution; although it is difficult to visually degrade their spectra to a 0.3 nm resolution, it appears that a ΔσClO of about 7 × 10−18 cm2 would result, which is about 20% higher than the value used for Table 2. The importance of spectral resolution can be seen by comparing these three literature values with the present results; ΔσClO/(10−18 cm2) increases from 4.3 to 4.8 to 5.6 to about 7 with increasing resolution. For this reason, we favor using the value of ΔσClO determined in the present study because it was done with exactly the same resolution used when determining Kabs. Even though the various error estimates given above are not all random errors, a propagation of errors approach can suggest the possible consequences for their combined effects on Keq.

Using propagation of errors on eq 2 and error estimates for Kabs (6%), σClOOCl (17%), ΔσClO (10% and 12%), and L (