J. Phys. Chem. 1983, 87, 118-120
118
other will be pushed outward to create a region of higher density and ensuing ripples. The volume of the system will, in general, change during this relaxation process. Relation A9 states that when the volume of the initial (prerelaxation) cavity is equal to I&, the PMV of the solute, the volume of the system does not change during this relaxation process. Thus, as far as the volume relation is concerned, one may ignore the relaxation process and consider the solution as consisting of two regions sharply divided by C, the outer region of which consists of the unaltered, uniform pure solvent (i.e., &(r)= E,") while the inside consists of the solute and a shell of void or "border". For the computation of the local average packing density for the solute, t2,it is natural in this sense t o choose the region enclosed by C as the locality over which the average is to be taken. Obviously
Location and Shape of the Cavity Surface Relation A9 is only a necessary condition for the construction of C. Obviously, C can have many different shapes and still satisfy condition A9. As a way of reducing this redundancy, we shall arbitrarily stipulate that tl(r) be constant over C.29 We also assume that the solvent
molecule is spherical. The largest surface over which tl(r) = 0 defines the "molecular surface" introduced by Richa r d ~ It. ~is ~made of convex patches that are the accessible parts of the physical surface of the solute smoothly joined with concave patches between two or more atoms of the solute, inside of which no part of solvent can penetrate. The "accessible surface" is the locus of the center of the spherical solvent in contact with the s ~ l u t e . ~The ~ ~ first ~' and most dominant peak in El(r)occurs near this surface. Because El(r) oscillates only mildly about f:beyond the first peak (see Figure 4), condition A9 cannot be satisfied unless the first peak is outside of C. In other words, the surface C lies between the molecular and the accessible surface. Since tl(r) varies rapidly along the radial direction in this region, C will also look parallel to these limiting surfaces. In the case of a spherical solute in water, surface C lies about 0.5 A from the surface of the solute while the accessible surface is about 1.4 A away. (29) This may not be possible if the solute interacts strongly with the solvent so as to essentially immobilize one or more solvent molecules on its surface. A three-dimensional contour plot of &(r)for such a system will contain one or more "islands" of high Sl(r) region and a plot of the value of Sl(r) as a function of the volume enclosed by the constant El(r) surface will show one or more horizontal gaps. If the value of d, happens to lie in the range of one of these gaps, the corresponding constant t,(r) surface will not exist. Such complication will not arise for our system where strong specific solvation is assumed not to occur. (30) B. Lee and F. M. Richards, J . Mol. Biol., 55, 379 (1971). (31) R. B. Hermann, J . Phys. Chem., 76, 2754 (1972).
Reaction of Hydroxyl Radicals with Hydrogen at Elevated Temperatures. Determination of the Activation Energy Hilbert Christensen' Studsvik Energiteknik AB, S-611 82 Nykoping, Sweden
and Knud Sehested Accelerator Department, Ris 0 National Laboratory, DK-4000 Roskilde, Denmark (Received: June 2, 1982)
The reaction of hydroxyl radicals with hydrogen in aqueous solutions has been studied in the temperature range 20-230 "C. The rate constant of the reaction was determined either by competition kinetics, using Cu2+as the competitor, or through the measurement of the formation rate of 02in aqueous solutions of oxygen and hydrogen. The rate constants at 20 and 230 OC are (3.4 f 0.3) X lo7 and (7.7 f 1.5) X lo8 dm3 mol-' s-', respectively. The activation energy of the reaction is 19 kJ mol-' (4.6 kcal mol-').
Introduction With the object of studying the radiation chemistry of water a t the operating temperatures in power reactors, we have continued a program to determine reaction rates and activation energies of radiolytic reactions by pulse radiolysis at elevated temperature^.'-^ In the present investigation the reaction of the hydroxyl radical with hydrogen in aqueous solutions was studied. Schmidt4measured the rate constant of this reaction at temperatures K. Radiat. Phys. Chem. 1980, 16, 183. (2) Christensen, H.; Sehested, K. Radiat. Phys. Chem. 1981, 18,723. (3) Christensen, H.; Sehested, K.; Corfitzen, H. J P h y s . Chem. 1982, 86, 1588. (4) Schmidt, K. H. J . Phys. Chem. 1977, 81, 1257.
between 15 and 90 O C , and, on the basis of these data, he calculated an activation energy of 3.18 0.5 kcal mol-'. Previous calculations of the radiolysis at elevated temperatures have been based on an estimated value of the activation energy of 7 kcal mol-' for the reaction OH + H2 H 2 0 + H.5-7 In calculations of the concentrations of H2, H202,and O2 at the operating temperatures of a boiling water reactor after additions of h y d r ~ g e nit, ~was found that the calculated values were either 1 order of magnitude higher or lower than the experimental results when acti-
*
-
Q) Chfistensen, H.; Sehested,
(5) Jenks, G. H. Report ORNL-4173; Oak Ridge National Laboratory: Oak Ridge, T N , 1967; p 10. (6) Burns, W. G.; Moore, P. B. Radiat. Eff. 1976, 30, 233. (7) Christensen, H. Radiat. Phys. Chem. 1981, 18, 147.
0022-3654/83/2087-0118$01.50/00 1983 American Chemical Society
Reaction of Hydroxyl Radicals with Hydrogen
vation energies of 3 or 7 kcal mol-' were used. However, good agreement between measured and calculated values was obtained by using an activation energy of about 5 kcal mol-'. Due to the importance of this reaction in the radiolysis of water, and the uncertainty in the activation energy, we have performed a pulse radiolysis study of this reaction in the temperature range 20-230 "C.
Experimental Section The experimental procedure has been described in detail previously.' The experiments at elevated temperatures were performed in the high-temperature-pressure cell.' The HRC linac at Rise, delivered 10-MeV electrons in a single pulse of maximum 1.1A and with a pulse length of 0.5-1 ps. The dose varied from 0.5 to 3 krd/pulse and was measured with the hexacyanoferrate(I1) dosimeter using €420 = 1000 dm3 mol-' and G = 6.0. The optical system consists of a 150-W Varian high-pressure xenon lamp, a Perkin-Elmer double quartz prism monochromator,a 1P28 photomultiplier, and a Nicolet Explorer I11 digital storage oscilloscope. The data were treated on an on-line PDP8 computer. CuS04of Merck p.a. quality was used without further purification. The gases 02,NzO, and Hz, in steel bottles from AGA, were used without purification. In some experiments a gas mixture of 10% H2 in Ar was used. Results and Discussion The rate constant of the reaction OH + H2 H2O + H
-
(1) can be measured following the decay kinetics of the OH radical in hydrogen-containing aqueous solution^.^ The extinction coefficient for the OH radical is, however, low and the absorption spectrum is situated in a wavelength region in which it is difficult to measure (emm = 540 dm3 mol-' cm-' at 230 nm).4!8 For this reason we decided to use two other methods for the determination of the rate constant for reaction 1. The first method was based on competition kinetics, and among a number of competitors we found Cu2+ions to be useful up to temperatures of 150 "C. Cu2+ions are oxidized by hydroxyl radicals forming CU(III),~ which absorbs at 300 nm with an extinction coefficient of 5700 dm3 mol-' cm-'.
OH
-
+ Cu2+
CU(OH)~+
(2) By measuring the yield of CU(OH)~+ at different ratios of hydrogen and Cu2+,one can determine the rate constant ratio from a competition plot. As the activation energy for reaction 2' and thereby the rate constants at various temperatures up to 220 "C are measured previously, the rate constant for reaction 1can be determined at different temperatures by making a competition plot at each temperature. An example at 50 OC is shown in Figure 1. The ratio H2/Cu2+was varied in the range 3-15 and the Cu2+concentration was either 1 X or 5 X mol dm-3. Each rate constant determination was carried out with at least three ratios of H2/Cu2+. The cell was pressurized with 1 MPa of N 2 0 and various pressures of hydrogen. In order to dissolve the hydrogen in the water, we stirred the solution for about 20 min. The hydrogen concentration was calculated from data on Henry's constant (Table I.)l0 The hydrogen pressure was measured (8) Pagsberg, P.; Christensen, H.; Rabani, J.; Nilsson, G.; Fenger, J.; Nielsen, S. 0. J. Phys. Chem. 1969, 73, 1029. (9) Baxendale, J. H.; Fielden, E. M.; Keene, J. P. "Pulse Radiolysis"; Academic Press: London, 1965, p 217. (10) Himmelblau, D. M. J . Chem. Eng. Data 1960, 5, 10.
The Journal of Physical Chemistry, Vol. 87,No. 1, 1983
119
9
OD,-OD
I T
t
/
l5
Figure 1. Scavenger plot for the reactions of OH radicals with hydrogen and Cu2+ at 50 OC. OD, Is the optical density of Cu(OH)'+ in the absence of hydrogen. OD Is the observed optical density at the mol dm-3 desi nated H2/Cu2+ratio on the graph. Solution: 5 X CuzB, pH 5.1-5.6; 1 MPa of N20, 0.5 krd, 2.5-cm cell.
TABLE I: Rate Constants ( h ) for the Reaction of OH t H, at Various Temperaturesa 104 x
temp, "C
(solubility of H, a t 0.1 MPa),b mol dm-3
20 50 60 80 90 100
7.9 7.3 7.2 7.2 7.5 7.9 8.1
130 150
9.3 10.4
200 230
14.5 17.8
40
lO-'h, dm3 mo1-ls-l oxygen soln 3.2 6.2
cua+ soln 3.6 8.0
8.4 12.2 17.5 20 27 42 55 77
43
a Dose, 5.3 Gy. Actual hydrogen pressures varied from 1.2 MPa at the lowest temperature t o 0.05 MPa at the
highest temperature.
Reference 10.
at room temperature. At the higher temperatures the pressure increased partly due to the increasing vapor pressure of water and partly due to the expansion of the gas. The calculations of the concentrations of hydrogen at high temperatures were thus based on the calculated pressure of hydrogen at the actual temperature. The concentration of Cu2+was corrected for the change in water density with temperature. The pH of the solution was 5.1-5.6. The high concentration of N20 (-0.2 mol dm-3) is sufficient to scavenge more than 90% of the hydrated electron even with a copper concentration of 5 X mol dm-3. The hydrogen atom formed in reaction 1 reacts with Cu2+forming Cu+, which in turn reacts with the complex formed in reaction 2. By minimizing the dose, one can suppress this second-order reaction. The results are shown in Figure 2 and Table I. The Cu2+system can only be used up to 150 "C and already at this temperature the determination is less reliable due to a slow thermal reaction apparently taking place between Cu2+and hydrogen Cu(I1) + H2 Cu(1) (3) After a number of pulses, Cu(II1) is formed in predicted
-
120
Christensen and Sehested
The Journal of Physical Chemistry, Voi. 87,No. 1, 1983
TABLE 11: Reactions in Aqueous Solutions of 0,, N,O,and H, at 20 and 200 "C After a Dose of 5.3 Gy 20 "C 200 "C reaction O H t H,+H H t O,-+O; OH t OH -+ H,O, e - t 0 , -+ 0,e,aq.. N , O + O H 08 t 0,- + OH+
a
c , mol ~ dm-3
- lo-, 1 . 2 5 x 10-3 +
Concentration.
0,
-3 x 1.25 x 10-3 0.2 -3 x
-
Reference 5 .
k , dm3 mo1-ls-l 3.4 x 107 2 x 1010 5 x 109 2 x 1010
6
X
lo9
1010
k c , s-l 3.4 x 105
2 . 5 ~ io7
1 . 5 x 104 2.5 x i o 7 1.2 x io9 3 x 104
E,, kcal mol-' c , mol ~ dm-3 3b
1.1 x 10-3 1.8x 10-3
2c
2.5 X
4.6 3b
3b 3b
1.8x 10-3 -0.2 -2
x 10-6
k , dm3 mol-' s-I 5.5 x
io*
3 x 1011 2 X 10"
3 x ioil 1 x 10" 10"
k c , s-'
-
6 X lo' 5 x lo8 5 x 104 5 x lo8 - 2 x 1O'O - 2 x 105
Reference 2.
To avoid serious influence from second-order reactions (Table 11),we kept the dose low (0.5 krd in 0.5 ps) and adjusted the buildup of 0; to last only a few microseconds by varying the hydrogen concentration. A small correction, however, for second-order reactions was applied by computer simulatons, using a computer program based on DIFSUB.'~ The corrections were of the order of 1O-15%. pH 9 was chosen to minimize the influence of the decay of 0 2 - . From the Arrhenius plot in Figure 1an activation energy of 19 kJ mol-' (4.6 kcal mol-') can be calculated. The expression for the rate constant in the gas phase has recently been given as k = (4.12 X 10-'9)P.44 exp(-1281/T) cm3 molecule-'s-l (ref 12) and k = (1.83 X 10-15)P3exp(-1835/T) cm3 molecule-' s-l (k = (1.10 X 106)T1.3exp(-1835/T) dm3 mol-' s-l)13 (non-Arrhenius behavior). In the latter case an activation energy, at 300 K, was estimated as 4.4 kcal mol-'. The activation energy in aqueous solutions is thus only slightly higher than in the gas phase.
-
Figure 2. Rate constant for the reaction between OH and H, as function of the reciprocal Kelvin temperature: (0)oxygenated solutions, pH 9; (X) Cu2+ solutions, pH 5.1-5.6.
quantities, but the concentration of hydrogen is then uncertain. In the second set of experiments kl was measured in a system containing both oxygen and hydrogen. The rate of formation of 02-in reaction 4 is measured by its abH O2 02-+ H+ (4)
+
-
sorption at 265 nm. Oxygen is added in sufficient amounts (1.25 X mol dm-3, k4 = 2 X 1O'O dm3 mol-l 9-l) to let reaction 1 be rate determining (Table 11). In order to scavenge all eeq-,we added 1MPa of N20 to the solutions. The hydrogen was supplied from a bottle containing 10% H2 in Ar. A t each temperature about 20 determinations were carried out; the mean values are given in Table I.
Acknowledgment. T. Johansen and H. Corfitzen are greatly acknowledged for technical assistance and E. Bjergbakke for assistance in the computer calculations. H.C. gratefully acknowledges a grant from Asea Atom, OKG, Sydkraft, and The National Swedish Power Administration. Registry No. H20, 7732-18-5; OH, 3352-57-6; H2, 1333-74-0; CU'+, 15158-11-9; 0 2 , 7782-44-7; N20, 10024-97-2. (11) Gear, C. W. Commun. ACM 1971,14,185, algorithm 407, DIFSUB. (12) Ravishankara, A. R.; Nicovick, R. L.; Thompson, R. L.; Tully, F. P. J . Phys. Chem. 1981,85, 2498. (13) Cohen, N.; Westberg, K. J . Phys. Chem. 1979, 83, 46.
