J . Phys. Chem. 1984, 88, 3643-3647
3643
predicted depletion in ozone (due to 9.4 ppb of Cl,) to change by 40%, from 6.8% to 4.1% in the Cicerone et al.' model M.8v2i This stronger than linear effect is unusual for stratospheric reactions since feedback effects generally decrease the sensitivity: the initial NAS report22on ozone depletion found only an 0.26 power dependence of A 0 3 / 0 3 on Akl/kl. The present sensitivity arises because the depletion of ozone in the upper stratosphere is largely balanced by production in the lower stratosphere. This production arises from increased penetration of solar UV and C10, suppression of NOJnduced ozone depletion through the C10 NO2 M recombination to form chlorine nitrate. The net change in the ozone column (AO,) is thus the difference between production and loss terms. Since reaction 1 is the rate-determining step in the C1,-ozone loss, changes in kl are directly reflected in the loss of ozone, currently with less than linear ~ensitivity.~~ The
ozone production in the lower stratosphere, however, is not directly affected by changes in kl since that reaction is relatively unimportant in that region. In the case in which the production largely balances the loss, the net column depletion is only a small fraction of the total loss. Thus, a small fractional change in the total loss, with no change in the production term, can result in a large relative change in the net depletion. If the results of Cicerone et al.' reported in Leu8 are interpolated, the value of k, recommended here would change the model M ozone depletion from 6.8% to -5%. This change probably also shifts the time at which the AO, actually becomes a depletion to later years (see Cicerone et al.'). The effect of a new value for kl on the calculated C10 profile is minor and within the uncertainties inherent in the measurement vs. computation comparison: the calculated [ClO] at 40 km will only increase by -5%.
(21) The characteristics of the various models are discussed in ref 1. Model M uses basically standard kinetic data (ref 6) except for a slightly different value of the OH HCI rate constant and also uses the 'fast" chlorine nitrate formation rate (see ref 6 and 9b). The quoted percent depletions in column ozone are for a total odd chlorine mixing ratio of 9.4 ppb. (22) National Academy of Sciences, "Halocarbons: Effects on Stratospheric Ozone", Washington, DC, 1977. (23) L. Froidevaux, Ph.D Thesis, California Institute of Technology, Pasadena, CA, 1983.
Acknowledgment. The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Helpful discussions with other members of the Chemical Kinetics and PhotochemistrY Group, especially M. J. Molina, were appreciated.
+
+
+
Registry No. 0, 17778-80-2;CIO, 14989-30-1.
Equilibrium between Hydroxyl Radicals and Thallium( I I ) and the Oxidation Potential of Waq) H. A. Schwarz* and R. W. Dodson Chemistry Department, Brookhaven National Laboratory, Upton, New York I I973 (Received: December 8, 1983)
The oxidation potentials Eo(OH/OH-) and Eo(OH,Hf/H20) are found to be 1.89 and 2.72 V, from which AGof(OH),, and the free energy of solution of OH are found to be 6.0 and -2.4 kcal/mol. These results are obtained from pulse radiolysis studies of two equilibria, T1+ OH * TIOH+ and TlOH+ + H+ + TIZf + H20, and the known E0(Tl2+/T1+)of 2.22 V. The first equilibrium constant is 3.4 X lo3 M-' and the second is 8.6 X lo4 M-' in 1 M (LiC104) ionic strength at 25 OC. Oxidation potentials for ClOH-, Cl,, C1, BrOH-, Br2-, and Br are given. A model of H-bond formation in water is developed which satisfactorily correlates AGosolnfor HzO, OH, H20,, CH,OH, and C2H50H.
+
Several estimates of the oxidation potential of the O H radical in aqueous solution have been published.I4 The values are based on gas-phase measurements of AGOXOH) and assumptions about the free energy of solution of O H and range from 1.83 to 2.0 V. An experimental measurement relating the O H oxidation potential to that of Tl2+ is reported here. It is known that OH oxidizes T1+ to T12+in acid solution at a diffusion-limited rate,5 and the oxidation potential of T12+has been established by several workers to be 2.20-2.22 V.5-7 Since OH is not a strong enough oxidant to oxidize T1+ to T12+without an assist from H+, the mechanism was assumed to be
T1+ + OH + TlOH'
(1)
(1) W. M. Latimer, "Oxidation Potentials", 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1952, p 48. (2) H. A. Schwarz, J . Chem. Educ., 58, 101 (1981). (3) A. J. Frank, M. Gratzel, and A. Henglein, Ber. Bunsenges. Phys. Chem., 80, 593 (1976). (4) J. H. Baxendale, M. D. Ward, and P. Wardman, Trans. Faraday Soc., 67, 2532 (1971). ( 5 ) H. A. Schwarz, D. Comstock, J. K. Yandell, and R. W. Dodson, J . Phys. Chem., 78, 488 (1974). (6) B. Falcinella, P. D. Felgate, and G. S. Laurence, J. Chem. Soc., Dalzon Trans., 1367 (1974). (7) R. W. Dodson, J . Radioanal. Chem., 30, 245 (1976).
0022-3654/84/2088-3643$01.50/0
+ H20
TlOH+ + H+ + T12+
(2)
In subsequent pulse radiolysis experiments TlOH' was observed to have an absorption maximum at 360 nm with an extinction The pK of the species was found coefficient of 3000 M-' to be 4.65 by two independent methods, the change in absorption with pH and the change in conductivity with pH (one H+ per TlOH' was consumed at low pH). A second pK of 7.7 for T1(OH), was also foundg but is not pertinent to this work. The pK of T12+and the range of oxidation potentials for T12+/+ indicate that reaction 1 must be a reversible reaction with the equilibrium ratio [OH]/[TlOH+] equal to unity at some T1+ M. Pulse radiolysis has been concentration between and used in this work to measure K1 and remeasure K2 and thus establish the oxidation potential of OH in aqueous solution. Experimental Section The pulse radiolysis was carried out with 2-MeV electrons from a Van de Graaff accelerator. Pulse lengths varied from 100 to
0 1984 American Chemical Society
3644
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
300 ns, which produced 10"-3 X 10" M radicals. The.analyzing light source was a Xe arc, pulsed for 2 ms for reaction lifetimes shorter than 3 p s , and the optical path length was 6 cm. Signals were sometimes averaged over 3-10 pulses, particularly at short s) and low absorbance changes (below lifetimes (less than 0.01). Stock solutions of sodium perchlorate, perchloric acid, and thallous perchlorate were as described earlier.5-7 Lithium perchlorate was prepared from Baker analyzed LizCOJ and perchloric acid and was recrystalized from water. The acid concentration of the solutions, after bubbling with Ar, was adjusted to about lo6 M with NaOH, LizC03,or HClO,. About 50 cm3 of reaction mixture was prepared at a time in a reservoir attached to the pulse radiolysis cell and was saturated with NzO or Ar by bubbling. All samples were thermostated at 25 OC. Hydrogen ion concentration was determined by using a glass electrode in the solution remaining in the reservoir after completion of the experiments, to avoid contamination of the solution with C1- from the electrode during the run. The salt bridge solution in the outer envelope of the combination electrode was changed from saturated KC1 to 1 M NaCl for the work at 1 M perchlorate in order to avoid precipitation of KClO,. The glass electrode was used in the pH mode, but the H+ response was directly calibrated on solutions in the reservoir by adding known amounts of H+. It was found that a weak base was present which consumed 3 X 10" M H+ in Ar-bubbled solutions and 8 X lo4 M H+ in N20-bubbled solutions. After correction for this, - log [H+] = C pH(obsd), where C contains activity coefficients and junction potentials and varied from 0 in 0.01 M NaClO, to 0.38 in 1 M LiClO,. Reproducibility of the pH measurements was 0.01 pH unit.
+
Results The pulse radiolysis of water produces hydrated electrons and hydroxyl radicals in about equal amounts and a smaller yield of H atoms. When NzO is present, reactions 3 and 4 occur. The
+
eaq- NzO 0-
+ HzO
-
-
+ 0OH + OHN,
(3)
(4)
rate constant k3 is 8.7 X lo9 M-' s-' (r ef 10) and k4 is lo8 s-' (ref 1l), so in NzO-saturated solution (0.023 M) the time scale for reactions 3 and 4 is 10-20 ns. In the absence of other e,; scavengers the yield of O H available for reaction with solutes is 6.0 per 100 eV and H atoms are the only other radical species present (OS per 100 eV). The yield of OH radicals which react with solute is a relatively insensitive function of the solute concentration. We determined the sensitivity to T1+ concentration by measuring TIZ+production M Nain solutions with 5 X lo-' M H+ present (and 5 X C104). The acid was present to drive the overall reaction, which is the sum of reactions 1 and 2, to the right:
T1+
+ O H + H+
TlZf
+ HzO
(5)
The absorbance of T12+at 290 nm at constant dose to the sample M, 0.0353 in 1 X M, and 0.0336 in was 0.0356 in 5 X 2X M T1+, which indicates that the reaction yield is constant &3% and that the reaction is quantitative, at least above 5 X lo-' M T1+. The pseudo-first-order rate constant observed for TIZ+ production was 1.0 X 101o[T1+]S-I, which agrees with earlier value~.~J~ Reactions of err; and H with TI+. It is known that e,; reacts rapidly with T1+ eaqwith k6 = 2.8 X 1O'O M-'
+ T1+
s-l
-
T1°
(6)
at low ionic strength.1° W e found
(10) M. Anbar, M. Bambench, and A. B. Ross,Natl. Stand. Ref.Data Ser. (US., Natl. Bur. Stand.), No. 43 (1973). ( 1 1) G . V. Buxton, Trans. Faraday SOC.,66, 1656 (1970). Dalton Trans., (12) B. Cerek, M. Ebert, and A. J. Swallow, J. G e m . SOC., 612 (1966).
Schwarz and Dodson the rate constant to be 2.0 X 1O'O M-l s-l in 1 M LiClO, by measuring the disappearance of e,; absorption at 600 nm. The highest T1+ concentration used elsewhere in this work was 2 X M at 1 M ionic strength at which concentration 16% of ea; are expected to react with T1+, or about 8% of the total radicals. Consequently it is necessary to know something of the fate of TlO. Cercek et a1.lZpresent evidence that T1° reacts rapidly with T1+ to form Tlz+ which absorbs light in the neighborhood of 400 nm. We have recently determined the equlibrium constant for T1,+ formation from T1° and T1+ to be 160 M-I, so 75% or more of the reduced thallium in our reaction mixtures was present as TlO. At 360 nm, where most of the,work on TlOH' was done, t = 3000 M-] cm-', which is coincidentally nearly the same as for TIOH'. In Ar-saturated solution, T1° disappeared in processes which were second order in dose, but it reacted much faster in N,Osaturated solution. The rate constant for reaction with NzO was 5 X lo6 M-I s-I. This rapid decay was not seen at 360 nm where the extinction coefficients for TIOH+ and T1° are nearly the same, indicating that the reaction with NzO produces OH which oxidizes
T1'. At the highest T1+ concentration, 2 X M, about 8% of the observed absorption may be due to T1° but this is produced at the expense of the same amount due to TIOH+, since the extinction coefficients are so similar. In the measurement of Kz this absorption should show up, as it does, as a small pH-independent component, since [T1+] is kept constant. The active region of T1+ concentration for measurement of K1is around 2 X 104-5 X lo4 M, at which concentrations only 1-2% of the radicals produce TlO. The conclusion is that T1° does not affect the determination of K , and Kz to more than 5%. In Ar-saturated M H+ solution most e,< react to form H atoms. These were observed to react with T1+ with a rate constant of 5 X lo7 M-' s-l to form TlO,which can be compared with an earlier value of 3.8 X lo7 M-' s-l obtained by a competition method.lZ Thus, in 2 X lom3M T1+ the pseudo-first-order rate constant for H atom reaction is 1 X lo5 s-I, which is 3 times slower than the slowest relaxation times measured for reactions 1 and 2, and consequently of little importance, particularly in view of the small yield of H atoms (8% of total radicals). Kinetic Treatment. In the study of reactions 1 and 2 it was convenient to use experimental conditions which greatly simplified the kinetics. The concentrations of H+ and T1+ were each high enough to be treated as constant. Reaction 1 was observed at sufficiently high pH that reaction 2 occurred to a negligible extent. Reaction 2 was followed beginning 50 ns after the pulse in solutions containing sufficient thallous that reaction 1 was complete during the interval. In each case the kinetics can be treated by
+
+
A = Ale-kobsdt Afinal at
(7)
+
where A is the absorbance, A I Afinalis the "initial" absorbance, and a t is a small catchall term to partially correct for slower reactions such as the second-order reactions between products, H atom reactions, and the like. The rate constant kom was always at least 30 times larger and a/Afinal.The slope a was evaluated on a time scale 2-5 times longer than was used to find kobsd.This method of data analysis gives values of kobsdwhich include k[RIo terms, where k is the OH + O H or O H + TIOH' or TlOH' + TIOH+ reaction rate constant and [R], is the initial radical concentration. These terms amount to at most 3% of the total and were ignored. The calculated final absorbance would be independent of radical disappearance reactions if they occurred randomly among the species. The worst case would be if one recombination reaction was totally dominant, in which case the error introduced by this method into Afinalwould be at most 1.5% at the midpoint of the equilibria. OH-TIOH+ Equilibrium. At pH 5.7, well above the pK of Tl2+, the absorbance at 360 nm of a pulsed solution containing 5 X 10-5-2 X M T1+ rises rapidly in a first-order process. Since only reaction 1 is occurring to an appreciable extent (8) kobsd = kl[Tlf] + k-l The form of eq 8 was observed experimentally and the rate
Equilibrium between Hydroxyl Radicals and Thallium(I1) TABLE I: Equilibrium and Rate Constants ionic strength, M 10-3K1,M-I 0.01 (NaC104) 1.OO (NaC10,) 1.00 (LiC104) \I
I
10-IOkl,M-1
5.8
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3645
10-6k-,, s-1
S-1
1.2
1.o
3.4 I
I
10-1°k2, M-'
M-'
1.8
6.0
2.8
8.2 8.6
10-5k_2,s-I
s-I
1.o 1.4 1.3
2.0
1.4 1.7
I 0
0
o
v
o
0
'TIOH+
=
0
32-
c0
-aa X
3
E2 2 -
0
X
Y
Y,
-
0 0 D
I-
YI
l -
"
3
4
I'
5
6
-Log I0 [H
0 -?\ -6)
'
-5
I
-4
I
I
-3
-2
Figure 2. Dependenceof apparent extinction coefficient of TIOH+ at 360 nm on hydrogen ion concentration in 1 M (LiCI04)ionic strength; [TIt] is 2 X lo-) M: ( 0 ) c(app) at end of reaction, (0) €(init). Curve is calculated from eq 12.
and k2 and k-z are given in Table I. e'(app) is defined as Afi,,~/(l[OH],), similar to eq 9, and €'(app) = Kz[H+]~TIOH+/(l + l/K1[T1+]
+ Kz[H+]) + e(TlO)
(12) where e(TlO)represents residual absorption at high acid concentration and is assigned to TlO,though it may contain a contribution from T12+. For 0.002 M T1+ e'(app) = Kz[H+]qIoH+/(1.15+ K2[H+]) + e(TlO)
The variation of c(app) with [T1+] in 1 M (LiClO,) ionic strength is shown in Figure 1 with the curve calculated for K, = 3.4 X lo3 M-I. This value and a similarly measured one for M ionic strength are given in Table I. The ratio of kl/k-l in Table I should also be equal to K1 and is 3.6 X lo3 M-' for the 1 M solution M). The calculated values of eTIOH+ (similar agreement for were 3300 and 3260 M-I cm-' at the two ionic strengths. eOH is a small term which may represent OH or any other absorptions at low [Tl']. A value of 75 M-I cm-I was found in pure H20. TlOH+-TIz+ Equilibrium. Reaction 2 was studied at 1 M ionic strength, 0.002 M Tl+, and varying acid concentrations. The pseudo-first-order rate constant for equilibration of reaction 1 at this concentration is 2.3 X IO7 s-l and the pulse length used for the radiolysis was 1.5 X s, so reaction 1 was 70% equilibrated by the end of the pulse and 90% equilibrated 50 ns later. The steady-state assumption may be made for O H for the duration of kinetic measurements of reaction 2. Consequently
In 1 M ionic strength, K1 = 3.4 X lo3 M-l, so at 2 X low3M T1+, Kl[T1+]/(l + Kl[Tl+]) is 0.87. The H+ dependence was observed
The variation of c'(app) with [H'] is shown in Figure 2 for 1 M (LiC104) ionic strength. Also shown is an €(init) calculated from the "initial" absorbance, A' + Afinal,which is that of TlOH+ equilibrated with OH but not H+. The initial absorbance is corrected for the decay of TIOH' via reaction 2 during the pulse (a 6% correction at the highest acid concentration shown). €(init) is independent of acid concentration, as it should be, and is related to tT1OH+ by eq 10. Values of tT1oH+ so calculated were 3160 f 30 at the three ionic strengths. The curve in Figure 2 is calculated for Kz = 8.6 X lo4 M-I; the corresponding value of k?/kyz is 9.2 X lo4 M-l. The results for other media are given in Table I. M T1+ (1 M ionic Values of e(TlO)were 440 M-' cm-' at 2 X M T1+ (0.01 M ionic strength) and 280 M-' cm-I at 1 X strength). The expected values, based on e,; rate constants and c(Tl0) = 3000, are 240 and 150, which indicates that at least half of the residual absorption is indeed due to TlO. The most accurate value for Kz in the literature is from BonM T1+ (about ifacic and as mu^,^ who found a pK of 4.65 in M ionic strength). This K is actually K2/(1 (K1[Tl+])-'), so their corrected K2 is 5.2 X lo4 M-I (pK of 4.72), which compares very well with the value at loVzM ionic strength given in Table I (6.0 X lo4 M-l), particularly in view of the expected 7% effect of ionic strength between the two media.
+
Discussion The equilibrium constant for the overall reaction 5 is K5= K,Kz, which is 2.9 X lo8 M-2 in 1 M (LiC104) ionic strength. This value corresponds to a AGO of -11.6 kcal/mol or -0.50 eV. Since E0(Tl2+/T1+) is 2.22 V5 in 1 M (HClO,) ionic strength, E O (OH,H+/H20) is 2.72 V. The activity coefficient of 1 M HC104 is 0.82 and that of LiC104 is 0.89, so that of dilute HClO, in
3646
The'Journal of Physical Chemistry, Vol. 88, No. 16, 1984
TABLE II: Oxidation Potentials of Halogen Radicals reaction Kcs O H + C1- F= CIOH0.7 M-' CIOHCI + OH3 x 10-9 M 1.9 x 105 M-' c1 c1- == Cl2320 M-' OH + Br- ==BrOHBrOH- Br- * Br2- + OH70 Br BrBr21.1 x 105 M-'
+
+
+
LiC104 is likely to be about 0.85, which would introduce a correction of 0.004 V in going to infinite dilution, less than the round-off error. With the ion product of water, lo-', M2, the value of Eo(OH/OH-) is 1.89 V. The free energy of formation of aqueous OH- in its standard state is -37.595 kcal/mol, so AGof(OH) is 6.0 kcal/mol in aqueous solution. The value in the gas phase (1-atm standard state) is 8.37 kcal/mol,14 so the partial molar free energy of solution of O H is -2.4 kcal/mol. The pK of O H is 11.9,15so AGf(O-) is 22 kcal/mol and Eo(0-,H20/ 20H-) is 1.76 V. The value of E0(Tl2+/Tl+)to be discussed here for error analysis was based on literature values of Eo(Fe3+/FeZ+)and E0(T13+/ TI+)' in 1 M HC104 and the equilibrium constant for
+
+
T12+ Fe3+ T13+ Fe2+ (13) as determined from measurements of the forward and reverse rate constants in the same m e d i ~ m .The ~ error in the Fe3+ and T13+ Eo measurements is at most 1 or 2 mV and is neglected. The value of K13was believed to be accurate to f25%, which corresponds to f 6 mV. In this work the scatter in measurements is small, about f 5 % in K 1and K,, but systematic errors may be larger. One source of systematic error in K , would be in estimating tTIOHt by first extrapolating rate curves to long times (about 2-3 ps) during which time minor reactions could interfere, and then to "infinite" [Tl+]. The values of CTlOHt so obtained are only 4% greater than those obtained by extrapolating data to "zero" time to obtain €(init) when following reaction 2, which suggests that such systematic errors are probably less than 5%. Another test for systematic errors lies in the internal consistency of the data. The ratio of the various K I and K2 in Table I to the corresponding values of k l / k - l and k2/k-, can be averaged to give 0.99 f 0.16. The kinetic and equilibrium measurements of Kl and Kz are largely independent, so it would seem that 16% error in each K is an outside limit to their reliability (in fact the 16% error is near the reliability of the k-l and k-2 measurements). It is believed that the errors in K , and K2 are equal to or less than 15% each, or f3.5 mV. Another error, difficult to assess, is the difference in activity coefficients, principally that of T12+,between 1 M HC104 and 1 M LiClO,. Specific effects on TTIzt,etc., should be governed largely by the counterion, C104-, which is the same in the two media. The agreement of K2 in 1 M NaC104 with that in 1 M LiClO, ( 5 % ) is evidence that specific ion effects of Na+, Li+, and probably H+ on the activity coefficients are not large, and a possible error of f 2 5 % is assigned (6 mV). If the errors add randomly, they would lead to an expected error of f 1 0 mV. It is conservative then to give the errors on the Eo values as f0.02 V, or f0.5 kcal in free energies. Oxidation Potentials of Some Halogen Radicals. Equilibria associated with the O H + C1- (ref 16) and O H + Br- (ref 17) reactions have been studied and data from the literature are presented in Table I1 with oxidation potentials calculated from (13) Deleted in Proof. (14) "JANAF Thermochemical Tables", 2nd ed., Dow Chemical Co., Midland, MI, 1971, Natl. Stand. ReJ Data Ser. ( U S . , Natl. Bur. Stand.) No. 37. (15) J. Rabani and M. S. Matheson, J . Am. Chem. Soc., 86, 3175 (1964). (16) G . G. Jayson, B. J. Parsons, and A. J. Swallow, J. Chem. SOC., Faraday Trans. 1 , 69, 1597 (1973). (17) A. Mamou, J. Rabani, and D. Behar, J . Phys. Chem., 81, 1447 (1977). (1 8) "Selected Values of Chemical Thermodynamic Propertes", US.Department of Commerce, National Bureau of Standards, Washington, DC, 1968, NBS Tech. Note (US.) No. 270-3.
Schwarz and Dodson
ref
oxidn potential
16 16 16 17 17 17
Eo(CIOH/CI-,OH-) = 1.90 V Eo(CI/CI') = 2.41 V E0(C12-/2C1-) = 2.09 V EO(BrOH-/Br-,OH-) = 1.74 V Eo(Br2-/2Br-) = 1.63 V Eo(Br/Br-) = 1.93 V
them and based on Eo(OH/OH-) = 1.89 V. The errors are expected to be f0.03 to f0.04 V from error limits quoted in the papers. The Eo(C1/CI-) and Eo(Br/Br-) can be combined with AGOf values in standard reference^'^,'^ to give the free energies of solution of $0.9 f 1 and -0.1 f 0.8 kcal/mol for the chlorine atom and bromine atom. Hydrogen Bonding of OH in Aqueous Solution. The free energy of solution of O H is defined by
OHW
OWaq)
AGosoln(OH) = -RT In I[OH(aq)I/[OH(g)IJ
(14)
In order to compare this free energy with the corresponding quantity for water HzO(g)
F=
H,O(aq)
AGosoln(HP) = -RT In ([H@(aq)l /[H,O(g)IJ
(15)
it is helpful to introduce the concept of the non-hydrogen-bonded species, OH(0) and HzO(0). It is assumed that these species exist as minor constituents of OH and HzO in solution. The free energy of solution of OH(0) is defined by OH(g) + OH(0) AGoso~n[OH(0)I= -RT ln ([OH(O)I/[OH(g)lJ
(16)
If all dipolar interactions are included as hydrogen bonding, then a reasonable model for OH(0) would be a nonpolar molecule of similar size. The free energies of solution of Ne, Ar, CH4, C2H6, and C3H818are all close to +4.0 kcal/mol with N e the largest (+4.6) and C2H6 the smallest (+3.7). AGosOIn[OH(O)]and AGo,,1n[H20(0)]will each be taken as +4.0 kcal/mol. A free energy of hydrogen bonding may then be defined as AGOHB(OW = AGOsoIn(0H) - AG'so~n[oH(O)l
(17)
Thus, AGOHB(0H) = -6.4 kcal/mol, and similarly AGoHB(H20) = -8.4 kcal/mol.19 From eq 14, 16, and 17 AGoHB(OH) = -RT In ([OH(aq)]/[OH(O)]J
(18)
and similarly for HzO AG0HB(H20) = -RT In ([H20(aq)l/[H20(o)ll
(19)
Several assumptions will be made in order to compare these free energies of hydrogen bonding. (1) Each water molecule has four H-bonding sites, one on each proton and two on the oxygen. (2) Each OH radical has three H-bonding sites, one on the proton and two on the oxygen. (3) All H bonds in water and between water and OH are equivalent. The first assumption is true in ice, but in water a proton might bind simultaneously to several oxygens.20 If so, it is reasonable to propose that multiple bonding will average out to about the same energy as a single H bond. The third assumption is also more reasonable when average bonds are considered. A hydrogen bond between two water molecules should be stronger than one between a water and an O H when the water (19) The difference in AGOf of H 2 0between the gas and the liquid is -2.06 kcal/mol (ref 15), which gives AGo,,,n(H20)= -4.4 kcal/mol by adding -RT In 55 to convert to molar units for the liquid. (20) See C. M. Davis and J. Jarzynski in "Water and Aqueous Solutions: Structure, Thermodynamics, and Transport Processes", R. A. Horne, Ed., Wiley-Interscience, New York, 1972, p 377. (21) A. L. Rotinyan and A. I. Anurova, Elektrokhimiya, 5, 1352 (1969).
Equilibrium between Hydroxyl Radicals and Thallium(I1) acts as proton donor because the 0 in water is more negative, being polarized by two protons, than is the 0 in OH. But when O H acts as proton donor, the resulting H bond will be stronger than one between two waters because the proton on OH is more positive than is either one on water. These assumptions suggest that O H forms three H bonds, H,O forms four H bonds, and since they are all equivalent the free energies of H bonding might be expected to be in the ratio of 3:4, which is close to the observed ratio. It is not obvious, however, why the free energies should be proportional to the number of bonds without further development of the argument. The three assumptions made before suggest the following equilibria, where the parameter in parentheses refers to the number of H bonds in which the species is participating:
+ H2O(4) * HzO(1) + HzO(3) HzO(1) + H2O(4) + HZO(2) + HzO(3) HzO(2) + HzO(4) HzO(3) + H,0(3) OH(0) + HzO(4) =+ OH(1) + H20(3) OH(1) + HzO(4) + OH(2) + H20(3) OH(2) + HzO(4) + OH(3) + HzO(3) HzO(0)
f
(20) (21) (22) (23) (24) (25)
Since all H bonds are assumed equivalent, the equilibrium constants for reactions 20-25 will all be 1. (Actually there should be statistical factors of 2-4 for reactions 20,21,23, and 24 because there are multiple ways to form the incompletely bonded species, but they are not important here.) Thus, reactions 20-22 each give an equation of the type [H2O(i)l [H20(3)1 /I[HzO(i-1)1 [H20(4)1) = 1
(26)
If each concentration is expressed as the ratio of [H,O(i)] to total water w(i)
= [HzO(i)l/ [HZ01
(27)
then w(i) = [w(4)/~(3)]w(i-l)
(28)
or w ( i ) = [w(4)/w(3)1"0
(29)
The sum of all w(i) is 1, so wo(1
+ w ( 4 ) / ~ ( 3 )+ [w(4)/w(3)lZ + [w(4)/w(3)I3 + [w(4)/w(3)I4) = 1 (30)
The value of wo is 7 X lo-' (from eq 19 and AGOHB(H~O) = -8.4 kcal/mol). The solution of eq 30 is w(4)/w(3) = 34.32, so the fourth-power term is strongly dominant, and w(4)/w(3)
= [l/w(0)]'/4
(31)
The Journal ofPhysica1 Chemistry, Vol. 88, No. 16, 1984 3647 TABLE III: Free Energies (kcal/mol) of Solution and of Hydrogen Bonding compd AG',l: AGO HB AGO H B I ~ H B ~ HzO OH CHSOH C2H50H H202
-8.4 -6.4 -7.2 -7.2 -10.8, -12.4
-4.4c -2.4d -3.2 -3.2 -6.8, -8.4'
-2.1 -2.1 -2.4 -2.4 -1.8, -2.1
"From ref 18 unless otherwise noted. bSee text. cSee footnote 19. dThis work. 'Reference 21.
Reactions 23-25, with h(i) defined as [OH(i)]/[OH(aq)], lead to
h(i) = [w(4)/w(3)lih(0)
(32)
and, as the sum of all h(i) is 1 l / W ) = [w(4)/w(3)I3 + [w(4)/w(3)I2 + w(4)/w(3)
+1 (33)
The cubic term is strongly dominant, so with eq 31 i/h(o)
= [i/w(0)13/4
(34)
and with eq 18 and 19 AGOHB(0H)
3/AGo~~(H20)
(35)
which can be compared with the observed ratio of 0.76. The above development should apply equally well (per OH group) to other ROH compounds as long as R is not strongly electronegative or electropositive. The data for HzO, OH, C H 3 0 H , C 2 H 5 0 H ,and H z 0 2are given in Table 111. AGoSoln[ROH(O)] is taken as 4 kcal/mol in each case. The comparison is made in the last column of the table, in which AGOHB is divided by the assumed number of hydrogen bonds in which the molecule participates, four for H 2 0 , three for OH and the alcohols, and six for HzO2. AGoHB/nHB is reasonably constant at -2.2 & 0.2 kcal/mol (using -2.1 for H2O2). Since two molecules participate in each H bond, the free energy of formation of one H bond is twice this value, or -4.4 kcal/mol. It is seen that the free energy of solution of OH fits in well with those of water, the alcohols, and H 2 0 2 . Acknowledgment. We thank F. Silkworth for preparation of reagents, E. Norton for standardization of solutions, and M. Newton for helpful discussion. This research was carried out at Brookhaven National Laboratory under the auspices of the US. Department of Energy under contract DE-AC02-CH00016 and supported by its Office of Basic Energy Sciences. Registry No. TlOH', 61003-61-0; NaClO,, 7601-89-0; LiClO,, 7791-03-9; TI, 7440-28-0; OH, 3352-57-6; H', 12408-02-5; ClOH-, 34524-94-2; C1,22537-15-1; Cl,, 12595-89-0; BrOH-, 36505-09-6; Br,, 12595-70-9; Br, 10097-32-2.
