J . Phys. Chem. 1989, 93, 3360-3363
3360 I
7..
I
I
I
I
. I
d n / 1
Figure 5. Comparison of the experimental vapor pressure ratio ( r ) of isotopically pure methanes (CDd/CH,) with that calculated from theory: (-) experimental from this work (eq 2); (--) simple-cell model;" (-.-) medium-cluster nod el.'^^'^
into log ( p / p ? with use of eq 13. The necessary thermodynamic quantities needed to perform this calculation were obtained from the same literature sources used for the enthalpies of vaporization. The results are shown in Figure 5 (dashed line).
Similarly, Pollin and IshidaI2*I3modified slightly the SCM, considering that the cell can now hold not only one simple molecule but a cluster of m molecules freely translating and rotating, with a central molecule and m - 1 shell molecules. With this medium-cluster model (MCM), Pollin and Ishida deduced similar equations, suggesting that the entire liquid phase can be represented by the central molecule of the cell. They applied them to the methane isotopes using a cluster of size m = 9 in a gear geometry. This approach leads to the dot-dash line shown in Figure 5 . Considering the approximations involved, and the fact that anharmonicity was ignored, it is somewhat surprising that both models predict the experimental results quite well. They give similar overall agreement with experiment, with SCM giving results that are too high, whereas those obtained with the MCM are too low; MCM performing slightly better perhaps because of the isotropic nature of the SCM. Acknowledgment. This work has been supported by a grant from the Volkswagen Foundation, which we gratefully acknowledge. Registry No. CHI, 74-82-8; CD4, 558-20-3.
Reduction Potential of the 'COP- Radical Anion in Aqueous Solutions Parminder S. Surdhar, Stephen P. Mezyk, and David A. Armstrong* Department of Chemistry, University of Calgary, Calgary, Alberta, Canada T2N 1 N4 (Received: January 26, 1988; In Final Form: August 29, 1988)
The reduction potential for the 'COT radical anion has been determined by equilibration of formate with sulfhydryl radicals of @-mercaptoethanol,penicillamine, and lipamide in aqueous solutions at pH 3-6. The reaction 'CO, + e- + H+ = HCOT yields the value Eo9 = 1.49 V with an uncertainty of i0.06 V. On the basis of this value and the known free energies of C02(aq) and HC0,-(aq), E O 1 9 for C 0 2 + e- = T O 2 - was found to be -1.85 V.
Introduction Hydrogen-transfer reactions have frequently been utilized in the gas phase for obtaining bond dissociation energy data.'S2 Unfortunately the halogen systems, such as HI, which are so useful in the gas phase, cannot be employed in solution because of the very small pK values of these hydrides. However, sulfhydryl systems do not suffer from this restriction, the pK value of a typical aliphatic sulfhydryl being about 8-10.394 Recently, we determined the redox potentials of several sulfhydryls in aqueous s o l ~ t i o n . ~ In the current investigation, we have utilized the equilibrium depicted in reaction 1 to obtain values of the redox potential for PSH 'C02- = PS' + HC02(1)
+
the 'COY radical anion. Three different sulfhydryl molecules were used: P-mercaptoethanol, penicillamine, and lipoamide. The latter species has the advantage that the PS' radical form has a high absorbance coefficient, similar to that of the protonated lipoic acid radical studied earlier by Hoffman and Hayon6 and Farragi, Redpath, and Tal.' This is also true of the penicillamine radical,*v9 (1) Benson, S. W. Thermochemical Kinetics 2nd 1976.
ed.; Wiley: New York,
(2) Alfassi, 2.B.; Golden, D. M. J . Phys. Chem. 1972, 76, 3314. (3) Szajewski, R. P.; Whitesides, G. M. J . Am. Chem. Soc. 1980, 102,
2011.
(4) Whitesides, G. M.;Houk, J.; Patterson, M. A. K. J . Org. Cbem. 1983, 48, 112. ( 5 ) Surdhar, P. S.; Armstrong, D. A. J . Phys. Cbem. 1987, 91, 6532. (6) Hoffman, M. Z.; Hayon, E. J . Am. Chem. Soc. 1972, 94, 7950.
0022-3654/89/2093-3360$01.50/0
and for these two radicals, their concentration was followed by observing the absorbance of the pulse-irradiated solutions. For the P-mercaptoethanol system, the equilibrium was observed from the photostationary ratio of flavin and dihydroflavin in a y-radiolysis experiment, which will be described under the Experimental Section and Results. Since the equilibrium of reaction 1 lies to the right, the formate ion was present in excess (0.01-0.5 M) and the sulfhydryl molecule M). Radicals at a considerably lower concentration ( 104-2 X were produced by the radiolysis of the aqueous solutions, with N20 added to convert the solvated electrons to hydroxyl radicals.I0 The following reactions then take place: 'OH (or 'H)
+ HCOT = H 2 0 (or H2) + T O 2 -
'OH (or 'H)
+ PSH = H,O
(or H2)
+ PS'
(2) (3)
Experimental Section Penicillamine (PenSH) and @-mercaptoethanol(P-RSH) were purchased from Sigma Chemical Co. Dihydrolipoamide [L(SH),] was prepared from lipoamide (LS,) by the method of Reed et al." (7) Faraggi, M.; Redpath, J. L.; Tal, Y. Radial. Res. 1975, 64,
452.
(8) Purdie, J. W.; Gillis, H. A.; Klassen, N. V. Can. J . Cbem. 1973, 51, 3132. (9) Hoffman, M . Z . ; Hayon, E. J . Pbys. Cbem. 1973, 77, 990. (IO) Draganit, I. G.; Draganit, Z . D. The Radiation Chemistry of Water; Academic Press: New York, 197 1.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3361
Reduction Potential of T O 2 -
,
1
I
b
TABLE I: Summary of AGO Data and Equilibrium Constants reaction DH KI -AG,/kJ mol-' 'CO2- + L(SH)2 4 61 f 15 10.1 f 0.5 53 f 20' 9.8 f 1.0 TO2- + PenSH 6 256 f 30 13.6 0.3
'COT
+ P-RSH
3
*
2000 f 250
18.8 f 0.5
'From kinetic data. Others from equilibrium data.
x
I
1
I
4
8
12
L(SH~ x 1041~
Figure 1. (a) Oscilloscope trace illustrating the growth of the LStH absorbance at 390 nm for a solution containing 0.01 M formate and 5.0 X IO4 M L(SH)2at pH 4. (b) Plot of pseudo-first-order rate constants derived from absorbance changes at 390 nm, vs L(SH)* concentration at pH 4.
All other chemicals were the purest grade available and were used as supplied. The water for preparing the solutions was purified from a Millipore Milli-Q system purged of air. All experiments were conducted at room temperature (23 f 2 "C) with solutions containing phosphate buffer that was 10 mM and saturated with N20. The pH was adjusted with HC104 or NaOH and measured with an Orion 81 1 pH meter calibrated against Orion reference buffers. The pulse radiolysis experiments were performed by utilizing 10-ns pulses of 8-MeV electrons with doses of -4 Gy/pulse from the linear accelerator at the Notre Dame Radiation Laboratory. Details of the pulse radiolytic methods and dosimetry have been previously described.I2 The equilibrium ratio [PS']:['CO,-] was also determined at pH 3 with PSH = @-mercaptoethanol with lumiflavin (FI) as an indicator.l3*I4 The experiments were carried out by irradiating the solutions with an AECL Co T-cell, giving a steady dose rate of 6.0 Gy min-I. The dose rate was determined accurately by ferrous sulfate dosimetry, assuming G(Fe3+) = 15.6 molecules/ 100 eV,l0 and checked periodically. The changes in absorption (AA) due to conversion of lumiflavin to dihydrolumiflavin (F1H2) were measured with a Cary-219 spectrophotometer and were followed until a photostationary ratio [ Fl]: [F1H2] had been obtained.
Results Section I : Pulse Radiolysis Experiments. Lipoamide. The sulfhydryl PS' radical form of lipoamide is known to cyclize to the species shown in reaction 4, which has a much stronger absorbance c o e f f i ~ i e n t ~ ~than ~ ~ 'uncomplexed ~*'~ sulfhydryl radicals.
"3
e
R
ISH
CkSH R
(4)
/
R = (CHa),CONH2
Reaction 4 takes place on a relatively fast time scale and may be assumed to have a half-life similar to that of the radical of the related disulfhydryl molecule dithiothreitol, which cyclizes with a half-life of about 1 ps.I7 This process is therefore fast relative
to reaction 1. On the basis of available rate constant data, the sulfhydryl molecule concentrations were chosen so that equilibrium 1 would have been reached within a period of 100 ps. The concentration of formate was 0.01 M, and that of lipoamide was varied M. The pH was 4.0. from (1-15) X Formate radicals do not absorb strong1.y at the wavelength of maximum absorbance of the lipoamide LS2H radical. Thus, the opltical absorbance increases as the reaction proceeds and produces LS2H. This is illustrated by the data in Figure l a , which shows the change in absorbance, at 390 nm, with time after pulsing a solution.containing 5 X lo4 M lipoamide. It should be noted that some LS2H radicals will be formed by direct reaction of the 'OH with the sulfhydryl molecule (reaction 3). The absorbance of those species, together with the weaker absorbance of the formate radicals, was responsible for the observed sharp initial rise, which became more pronounced as the sulfhydryl molecule concentration increased. Because of the fact that the sulfhydryl concentration is much larger than that of the radicals, reaction 1 is a pseudo-first-order process. Values of the pseudo-first-order rate constant, kf = k , [PSH], for the reaction of formate radicals have been plotted against the concentration of lipoamide in Figure 1b. The slope gives the value of k , , in the forward direction, while the intercept divided by the formate concentration is a measure of the second-order rate constant in the back-reaction, k-,. The values of these two parameters were k l = (6.4 f 0.4) X lo8 M-l s-l a nd k-, = (1.2 f 0.3) X lo7 M-' s-l. The ratio of the forward rate constant to the back rate constant, obtained from the results in Figure 1b, leads to an equilibrium constant of 53 f 20. Under the present conditions, where the yield of T O 2 - radical is independent of sulfhydryl molecule concentration, it can be shown that at equilibrium the absorbance should be related to the ratio of the formate and lipoamide ncentrations by
-- 1 e - c.co2-
-
1 CPS. - e.co*-
+-[HCO,
[PSI31
1 KIkPS.
(5)
where t is the absorbance per mole for the mixture of lipoamide and formate radicals occurring at a given f ~ r m a t e : L ( S H ratio )~ and and cps. are those of the formate and LS2H radicals, respectively. The intercept divided by the slope of a plot in accord with eq 5 yielded a value of 61 f 15 for K I , in agreement with the result derived from Figure lb. Also, the reciprocal of the intercept was in agreement with the value of em. The values of the equilibrium constants and the AGl values calculated from them are summarized in Table I. The value of the standard free energy for the reaction LS2- + 2H+ + e- = L(SH)2
(6) relative to the standard hydrogen half-reaction (Le., -FEo6) is AGO6 = -168.9 kJ/mol, from the data in ref 5. Addition of AGO7 (=31.2 kJ/mol, from the known value of the pK 5.516) for LS2H = L S F
+ H+
(7)
yields AGO8 = -137.7 kJ/mol. LS2H + H+ + e- = L(SH)2
(1 I ) Reed, L. J.; Koike, M.; Levitch, M. E.; Leach, F. R. J . Biol. Chem. 1958, 232, 143. (12) Alfassi, Z. B.; Schuler, R. H. J . Phys. Chem. 1985, 89, 3359. ( I 3) Ahmed, R.; Armstrong, D. A. Can. J . Chem. 1984, 62, 17 1. (14) Surdhar, P. S.; Armstrong, D. A. I n f . J . Radiaf. Biol. Relaf.S f u d . Phys., Chem. Med. 1987, 52, 419. (15) Simic, M.; Neta, P.; Hayon, E. J . Phys. Chem. 1969, 73, 3794. (16) Wu, 2.;Ahmad, R.;Armstrong, D. A. Radiat. Phys. Chem. 1984, 23, 251.
- '.co2-l
(8)
Utilizing this result one can calculate the magnitude of AGO9 for the formate radical from the free energy change for reaction 1 . T O 2 - + e-
+ H+ = HC02-
(9)
(17) Akhlaq, M. S.; von Sonntag, C . Z . Nafurforsch., C: Biosci. 1987, 42C. 134.
3362 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Surdhar et al.
TABLE 11: AGO (kJ/mol) and E o (V) Values PSH PenSH
L(SH)2 146.7
AGO,
0-RSH
141.0 143.6 f 3.0 1.49 f 0.03
mean AGO9 EO9
y: 0 2
143.0
E
-
500
0
[HCO;]
1000
/ [PSH]
.
2
I
4
7
2
0.5 I
I
500
Figure 3. (a and b) Plots of [FIH,]/[FI] vs [HCO,]/[PSH] and l/fnH2 vs [PSH]/[HCOc], respectively, at pH 3 for PSH = 8-mercaptoethanol. Key: 0, solutions containing 10 pM FI and 0.20 M HCO,; 0 , solutions containing 20 pM FI and 0.20 M HCO,; A,solutions containing 10 pM FI and 0.10 M HCO;; 0,solutions containing 20 pM FI and 0.25-0.3
I
1000 1500 [HCO;] / [PenSH]
Figure 2. Plot of I / c vs ratio of [HCOF] to [PenSH] at pH 6
One should note that in the range pH 0-5 it is necessary to take into account protonation equilibria for both the formate radical (pK, = 1.418)and the formic acid molecule (pKf = 3.8419). The equation for the free energy change of the overall reduction of formate radicals to formate and formic acid is then given in terms of AGO9 by eq IO, in which [F] and [F,] refer to the total con-
M HCOZ-.
3. The formate concentration was 0.1-0.3 M, and the @-mercaptoethanol concentration was varied in the range (4-20) X M. This causes the ratio of PS' to 'COT, at equilibrium, to shift. While the former radicals are known to oxidize dihydroflavin (F1H2) at this pH, 'COT radicals reduce flavin (F1).13~14~20 Under these circumstances, it can be shown that in a mixture of formate and PSH, containing micromolar concentrations of flavin, a steady-state ratio [FI],,:[F1H2], is reached,I3J4and this is given by
-[ F l H 2 1 ~- k13['C02-1 [F11S5 centrations (i.e., both ionized and un-ionized) of formate and formate radical, respectively. Since the present analytical method does not determine TO2- and H C 0 2 - explicitly, it is necessary to use eq 10 to correct AG9 to AGO9. This was done with the data from L(SH)2 and @-RSH. Utilizing the value of AGO8 above, the appropriate pK values, and the magnitude AGI = 10.0 f 1 .O kJ mol-' for L(SH)2 from Table I, one obtains -146.7 kJ/mol for AGO9, the free energy of reaction 9 relative to the hydrogen standard. This result has been given in Table I1 along with the value of AGO9 for the other experiments. Penicillamine. Experiments similar to those with lipoamide were performed with penicillamine at pH 6. The formate concentration was 0.1 M while the penicillamine was varied from (5-40) X M. The weakness of the absorbance signal of the penicillamine radicals meant that a large number of experiments had to be averaged. A plot of the absorbance data at equilibrium in accord with eq 5 has been shown in Figure 2. Here, the intercept was determined by the reciprocal of the difference in the values of 6 for the PS' radical (1220 M-l cm-' 8,9) and the T O 2 - radical (200 M-I cm-l I s ) . Although some rate data were obtained, it was felt that the absorbance plot would provide a more reliable value of k l , and the kinetic data were not used. From the results in ref 5, the value of AGOII PS'
+ H+ + e- = PSH
(11)
relative to the standard hydrogen half-reaction is -1 27.3 kJ/mol for penicillamine. Combination of this with AGI for penicillamine in Table I leads to AGO9 = -141.0 kJ/mol. Section I I y-Radiolysis Experiments with p-Mercaptoethanol. With the 6oCocontinuous radiolysis source, formate radicals were equilibrated with the PS' radicals of 8-mercaptoethanol at pH (18) Buxton, G. V.; Sellers, R. M. J. Chem. Soc., Faraday Tram. I 1973, 69, 5 5 5 . (19) Selected Values of Chemical ThermodynamicProperties. NBS Tech. Note (U.S.)1965, No. 270-1. The AEO values for reaction 20 are actually
2E'.
1.0 1.5 18 x (PSH] I (HCO;]
kl6[PS']
+ k14['C02-1 ['FIH], - kis['FlH]~ k16[PS'l [F1lss I6 [ F1l ss (12)
where the reactions referred to are F1 + 'C02- (+H+) = 'FlH 'FIH
+ CO2
+ T O 2 - (+H+) = F1H2 + C 0 2 'FlH + PS' = F1 + PSH F1H2 + PS' = 'FIH + PSH
(13) (14) (15) (16)
The terms containing the 'F1H radical concentration are small, provided the F1H2 and F1 equilibrium concentrations are kept sma11.13J4 Writing ['C02-]/[PS'] = [HC02-]/Kl [PSH] and neglecting the terms with ['FlH], eq 12 can be written as
or
wherefW2is the fraction of Fl in reduced form when the stationary state is reached. The first form is convenient for treating data for lower [FlH,], and the second for higher. Plots of data obtained in 0.1-0.3 M formate solutions with varying amounts of flavin in accord with these equations have been presented in parts a and b of Figure 3, respectively. The average value of Kl from the slopes of the lines and the known values k I 3(=3.6 X lo9 M-' S-')~O and k16 (=4.0 X lo9 M-l s-')~~give a value of K I = 2000 f 250. This corresponds to a AGO1 value of -18.8 kJ/mol. The magnitude of AGOll for P-mercaptoethanol relative to the standard hydrogen half-reaction was shown in ref 5 to be -128.8 kJ/mol. From this, the above value of AGl at pH 3, and eq 10, (20) Ahmed, R.; Wu,Z.; Armstrong, D. A. Biochemistry 1983, 22, 1806. (21) Surdhar, P. S.; Armstrong, D. A,, unpublished work.
J . Phys. Chem. 1989, 93, 3363-3368 the magnitude of AGO9 was calculated to be 143.0 kJ/mol. At this pH, a small ionic strength correction was also made. This was the only case where such a correction was necessary.
Discussion Kinetics. The value of k, = 6.4 X lo8 M-' 6' for 'COT radicals abstracting hydrogen atoms from L(SH)z at pH 4 may be compared with 8.3 X lo8 M-I s-l reported by Akhlaq, Schuchmann, and von SonntagZZfor the same radicals abstracting hydrogen atoms from dithiothreitol, which is another disulfhydryl molecule. The existence of a back-reaction, that is, the abstraction of hydrogen from organic molecules by S-centered radicals, has been propqsed in earlier s t ~ d i e s . ~For ~ *the ~ ~specific case of the reaction of LS2H with formate, concrete evidence came from the observation of a chain reaction16 in the reduction of oxidized lipoamide, which is a cyclic disulfide, in irradiated formate solutions. Chain reactions were also reported by Elliot, Simsons, and SopchyshynZ5 for the reduction of other disulfides in formate solutions. The propagating steps in these chain reactions consist of the reverse of reaction 1 plus reaction 18. Relatively few reports of rate PSSP
+ T O 2 - + H+ = 'PS + PSH + C 0 2
(18)
constants for the abstractions of hydrogen atoms by PS' radicals in aqueous solutions have been given, but similar reactions have been investigated in organic systemsz6 Free Energy Changes and Reduction Potentials. The values of Eo1 for penicillamine and P-mercaptoethanol were measured relative to chlorpromazine in ref 5 , with a relative uncertainty of f0.02 V. The value of Eo8on the same basis is known to within k0.04 V. The values of AGO9 for T O 2 - , in Table 11, from the three sulfhydryl systems lead to an average of -143.6 f 3 kJ/mol or an Eo9value of 1.49 f 0.03 V, which is within the error range for the three PSH standard compounds. The accuracy of the chlorpromazine potential in the pH used here is probably about f 0 . 0 3 V, for a total uncertainty of f0.06 V. (22) Akhlaq, M. S.;Schuchmann, H.-P.; von Sonntag, C. Inf. J . Radiat. Biol. Relat. Stud. Phys., Chem. Med. 1987, 51, 91. (23) Elliot, A. J.; Sopchyshyn, F. C. Radiat. Phys. Chem. 1982,19,417. (24) For earlier work see the review by: von Sonntag, C.; Schuchmann, H.-P. In Chem. Ethers, Crown Ethers, Hydroxyl Groups Their Sulphur Analogues; Patai, S . , Ed. Wiley: Chichester, England, 1980; Vol. 2. (25) Elliot, A. J.; Simsons, A. S.; Sopchyshyn, F. C. Radiat. Phys. Chem. 1984, 23, 377. (26) See for example: Pryor, W. A. Free Radicals Biol. 1976, 1, 1.
3363
Recently, Schwarz and Dodson2' obtained a value of -1.90 0.05 V for EO191
f
C 0 2 + e- = TOz-
(19) The sum of this and Eo9 yields a value of -0.41 V for the twoelectron reduction of C 0 2 , Le., reaction 20. C 0 2 + 2e-
+ H+ = HC0,-
(20) This agrees with -0.36 V calculated from standard free energy datal9 within the uncertainties of the Eo measurements. Combination of the present Eo9with EozOgives EO19 = -1.85 f 0.06 V. The free energy change of reaction 21 can be calculated from AGO9 and the known free energy of protonation of 'C0,- l 8 and HC02-.19 Since the present AGO values are all relative to the 'C0,H
+ e- + H+ = HCOZH
'C02H
+ 1/2H2(g) = HCOzH
(21) standard hydrogen half-reaction, the result, 157.2 kJ mol-', is also the AGO value of (22)
Assuming identical energies of solution of T 0 2 H and H C 0 2 H and taking the known free energy of dissociation of Hz(g)28and the entropies of the 'C02H(g), 'H(g), and HC02H(g) species from published data,2sq2gone can estimate the H-C02H bond dissociation energy. The result is 396.7 kJ mol-' or 94.8 kcal mol-', which agrees within -2 kcal mol-' with 92.7 kcal mol-' given in ref 1. Acknowledgment. We are grateful for the use of the linear accelerator facilities at the Radiation Laboratory at the University of Notre Dame and to the Director, Professor R. H. Schuler, and his staff for their help and support in this work. We also acknowledge very helpful discussions with H. A. Schwarz and thank the authors of ref 27 for making a preprint of their paper available to us. The financial support of the Natural Sciences and Engineering Research Council of Canada was under Grant No. A357 1. Registry No. PenSH, 52-67-5; P-RSH, 60-24-2; L(SH)*, 3884-47-7;
HCO,-, 71-47-6; 'CO*-, 14485-07-5. (27) Schwarz, H . A,; Dodson, R. W. J . Phys. Chem. 1989, 93, 409. (28) JANAF Thermochemical Tables, 2nd ed.Natl. Stand. Re$ Data Ser. (US., Natl. Bur. Stand.) 1971, NSRDS-NBS 31. (29) Golden, D . M.; Benson, S. W. Chem. Rev. 1969, 69, 125. ONeil, H. E.; Benson, S. W. Int. J . Chem. Kinet. 1969, 1, 221.
Effect of Mass Transfer Coefficient on the Elution Profile in Nonlinear Chromatography Bingchang Lin, Sadroddin Golshan-Shirazi, and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1600, and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (Received: June 20, 1988; In Final Form: September 28, 1988)
The kinetic model of chromatography includes a mass balance equation and a mass transfer rate equation. A numerical analysis of the system of partial differential equations obtained is presented, and the characteristics of the finite difference method used to solve it are described. The results obtained are discussed, and the influence of the value of the mass transfer coefficient on the chromatographic elution profile is analyzed. In the limit case of linear chromatography (linear equilibrium isotherm), the result is very similar to the result obtained from the solution of the ideal nonequilibrium equation. In the other limit case, at large values of the rate constant, the result is very similar to the one obtained by solving numerically the equations of nonlinear equilibrium chromatography.
Introduction The influence of the different sources of mass transfer resistance and of their kinetics on the shape of the elution profile of 'Author to whom correspondence should be addressed at the University of Tennessee.
0022-3654/89/2093-3363$01.50/0
large-concentration bands is an important problem in preparative ChromatograPhY. Although, in most cases involving small molecules, the chromatographic process takes place under experimental conditions that are near equilibrium, this is not true for large molecules, such as those found in biochemical "des where the rate of mass transfer is much slower, nor in the case of affinity 0 1989 American Chemical Society
