J. Phys. Chem 1983, 87, 149-153
value in the steady state as in the diffuse layer and so does not constitute a new parameter. Thus, we have only six instead of seven parameters. At the same time, we have the four boundary conditions of continuity at the interface located at 5 = 0 together with the integral condition on the space charge density connected with our approximative technique. These five conditions apply to the six parameters and leave only one undetermined new parameter. This can be added to the two that remain undetermined in the diffuse layer to leave us with a total of three undetermined parameters. Moving now to the final region, namely, the bulk membrane on the right in Figure 1,we discover three additional parameters. These are the uniform coion concentration n(L)”’ and the uniform counterion concentration p(L)”’ in the bulk membrane at 5 = L. These additional three parameters, together with the other three, leave us with six undetermined parameters in all. However, we have the four boundary conditions of continuity, now applied at 5 = L. The last of these conditions, shown in Figure 1, reflects the fact that, in view of the uniform concentrations n(L)”’ and p(L)”’,the Nernst-Planck equations require the field in the bulk membrane to be given by h”’/p(L)‘”. In the bulk membrane, we also have another condition based on the uniformity of n and p ; this is
g / h = -n(L)”’/p(L)”’
(B15)
as shown in Figure 1. This is the condition for the infinitely thick membrane eq 20 which is discussed at some
149
length in the text. Furthermore, in the bulk membrane we require electroneutrality which, in this case, takes the form N n(L)”’ = p(L)”’ (B16) These two conditions, with the four boundary conditions at 5 = L, provide six equations that finally determine the six undetermined parameters, accumulated up to this point. This closes the problem. As made clear in Appendix A, the accuracy of the approximate 4 obtained in this manner is generally much better than the accuracy of the guessed space charge form when its parameters are determined. Ultimately, the validity of the approximation can be evaluated by choosing more than one form for the original space charge and noting the insensitivity of to these various forms. Actually, as indicated by Figure 14 such insensitivity is the rule. Finally, the potential difference across the domain of interest will depend, among other things, on h and g. When h and g are set equal to zero in our final results, that potential difference should correspond to the equilibrium potential difference, a quantity that can be known exactly (even without solving the Poisson-Boltzmann equation). The result of the approximation can then be compared in this limit with an exact result. Again, in our case, the results go smoothly to the equilibrium value. The foregoing discussion was offered to indicate in a clear manner that the problem could be closed. In actual practice, however, the steps in the solution are performed in a different order.
+
Infrared High Pressure and Monte Carlo Study of OH Vibratlonal Relaxation of terf-Butyl Alcohol in Carbon Tetrachloride T. W. Zerda’ and J. Zerda Instltute of Physics, Uniwersytet Slaski, Uniwersytecka 4, 40-007 Katowice, Poland (Received: February 17, 1982; In Final Form: September 7, 1982)
The results of IR high-pressure measurements of the OH bandshape of tert-butyl alcohol in mixtures with CC14 and CS2are presented. The shift of the band center to the low-frequency side (a typical observation for OH groups) accompanied by an increase in the band intensity is the only change noted in the bandshape as the pressure is increased. Assuming that the band contour is determined by inhomogeneous broadening, we applied the Monte Carlo (MC) technique to explain the observed effects in solutions with CC14. The butanol molecule surrounded by 124 molecules of CC14is placed in a cube, and the intermolecular interactions are given by the Lennard-Jones potentials. The theoretical band shapes are constructed by evaluating the shifts of the OH frequency in different environments and are compared to the experimental ones. The computed bands are narrower than the measured bands indicating the possibility of homogeneous broadening. The pressure-induced red shift is also well reproduced by the MC technique.
Introduction Vibrational relaxation in liquids has been extensively studied over the past few years. Information on this process has been obtained by analyzing Raman and IR band contours, which are determined by both vibrational and rotational relaxation. The separation of these two components is practical only in Raman experiments.’I2 Information on vibrational relaxation is obtainable from the IR bands only if the rotational component is believed
to be insignificant. This is the case of molecules with large inertial moments or in liquids at sufficiently low temperatures or at high pressures. We have previously show3 that the OH bandshape of tert-butyl alcohol (t-BuOH) in dilute CC14solutions is determined completely by vibrational relaxation and that, at room temperature, the rotational component can be neglected. The same assumption was also made in another later IR study on this m~lecule.~
S. Bratos and E. Marechal, Phys. Reu. A , 4, 1078 (1971). (2) L. A. Nafk and W. L. Peticolas, J. Chem. Phys., 57,3145 (1972).
(3) R. Konopka, T. W. Zerda, Z. Gburski, A. Hacura, and E. Wilk, Chem. Phys. Lett., 30, 145 (1975). (4) F. G.Dijkman, Mol. Phys., 36, 705 (1978).
(1)
0022-365418312087-0 149$01.50lO
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The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
It is well established that the vibrational band broadening is caused mostly by dephasing processes which modulate the vibrational frequency and that energy relaxation and resonance transfer are generally negligible. Most theories related to the vibrational relaxation involve only a single mechanism for phase modulation. In the fast modulation limit, homogeneous broadening, several different theories have been developed, such as the binary collision model: the hydrodynamic model: the cell model? resonant energy transfer,6 intramolecular potential coupling: etc. These theories predict the temperature and pressure dependence of the vibrational bandshape (or correlation function, or correlation time), and some of them were successfully compared to the high-pressure experimental results. The inhomogeneous broadening is caused by environmental fluctuations; randomly oriented molecules at various distances produce a slightly different intermolecular potential for each molecule. In the slow modulation limit, when the molecular motion is slowly relaxing, the environment can be assumed to be static. The inhomogeneous broadening can thus be easily simulated by the Monte Carlo technique-each step of the calculation represents one molecular environment and thus one of the possible perturbing potentials leading to a molecular vibration at a frequency shifted from the unperturbed, free oscillator. Hence the environmental distribution corresponds to the frequency distribution of the calculated band intensity. Harris et al.'O using the picosecond pulse technique and Raman scattering experiments have shown that in a case when both broadening mechanisms decay on different time scales the vibrational bandshape is given by the convolution of homogeneous, I H ( w ) , and inhomogeneous, I I N H ( w ) , shapes:1°
IexP(u)=
JIH(~IINH(.
-
4 dw'
(1)
It has shown by Dijkman, that the experimentally observed broadening of the OH vibration of t-BuOH in inert solvents is mainly due to inhomogeneous broadening. In this paper we present an IR high-pressure study of the OH stretching vibration of t-BuOH. The measurements have been made in dilute solutions of CCl, and CS2 (0.02 M), so that the butanol-butanol interactions can be neglected due to large average distance between butanol molecules. Consequently, the structure of the CCl,-tBuOH solutions can be correctly determined by the CC4-CC1, and CC1,-t-BuOH interactions. Under this assumption, the molecular structure of the CC1,-t-BuOH mixture has been modeled by the Monte Carlo technique." After approaching equilibrium, the spectra were calculated according to the perturbation theory developed by Dijkman and Maas.12 The results of these calculations are compred to the experimental bandshapes at each pressure. Experimental Section IR spectra of t-BuOH were recorded in the region 3450-3750 cm-' with the Specord 75 JR spectrophotometer. (5) S.F. Fisher and A. Laubereau, Chem. Phys. Lett., 35, 6 (1975). (6) D. W. Oxtoby, J . Chem. Phys., 70, 2605 (1979). (7) D. J. Diestler and J. Manz, Mol. Phys., 33, 227 (1977). (8) G. Doge, Z. Naturforsch. A, 28, 919 (1973). (9) R. M. Shelby, C. B. Harris, and P. A. Cornelius, J. Chen.Phys., 70, 34 (1979). (10) S. M. George, H. Anweter, and C. B. Harris, J . Chem. Phys., 73, 5573 (1980). (11) W. W. Wood in "Physcis of Simple Liquids", H. N. V. Temperley, J. S.Rowlinson, and G. S. Rushbrooke, Ed., North-Holland, Amsterdam, 1968. (12) F. G. Dijkman and J. H. van der Maas, J . Chem. Phys., 66,3871 (1977).
TABLE I : Comparison of the Experimental a n d ComDuted Results for t-BuOH Dissolved in CCIAa exptl data Emax,
~
P,
bar
1 200 500
1000 1460
uo,
cm-'
3616.0 3615.6 3615.0 3614.1 3612.8
MC results
M-'
fwhh, cm-1 cm-'
139 142 150 167 174
15.5 15.2 15.5 14.8 14.5
p,
uo,
g/cm3
cm-'
fwhh, cm-'
0.885 1.2512 1.5748 1.6075 1.6483 1.7022 1.7443
3638.4 3625.1 3624.5 3624.2
13.0 15.8 11.0 12.8
3621.2 3620.1
13.0 12.9
a v o denotes the position of the OH band, emax its intensity a t the m a x i m u m , fwhh t h e band width, and p the simulated density calculated according to e q 2.
TABLE 11: Pressure Dependence of the O H Band Position, v o , Bandwidth, Fwhh, a n d Intensity at the Center, E", of t-BuOH in Solution with CS, at 300 K
1 200 500 1000 1500 2000 2500 3000
3606.0 3605.0 3603.0 3602.5 3601.0 3599.5 3598.5 3597.5
18.0 18.2 18.0 18.5 18.2 19.0 19.0 19.5
377 378 386 411 472 495 543 573
The high-pressure optical cell with sapphire windows used in this study was described previ0us1y.l~ The pressure was measured with resistance manometer built from a manganin coil and independently with a Bourdon type gauge. The optical length was adjusted to 3.05 mm and there was no evidence that it had changed even at the highest pressures. By using such a long cell we were able to record the spectra of very dilute solutions, 0.02 M, where butanol is believed to exists only as monomers. Indeed, there was no evidence in the recorded spectra that hydrogen-bonded complexes were formed. The slit width was about 2.7 cm-' for all measurements. The correlation functions of the OH stretching mode were obtained by Fourier transformation of the observed bands followed by division by the Fourier transform of the slit function. The slit function was approximated by a triangular function. Results Tables I and I1 contain the measured values of the band position, YO, the full-width at half-height (fwhh), and the intensity of the band maximum, em=, of the OH vibration of t-BuOH in CCl, and CS2,respectively. ,,e is given by (cd)-l In (Io/Imm),where c is concentration, d the cell length, Io the transmittance of the baseline, and I,, the transmittance of the band maximum. The intensity data, extinction, are corrected for the increasing number of absorbing molecules in the cell caused by the increasing density. CCl, solution measurements were taken only to 1460 bar because at higher pressures CCl, solidifies at room temperature. The pressure-induced red shift of the band center seems to be a typical behavior for monomer bands in alcohols. The same observations were noted by Fishman and Drickamer14and by Jacobsen et al.15 These studies were (13) T. W. Zerda, A. Hacura, J. Zerda, and E. Kluk, Acta Phys. Pol. A, 54, 55 (1978). (14) E. Fishman and H. G. Drickamer, J . Chem. Phys., 24,548 (1956).
The Journal of Physical Chemlsrry, Vol. 87, No. 1, 1983
OH Vibrational Relaxation in t-BuOH-CCI,
TABLE 111:
151
Parameters f o r t-BuOH and CCl, Used in t h e
Calculation#
.
c
OH Oscillator D = 4 2 7 0 0 cm-'
L
v g = 3 6 8 1 cm-' f , = 1.056
t 0 0
.......................
00
025
050
075
(00
125
150
175
Figure 1. The correlation functions for OH band of t-BuOH in CCI,: solid line, 1 bar; broken line, 1460 bar. No differences between the measured courses can be observed.
mainly dedicated to the measurements of the vo shift and not to the investigation of the bandshape. So, in this respect, our study is the first to discuss this matter. A rough examination of the band contour can be made by measuring the bandwidth; it can be seen from Tables I and 11that values are nearly constant and the observed changes are of the same range as the experimental error. More detailed information on the bandshape can be obtained from correlation function analysis. Figure 1illustrates the correlation functions for two extreme pressures, 1and 1460 bar in the t-BuOH-CC14mixture. They are so similar that no difference can be seen between the two curves. The same result was obtained for t-BuOH-CS, mixtures. In conclusion, there are no observed changes in the bandshape with the exception of the shift of the band center to lower frequencies and the pressure-induced increase of the extinction. These rather atypical results have been also observed in some other molecular liquids. In acetone16they have been explained qualitatively by a dipole-dipole coupling interaction; but it is impossible to apply this theory to the nonpolar solvents CC14and CS,. Similar observations in SF," have been explained as a motional narrowing process18 but again, this theory cannot be applied to our case, because it is concerned with the rotational motion of the molecule and the rotation is assumed to be very slow for butanol. For most liquids studied, the vibrational bands broaden with increasing pressure, for example, CH3CN,lgC6H6,,0 C6H12,21CH2C12,2,etc. In these cases the broadenings have been explained in terms of the Fisher-Laubereau binary collision model.5 In the t-BuOH-CC1, or t-BuOH-CS, systems, homogeneous broadening can be assumed negligible, so the Fisher-Laubereau model cannot be applied in our case. There are few theories dealing with inhomogeneous broadening. Examples of these include Bratos" stochastic and George et al.'s'O semiempirical model of local number density distribution. However, it is not easy to apply them in a high-pressure study. Thus in order to explain the (15)R. J. J. Jacobsen, Y. Mikawa, and J. W. Brasch, Appl. Spectrosc., 24, 333 (1970). (16)W. Schindler, P. T. Sharko, and J. Jonas, VI1 International Conference on Raman Spectroscopy, Ottawa, Canada, 1980. (17)T. W.Zerda, J. Schroeder, and J. Jonas, J . Chem. Phys., 75,1612 (1981). (18)S.R. J. Brueck, Chem. Phys. Lett., 50,516 (1977). (19)J. Schroeder, V. H. Schiemann, P. T. Sharko, and J. Jonas, J. Chem. Phys., 66,3215 (1977). (20)K. Tanabe and J. Jonas, J . Chem. Phys., 67,4222 (1977). (21)K. Tanabe and J. Jonas, Chem. Phys., 38, 131 (1979). (22)T. W.Zerda, Chem. Phys., 56, 213 (1981).
Lennard-Jones Parameters EH = 3 7 K E O= 355 K EB = 148 K ECC1, = 3 2 7 K
= 2.4 A 00 = 2.8 A O B = 5.0 A uCC1, = 5.88 A UH
f, = 1308 a = 2.212 .k-'
The data were taken from ref 4.
observed effects in butanol we decided to elaborate on the molecular simulation technique.
Theoretical Calculations Description of the Monte Carlo Systems. To start a Monte Carlo (MC) simulation one has to specify the positions of all molecules in a cube, their number, and the intermolecular forces. We have concentrated our interest on the CCb-t-BuOH system. The molecule of t-BuOH was modeled as three particles H, 0,and B with the HOB angle taken as 109O and the distance rOH= 0.958 and roB = 2.5 The molecule was placed in the center of a cube and surrounded by 124 atom like particles representing CC14. The normal boundary conditions were used." So that the real liquid densities were simulated only the cube length was changed. Because the solutions were very dilute-1 butanol was surrounded by approximately 2000 CCl, molecules-only a small mistake was introduced when the solution densities were assumed to be equal to the pure CCl, densities ( p ) . The p values for different pressures, p, were found from the Tait equation:
B ( n+P
v030,n - V(P,T) = a n log B ( T )
+
(2)
where C = 0.21290 and po = 1. B at the temperature T = 30 OC was calculated from the Gibson and LoefflerZ4 empirical relation B(T) =
0.8670 - 0.006808(T - 25) + O.O0001713(T - 25)2 (3)
In the MC simulation the system performs a random walk through coordinate space. In one MC step one particle in the system is transfered by a small, random displacement. The potential energy of the resulting configuration is then compared with the original potential energy. If the move lowers the potential energy, the move will be accepted, if it increases the potential energy by an amount AE,the displacement is rejected unless exp(AEIkT) > Q where Q is a random number between 0 and 1. MC simulations are not related to the time (especially to the time evolution of the system) and no dynamical information can be obtained in the course of the MC calculations. But this computer technique can give us information on the microstructure of the liquid and molecular environments which is especially important for asymmetrical molecules like butanol. The most important problem in the MC simulations is the proper selection of the intermolecular potentials. For the solution under consideration, CC14-CC14interactions were represented by a spherically symmetric LennardJones potential with e l k = 327O and u = 5.88 A, and the (23)J. Korppi-Tommola, Spectrochim. Acta, Part A , 33, 1077 (1978). (24)R. E.Gibson and 0. H. Loeffler, J. Am. Chem. SOC.,63, 898 (1941).
152
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Zerda and Zerda
interaction between CCl, and t-BuOH was given by a sum of three independent contributions: 0.-CC14, H.-CC14, and B-CCl,. The 6 and u for the potential terms were taken from Table I of ref 4 and are included in Table 111. The bandshape calculations were done after the systems had approached equilibrium. It was accomplished by plotting the total potential energy between t-BuOH and the surrounding CC14 molecules EB = C(Ei(CCl,...H) + E;(CC&-.O) + Ei(CCl4***B) (4) 1
and the total potential energy in the cube ET = CCEij(CC14.*.CClJ + EB i>j J
(5)
vs. the number of MC steps. The summations in eq 4 and 5 are over all C C 4particles in the cube. In all cases more than 150000 accepted displacements of CC14were needed before equilibrium was achieved. To save computation time one component of the solute molecule, particle B, remained fixed at the cube center throughout all the calculations. In order to improve the statistics the orientations of the B-0 bond in relation to b (angles B and 4) and then the 0-H bond in relation to the B-0 vector (angle cp) were also randomly changed. The MC calcuations based on different initial configurations, where the position of t-BuOH was randomly selected and then randomly moved during the MC runs, converge to unsignificantly different values. Although this test was made only for 64 molecules in the cube, it assured us that the chossen algorithm generates the equilibrium distribution of configurations of CC14 around t-BuOH. Following Dijkmaq4 we have considered the 0-H bond as a Morse oscillator perturbed by the external intermolecular potential Eg. The frequency shift relataive to the unperturbed gas-phase frequency, vg, has been shown by use of first-order perturbation theory to be
The Morse parameters a and D and values of fi and fi were taken from Table I of ref 4. The first- and second-order derivatives, (dEB/dr),(d2EB/dr2)(r is the vibrational coordinate), were evaluated analytically from eq 4, assuming that only the hydrogen atom moves during the 0-H oscillations. For each MC step the frequency shifts were calculated from eq 6 and stored in the computer. Results of more than 300 OOO calculations were then separated into sections 0.5 cm-' wide. The number of repetitions in each section was assumed to be equivalent to the intensity of the band at that frequency. The band contours evaluated in this manner for different pressures are presented in Figure 2. In two cases, 1 and 1460 bar, the number of MC steps was extended to 500000 but no changes in the bandshape were observed except that the contours were smoother due to the larger number of configurations averaged. Discussion In parts c and d of Figure 2 the experimental IR and computed bandshapes are also presented. The experimental conditions for parts c and d of Figure 2 were CC1, solutions at 30 "C and pressures of 1 and 1460 bar, respectively. We have compared the shapes despite the fact that the intensity axes are expressed differently-the experimental curve in terms of the normalized intensity and
F R E Q U E N C Y [cm-']
1
3580
3590
3600
3610
3620
FREPUENCY
3580
3590
3600
3610
3b10
3630
3640
3650
3660
3640
3650
3660
[cm"!
3630
FREQUENCY [cm.']
@
I
i
, I)
2
P
s
3580
3590
3600
3610
3630
3620
FREQUENCY
3640
3650
3660
[cm.']
Figure 2. Theoretical bandshapes of the OH vibration in solution with CCI, where the density of the simulated system corresponds to (a) 0.885, (b) 1.2512, (c) 1.5748, and (d) 1.7443 g/cm3. The intensity is expressed in terms of repetltkin number of similar frequency shifts for different MC steps. In the parts c and d the experimental I R contours are also shown (dashed lines) corresponding to 1 and 1460 bar. In this case the intensities are expressed in extlnction units. All contours are normalized to the unit area. The two first simulations, parts a and b, correspond to CCI, in the gaseous phase and thus cannot be compared to the experimental results.
J. Phys. Chem. 1083,87,153-159
the theoretical curve in terms of normalized repetition number. It can be seen that the maxima of the theoretical curves are about 8 cm-' higher than the experimental one. Even though the bandwidths are not identical (fwhhtheov = 13 cm-', fwhhe@= 15 cm-') they are rather close to each other. Similar observations have been made for the remaining bands at other pressures (Table I). In order to manifest the band shift we have also simulated the bandshapes for two densities corresponding to the discussed system in gaseous phase where no experiments have been carried out. The computed shift, 4.4 cm-l, corresponds nicely to the observed shift of 3 cm-' when the pressure is increased from 1 to 1460 bar. The differences in the positions of the centers are constant at each pressure, and small, considering the simplicity of the intermolecular interaction model. It is interesting to note that the calculated bands do show asymmetry on the high-frequency side like the experimental spectra. This simple model can also explain the nearly constant value of the bandwidth during the experiments. The computed bands are slightly narrower than the measured ones indicating the possibility of homogeneous broadening. We want to emphasize here that, although such processes like fast collisional effecta or intermolecular energy transfer cannot be neglected, their influence on the total bandshape is small-compare the discussion of eq 1. Previously Dijkman, indicated that internal rotation of the OH group might also be responsible for the total shape of the OH band. In such a case the internal rotation would cause additional broadening due to rotational relaxation and, on the other hand, would modulate the frequency perturbation leading to the motional narrowing effect.25 Which one of these contrary processes predominates is hard to state. Unfortunately, the MC technique cannot (25) W. G. Rothschild, J. Chem. Phys., 65, 455 (1976).
153
be used to answer this question-one can do it using molecular dynamic calculations. Some inaccuracies in the theoretical calculations originate from the oversimplified CCl, intermolecular interaction potential in which we have assumed that the CCl, potential is like that for a noble gas atom. However, it has been shown by Chandlerz6that, when this assumption is made, it is impossible to account for the observed density of liquid CCl, even under normal conditions. CCl, should be moldeled as four mutually overlapping van der Waals spheres, each representing one chlorine atom. Of course, at higher pressure this problem becomes more and more important, and the chlorine atoms from different molecules fit together with those from another molecule like gears. It is obvious that the MC system, discussed above, does not fully represent the actual solutions. Since taking the individual atoms into account would lead to very long computer calculations, we have approximated CCl, by one Lennard-Jones potential. Other inaccuracies are due to neglecting the possibility of hydrogen bonding between C1 and OH group. There are some indications that a weak specific type of interaction between carbon tetrachloride and hydrogen takes place; unfortunately, there are neither data nor a model for this effect. If there were, the method described in this paper of comparing the calculated and measured bandshapes might be a powerful technique for studying it.
Acknowledgment. Professor J. Jonas and Professor A. Rahman are acknowledged for discussions on the problem dealt with in this paper. Our thanks are due to Dr. S. Perry for reading the manuscript and correcting the English. This research was partially supported by the PAN 1-9 project. Registry No. t-BuOH, 75-65-0; CCl,, 56-23-5; CS2, 75-15-0. (26) D. Chandler, Annu. Reu. Phys. Chem., 29,441 (1978).
Structure of Ionic Micelles from Small Angle Neutron Scattering Dallla Bendedouch, Sow-Hsln Chen, Nuclear Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
and W. C. Koehler National Center for Small Angle Scattering Research, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 (Received: M y 14, 1982; I n Final Form: August 26, 1982)
Extensive small angle neutron scattering (SANS) measurements were carried out on aqueous solutions of small anionic micelles of lithium dodecyl sulfate (LDS). Scattered intensity distributions were studied by varying the contrast as a function of both micellar core and solvent scattering length densities. Information regarding size, shape, aggregation number, and polydispersity of the micelles, and also water content of the inner hydrocarbon core, was extracted. Furthermore, the spatial distribution of the end groups of the dodecyl chains in the micelles was deduced from a selective deuteration of the methyl groups. Packing of the alkyl chains in the core has a degree of order between that for a completely disordered liquid state and a fully ordered all-trans radial configuration. Introduction Hartley1 in 1936 was the first to propose a model explaining the solubilization properties of amphiphilic
molecules in aqueous solutions. A typical amphiphile is a molecule consisting of two parts. One portion is a hydrophobic hydrocarbon chain (or tail); the other is a hydrophilic polar head. The existence of these opposing (1) Hartley, G. S. 'Aqueous Solutions of Paraffin-Chain salts-; H ~ ~ - Properties in the same molecule when dissolved in water mann and Cie: Paris, 1936. is the origin of the thermodynamic driving force for mi0022-3654/83/2087-0 153901 S O / O
0 1983 American Chemical Society
