3674
J. Phys. Chem. 1984, 88, 3674-3678
theoretical w ~ r k . ~ Albeng * ’ ~ considered a lattice model in which only three discrete orientations are allowed ( X , Y, Z ) . This limitation is not present in the work of Rabin, McMullen and Gelbart’O who start from a free energy that implicitly contains powers of the concentration to all orders. In their treatment, however, they pay insufficient attention to the fact that the fractions of rods and disks in the coexisting isotropic and uniaxial nematic phase are not the same. Finally, Albeng and Saupe, Boonbrahm, and Yu6using a Landau expansion of the free energy find a phase diagram that qualitatively shows the same features as found here. (9) R. Alben, J . Chem. Phys., 59, 4299 (1973). (10) Y. Rabin, W. E. McMullen, and W. M. Gelbart, Mol. Cryst. Liq. Cryst., 89, 67 (1982).
A noteworthy feature is that the biaxial phase appears in a narrow gap between the n(+) and n(-) phase which may explain why it has been observed in some systems5*6but not in other quite similar systems.” Finally, the systems where the biaxial phase has been 0bserved~9~ have one inherent drawback the rodlike and disklike particles are aggregates (micelles) the size, shape, and concentration of which depend on the physicochemical conditions of the system, e.g., by varying the temperature or overall surfactant concentration one also varies the characteristics of the aggregates. Therefore, it would be worthwhile to look for a biaxial phase in a solution of stable rodlike and disklike particles that keep their identity upon dilution. One might for example think of a mixture of vanadium pentoxide (rods) and clay platelets (disks). (11) Y . Hendrickx and J. Charvolin, J. Phys., 42, 1427 (1981).
Rate Coefficients for Evaporation of Pure Liqulds and Diffusion Coefficients of Vapors G . Karaiskakis and N. A. Katsanos* Physical Chemistry Laboratory, University of Patras, Patras, Greece (Received: November 3, 1983)
The reversed-flow gas chromatography sampling technique, reported earlier, is now used to measure the diffusion coefficient of vapors from a liquid into a carrier gas and at the same time to determine the rate coefficient for the evaporation of the liquid. The mathematical expression describing the elution curves of the sample peaks is derived and used to calculate the above parameters for the liquids n-pentane, n-hexane, n-heptane, n-octane, methanol, ethanol, propan-1-01, butan-1-01, and pentan-1-01 evaporatinginto the carrier gas helium. The values of diffusion coefficients found are compared with those calculated theoreticallyor reported in the literature. An alternative mathematical analysis based on leads to profound disagreement with the experimental findings.
Introduction Reversed-flow gas chromatography is a new technique, first proposed for kinetic studies in heterogeneous catalysis,’V2applied to the dehydration of alcohols3 and the deamination of primary aminese4 The method has been extended to the determination of diffusion coefficients in binary5 and in multicomponent6 gas mixtures, the variation of these coefficients with temperature,’ as well as to other measurements like those of adsorption equilibrium constants* and of rate constants for removal of solvents from impregnated porous solid^.^ Finally, the method was used to study the heterogeneous kinetics of a complicated reaction with two gaseous reactants, namely the oxidation of carbon monoxide with oxygen over Co304-containingcatalysts.1° The reversed-flow gas chromatography has been reviewed recently.” This new method is carried out using simple gas chromatography instrumentation, and it is in reality a sampling technique based on perturbations imposed on the carrier gas flow. These perturbations consist of reversing the direction of flow of the carrier ( 1 ) Katsanos, N. A.; Georgiadou, I. J . Chem. SOC., Chem. Commun. 1980, 242. ( 2 ) Katsanos, N. A . J. Chem. SOC.,Faraday Trans. I 1982, 78, 1051. (3) Karaiskakis, G.; Katsanos, N. A.; Georgiadou, I.; Lycourghiotis, A. J . Chem. SOC.,Faraday Trans. I 1982, 78, 2017. (4) Kotinopoulos, M.; Karaiskakis, G.; Katsanos, N. A. J . Chem. SOC., Faraday Trans. 1 1982, 78, 3379. (5) Katsanos, N. A.; Karaiskakis, G. J. Chromatogr. 1982, 237, 1. ( 6 ) Karaiskakis, G.; Katsanos, N. A.; Niotis, A. Chromatographia 1983, 17, 310. (7) Katsanos, N. A,; Karaiskakis, G . J. Chromatogr. 1983, 254, 15. (8) Karaiskakis, G.; Katsanos, N . A.; Niotis, A. J. Chromatogr. 1982,245, 21. (9) Karaiskakis, G.; Lycourghiotis, A.; Katsanos, N. A. Chromatographia 1982, IS, 351. (10) Karaiskakis, G.; Katsanos, N. A.; Lycourghiotis, A. Can. J . Chem. 1983, 61, 1853. (1 1) Katsanos, N. A.; Karaiskakis, G. Adu. Chromatogr. (N.Y.),in press.
0022-3654/84/2088-3674$01 S O / O
gas from time to time. Each flow reversal creates perturbations on the chromatographic elution curve, having the form of extra peaks (“sample peaks”), and thus a slow rate process, taking place within the filled or the empty chromatographic column, is sampled at known times. The analytic mathematical dependence of the heights of the sample peaks on the time when the last reversal was made is given in the Theoretical Section. This function is used to calculate the relevant rate coefficient of the slow process being sampled. In the present work the reversed-flow technique was used to measure simultaneously the rate coefficient for the evaporation of pure liquids and the diffusion coefficient of these vapors into a carrier gas passing over the surface of the liquids at a remote point (cf. Figure 1). The liquids used were n-pentane, n-hexane, n-heptane, n-octane, methanol, ethanol, propan-1 -01, butan- 1-01, and pentan- 1-01, the carrier gas being helium.
Theoretical Section The general chromatographic sampling equation, describing the elution curves which follow the carrier gas flow reversals, is7 C
= C,(l’,to+t’+7) [ u ( 7 ) - u(7
U(7)
+ Cz(l’,to+t”7)[1 - U ( 7 - t ’ ) ] x
- tM’)] + c,(l’,to-t’+7) u(t0+7-t’)(u(t - t ’ ) [ l u(7 - tM’)] - u(7 - f ’ ) [ U ( 7 ) - u(7 - tM’)]) ( 1 )
where c is the concentration of vapor at the detector, cl(l’, ...), cz(l’,...), and c3(l’,,,,)are concentrations at the point x = I’(cf. Figure 1) for the times shown ( t o = total time from placing the liquid in column L to the last backward reversal of gas flow, t’ = time interval of backward flow, r = t - t M , t being the time t the gas from the last restoration of the carrier gas flow, and M hold-up time of column section 1 ) ; finally, the various u’s are unit step functions for the arguments shown in parentheses and tM’ is the gas hold-up time in the section 1’. 0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3615
Evaporation and Diffusion Coefficients
by exp(qL)/2. Then eq 10 becomes
which, for high enough flow rates, further reduces to
L
0 r
._
Z
4-
L
I
C
C(l’,Po) =
IC//
c(l:tn) =
Figure 1. Schematic arrangement showing the diffusion column L connected to the chromatographiccolumn 1‘+ I through which carrier gas flows from D2to D,or vice versa.
It has been pointed out7 that, for t’smaller than both t M and tM’,each sample peak produced by two successive reversals is symmetrical and its maximum height h from the ending base line is given by
2c(l’,t,)
ac,/ato = Da2c,/az2
(3)
where D is the diffusion coefficient of the vapor into the carrier gas. The solution of eq 3 is sought under the initial condition
=0
(5)
-D(ac,/az),=L = vc(i‘,t,)
(6)
--D(ac,/az)z=o = kJc0 - c m 1
(7)
where c,(O) is the actual concentration at the liquid interface at time to, co the concentration of the vapor which would be in equilibrium with the bulk liquid phase, and k, a rate coefficient for the evaporation process. Equation 7 expresses the equality of the diffusion flux for removal of vapors from the liquid surface and the evaporation flux due to departure of c, at the surface from the equilibrium value co. When the Laplace transform of eq 3 is taken with respect to to, a linear second-order differential equation results. This can be solved by using z Laplace transformation yielding C,l(O) +sinh qz 4
(8)
where 4 = (Po/D)1’2
(9)
and CJO) and C,’(O) are the to Laplace transforms of c,(0) and (ac,/az),,,, respectively. If one combines eq 8 with the to transforms of the boundary conditions (S), (6), and ( 7 ) , the Laplace transform of c(l’,to), denoted as C(f’,po),is found:
kcco 1 Po (Dq + vk,/Dq) sinh qL
+ kc(
:)”’] (13)
Finally, if one uses the relation erfc x N exp(-x2)/(xd2), which is a good approximation12 for large values of x, eq 13 becomes
-
c(l’,t,) = 2kcco( V : ) ‘ I 2 exp(
-&)(
2t01/‘
+ kct2/2 (14)
Coming now to the other extreme, i.e. long-time approximations, qL is small and the functions sinh qL and cosh qL of eq 10 can be expanded in McLaurin series, retaining only the first three terms in each of them. Then, from eq 10 one obtains
C(l’,PO) =
kcco 1 Po (Dq + vk,/Dq)qL + (u + k,)(l + q 2 L 2 / 2 ) (15)
and by using eq 9 and rearranging this becomes
c,(L,to) = c(l’,to)
where u is the linear velocity of carrier gas, and the boundary condition at z = 0:
c(1:Po)= -
V
(4)
the boundary conditions a t z = L
C, = C,(O) cash qz
%! exp( 7 k,L + T) k:to erfc [ 2( Dt,)
(2)
where c(l’,to) is the vapor concentration at x = l’and time to. This concentration can be found from the diffusion equation in column L (see Figure 1):
C,(Z,O)
exp(-qL) + kc/D)
q(q
Taking now the inverse Laplace transform of this equation, one finds pure liquid
h
2k,co
-
+ (v + k,)
cosh qL (10)
Inverse Laplace transformation of this equation to find c(l’,to) is difficult. It can be achieved by using certain approximations which are different for small or for long times. In the first case qL is large, allowing both sinh qL and cosh qL to be approximated
C(l’,Po) = k&O Lpo po[l + ( u
1
+ kC)L/2D]+ uk,/D + (u + k c ) / L (16)
For high enough flow rates k, can be neglected compared to u and 1 can be neglected in comparison with vL/2D. For instance, in a usual experimental situation it was calculated that vL/2D = 420. Adopting these approximations, eq 16 reduces, after some rearrangement, to
Finally, inverse Laplace transformation of this relation yields
c(l’,to) =
Wco 11 - exp[-2(kCL + D ) t o / L 2 ] ) (18) v(k,L + D )
By considering the maximum height h of the sample peaks in eq 2 and substituting in it the right-hand side of eq 18 for c(l’,to), one obtains h as an explicit function of time to. In order to linearize the resulting relation, an infinity value h, for the peak height is required:
h,
2k,D~o/[v(k,L+ D ) ]
(19)
Using this expression, we obtain In ( h , - h ) = In h , - [2(k,L + D ) / L 2 ] t 0
(20)
Thus, at long enough times, for which eq 18 was derived, a plot of In (h, - h) vs. to is expected to be linear, and from the slope -2(k,J D ) / L 2a first approximate value of k, can be calculated from the known value of L and a literature or theoretically calculated13 value of D. This value of k, can now be used to plot
+
(12) Boas, M. L. “Mathematical Methods in the Physical Sciences”;Wiley: New York, 1966; p 407.
3676 The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 six-port
Karaiskakis and Katsanos 80
valve
-
pressure regulator
needle
al L
gas controller
;/16”tu be
U 3
t I
v)
I
7 .m
-2
L
Po
C
X
g Q
F
2
ml b v c k $
E
$E
I
5
a
40-
o
E
$
s\
8a
;
- 0
-
%m
2% f 20-
0
0 %
2
4-
-
-
0
P 4-
E
-m
z
2
al
+-
L
0
9
60-
J
o r > i
z g
Q
-0 , 8 -
i s v)
2!
3
e
1
,
oven
-----
- - _ _baseline--__ _____-
-_
115
Figure 2. Gas lines and important connections for measuring simultaneously diffusion coefficients and rate coefficients for evaporation by reversed-flow gas chromatography.
small-time data according to eq 14 which is substituted now for c(l’,to)in eq 2. After rearrangement logarithms are taken, and there results
(21) Now a plot of the left-hand side of this relation vs. l/to will yield a first approximation experimental value for D from the slope -L2/4D of this new linear plot. This D value can be used back in the slope found from the plot of eq 20 to calculate a more accurate value for k,. In turn, the latter is utilized to replot eq 21, so that a more accurate value for D is found. These iterations can be continued until no significant changes in the k, and D values results. Experimental Section Chemicals. The various solutes used were either Merck “uvasol” (n-hexane, n-heptane, and ethanol), Merck “pro analysi” (methanol, propan- 1-01, and butan- 1-01), Fluka “puriss” (n-pentane and pentan-1-01), or Fluka “purum” grade (n-octane). The carrier gas was helium of 99.99% purity from Linde (Athens, Greece). Apparatus. The experimental setup for the application of the reversed-flow technique in the present problem is shown in Figure 2. A conventional gas chromatograph with a flame ionization detector (FID) was modified as to include a six-port gas-sampling valve, by means of which the carrier gas either enters at D2 and meets the detector at D1 (valve position indicated by the solid lines) or vice versa (valve switched to the dotted-lines position). All column sections of the cell of Figure 1 were inside the oven of the chromatograph. They were empty stainless steel ‘/&. chromatographic tubes with a 4-mm i.d. and lengths L = 116.2 cm and I = I’ = 100 cm. They were connected at the junction x = 1’by a ’/.&. Swagelok tee union. Another simple ‘/.,-in. union was used to connect a short tube (2 cm) containing 0.5 cm3 of liquid a t the end of the diffusion column L. A conventional two-stage reducing valve and pressure regulator was followed by a needle valve, a 4 A molecular sieve dryer, and a gas flow controller used to minimize variations in the gas flow rate. A restrictor placed before the detector served to prevent the flame from being extinguished, when the valve was switched from one position to the other. Procedure. After placing the liquid under study in its position (cf. Figure 2) and waiting for a certain time, during which no signal is noted, we recorded a monotonously rising concentra-
’t / I
5
0
0
I
I
I
40
I
f,
I 120
80
I
I 160
Imin)
Figure 4. The rise of the sample peak height with time for the diffusion of ethanol vapor into helium (t= 0.543 cm3s-l), at 336.8 K and 1 atm.
tion-time curve for the vapor of the liquid. When this rising continuous signal is high enough, the chromatographic sampling procedure is started by reversing the direction of the carrier gas flow for a time period shorter than the gas hold-up time in both column sections I and Z’(12 s). Then the gas flow is restored to its original direction. These flow reversals are done by switching the six-port valve from one position (solid lines) to the other position (dotted lines) and vice versa. After a certain dead time an extra signal is recorded having the form of a fairly symmetrical sample peak (cf. Figure 3). This double reversal of the flow is repeated several times with always the same duration (12 s) of backward flow. It gives rise to a series of Sample peaks corresponding to various times to from the beginning. The pressure drop along column I’ 1 was negligible, and the pressure inside the whole cell was measured by an open mercury manometer. Temperature variations inktheoven were less than fO.l K. The carrier gas flow rate (corrected at column temperature) was in the range 0.290-0.674 cm3 s-l.
+
Results and Discussion Figure 3 shows a section of a typical chromatogram with sample peaks created by flow reversals. In Figure 4 the height h of the sample peaks as a function of the time to,when the flow reversal was made, is plotted on a semilogarithmic scale. It shows the steep
Evaporation and Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3677
TABLE I: Rate Coefficients for Evaporation of Pure Liquids and Diffusion Coefficients of Vapors into Helium, at 1 atm i03D, cmz s-' liauid V, cm3 s-I T, K i04k,. cm s-I this work calcd ref 14 n-pentane 0.362 300.7 369 462 340 292 n-hexane
0.295 0.543 0.532 0.290 0.287 0.362 0.532 0.287 0.602 0.543 0.287 0.287 0.287
n-heptane n-octane methanol ethanol propan- 1-01 butan- 1-01 pentan- 1-01
295.2 293.2 332.5 293.2 342.7 322.8 373.9 322.5 322.5 336.8 342.3 358.1 374.0
228" 224 279 257b 257 25 1 340 330 340 279 364 268 203
334" 300 377 262b 332 254 320 705 663 576 473 423 398
311 308 381 308 343 265 340 688 688 593
268 265 33 1 265 245 317 642 642 614 467 438 409
accuracv.C % 26.4 (16.4) 6.9 (16) 2.7 (16.2) 1.1 (15.1) 17.6 (16.2) 3.3 4.3 (8.2) 6.3 (7.3) 2.4 (7.2) 3.8 (7.2) 3 (3.4) 1.3 3.5 2.8
"These values were determined by using a diffusion column with length L = 86.5 cm. bThese values were determined by using a diffusion column with length L = 120.2 cm. CThisis defined by eq 22. Numbers in parentheses are the accuracies of the respective literature values.
\
9 24
I
B
Figure 6. Data from evaporation of ethanol into helium, plotted according to eq 21.
\
\
t. \
t
40
I
I
I
60
80
100
b
I
I
Figure 5. Example of plotting eq 20 for the diffusion of ethanol vapor into helium ($' = 0.543 cm3 s-l), at 336.8 K and 1 atm.
rise and then the leveling off with time of the sample peak height. As an example of using eq 20 and 21 to analyze the experimental findings, the data of Figure 4 are treated as follows. Leaving out the first 3-4 points, which correspond to small times, the rest of the experimental points are plotted according to eq 20, as shown in Figure 5. As infinity value h , was taken the mean of the values found in the time interval 135-160 min, which differed little from one another. From the slope of this plot, which D)/Lz, according to eq 20,using a theois equal to -2(k,L retically calculatedI3value of D = 0.593cmz s-l, and the actual cm s-l for k, is value of L (116.2 cm), a value of 278 X calculated. This approximate value is now used to plot all but the few points close to h, according to eq 21 as shown in Figure 6. From the slope of this latter plot, which equals -L2/4D as shown in the Theoretical Section, a value of 0.576cmz s-l for D is found. If this is combined with the slope of the previous plot (Figure 5),a second value for k, = 279 X lo4 cm s-l is calculated and further used to replot the data according to eq 21. The new
+
48
loyt, (mln-')
9.5
9.0
36
value for D found coincides with the previous one (0.576cm2 s-l), and thus the iteration procedure must be stopped. Table I summarizes the results obtained with all liquids studied. In the same table the diffusion coefficients determined here are compared with those calculated the~retically'~ and those found In the latter case the values were reduced to in the 1iterat~re.I~ the temperature of this work by using the relation Dl/Dz = ( T l / T2)1,75 as suggested by the Fuller-Schettler-Giddings equation.I5 As one can see from Table I, the D values determined in this work are very close to the theoretical ones or those found in the literature. The accuracy given in the last column is the deviation of the experimental values from those calculated theoretically, except for the last three cases, in which instead of the Dcalcdwe used the literature values, Dlit.,because the appropriate constants for the determination of Dcalcdwere not available:
With the exception of two pairs, this accuracy is better than 7% in all cases. The accuracies of the respective literature values are expressed by bigger numbers, except for the first case (n-pentane). A comparison of the k, values determined in this work with others in the literature is impossible, because such values could not be found. With respect to the values of Table I two tentative conclusions can be drawn. First, that the k, is independent of the carrier gas flow rate, in accord with the presupposition that it is (13) Bird, R. B.; Stewart, W. E.; Lightfwt, E. N. "Transport Phenomena"; Wiley: New York, 1960; p 511. (14) Maynard,V. R.; Grushka, E. Ado. Chromatogr. (N.Y.) 1975,12,99. (15) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. Ind. Eng. Chem. 1966, 58, 19.
J. Phys. Chem. 1984, 88, 3678-3682
3678
a rate coefficient, and second, that these coefficients of evaporation increase with increasing temperature. Obviously, further experimentation is needed to confirm these conclusions. The boundary condition 7, adopted from Crank,16 is based on a departure from equilibrium at the liquid-gas interphase. This contrasts with the assumption of equilibrium at the interphase of Bird et al.,” but the latter is not supported by the present experimental findings. Thus, if eq 7 is abandoned and the boundary condition c,(O) = co, corresponding to equilibrium, is used, we obtain the relation
cz(”’po) =
CO
1 ( v / D q ) sinh qL
+ cosh qL
( 10’)
instead of eq 10. Adopting the same approximations as before, we find for small times
in place of eq 18. The above equation could also be obtained from eq 16 by assuming a large value for k,. Instead of eq 20 we would then have In ( h ,
- h) = In h,
-2
(;- + -;.)
to
(20’)
and the plots of In ( h , - h) vs. to, as that of Figure 5 , would have a slope -2(D/Lz v/L) which can be predicted from the known values of D, L, and v. For example, the slope of the plot in Figure 5 would be -0.074 s-l, whereas that found from the experimental plot is -5.65 X s-l, Le. 2 orders of magnitude smaller. The same applies to the slope of the plots for all other substances studied. Moreover, eq 20 predicts independence of the slope from carrier gas flow rate, while eq 20’ predicts a linear dependence of the slope on the flow rate. The experimentally determined slopes are independent of the flow rate and thus in agreement with eq 20 In conclusion, the assumption of equilibrium at the liquid-gas interphase leads to profound disagreement with the experimental evidence.
+
an equation which could also be obtained directly from eq 14 by assuming a large value for k,. Plots of the experimental data as vs. l/to, according to eq 14’, show that this equation In (hto1l2) does not fit the data so well as does eq 21 derived from eq 14. This is evidenced by the poorer linearity of the plots and the lower accuracy of diffusion coefficients calculated from their slopes. Coming to the longtime approximation of eq lo’, as we did for eq 10, we derive the relation
Acknowledgment. We thank Professors A. Jannussis and A. Payatakes for very helpful and stimulating discussions and also Anna Sinou-Karahaliou for assistance.
(16) Crank, J. “The Mathematics of Diffusion”;Oxford University Press: Oxford, 1956; p 34. (17) Reference 13, p 522.
Registry No. n-Pentane, 109-66-0; n-hexane, 110-54-3; n-heptane, 142-82-5; n-octane, 111-65-9; methanol, 67-56-1; ethanol, 64-17-5; propan-1-01, 71-23-8; butan-1-01, 71-36-3; pentan-1-01, 71-41-0; helium, 7440-59-7.
a
Molecular Complexes of Anionlc Porphyrin and Anionic Aromatics Toshinori Sato, Teiichiro Ogawa, Department of Molecular Science and Technology, Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816, Japan
and Koji Kano* Department of Applied Chemistry, Faculty of Engineering, Doshisha University, Kamikyo- ku, Kyoto 602, Japan (Received: November 16, 1983; In Final Form: February 22, 1984)
Absorption and,fluorescence emission spectra indicated that an anionic porphyrin, 5-phenyl-10,15,20-tris(p-sulfonatopheny1)porphine (TPPS’-), in water formed ground-state complexes with various anionic aromatics. The thermodynamic parameters (AH and AS) for complexation suggested that a van der Waals interaction is the predominant binding force and a hydrophobic interaction assists the stabilization of the complexes.
Introduction Water-soluble porphyrins tend to form molecular complexes in their ground states. Pasternack et al. have reported the spontaneous stacking of the anionic porphyrins such as 5phenyl- 10,15,20-tris(p-sulfonatophenyl)porphine(TPPS3-) and 5,10,15,20-tetrakis(p-carboxoxyphenyl)porphinein water containing inorganic salt.’ The same behavior was found for phthalocyaninetetrasulfonate, which forms a dimer with a binding constant of 6.2 X lo7 M-I at 26 O C . * Uroporphyrin, which has eight (1) Pasternack, R. F.; Huber, P. R.; Boyd, P.; Engasser, G.; Francesconi, L.; Gibbs, E.; Fasella, P.; Venturo, G. C.; Hinds, L. deC. J. Am. Chem. SOC. 1972, 94, 45 11. (2) Bernauer, K.; Fallab, S.Helu. Chim.Acta 1962, 45, 2487.
carboxylate groups, and its metal complexes associate with various organic cations and neutral bases in Mauzerall analyzed the electronic spectra of uroporphyrin free base in water in the presence of organic cations and bases and concluded that electrostatic and dispersive (or van der Waals) forces as well as hydrophobic interactions contribute to molecular complex formatione3 Quite recently, Shelnutt has demonstrated that charge-transfer interaction is a binding force for complexation of metal uroporphyrins with phenanthrolines on the basis of the results of laser Raman difference spectro~copy.~*~ Unfortunately, (3) Mauzerall, D. Biochemistry 1965, 4, 1801. (4) Shelnutt, J. A. J. Am. Chem. Sot. 1983, 105, 774. (5) Shelnutt, J. A. J . Phys. Chem. 1983, 87, 605.
0022-3654/84/2088-3678$01.50/0 0 1984 American Chemical Society
