J. Phys. Chem. 1981, 85, 2287-2291
2287
Kinetics of Conformation of Cryptand 222 in Water Herman Schnelder,* Sabine Rauh, and Sergio Petrucci” Department of Chemistry, Polytechnic Institute of New York, Brooklyn, New York 11201, and Max Planck Insfitute for Biophysical Chemistry, Goettingen. West Germany (Received: January 20, 198 1; In Final Form: March 30, 198 1)
Ultrasonic absorption data for aqueous cryptand 222 in the concentration range 0.15-0.40 M, temperature range 25-55 “C, and frequency range 1-300 MHz are reported. The ultrasonic spectra can be described by the sum of two Debye relaxation processes centered at 7 and 70 MHz at 25 ‘C, the relaxation frequencies being concentration independent. The data are interpreted by the scheme AI + Az +AB,namely, by the conformation changes involving three configurations of the cryptand. These are the so-called “out-out”, “out-in”, and “in-in” conformations and they correspond to the lone electron pairs of the nitrogen residing both outside, one inside, and both inside the bicyclic structure of the cryptand. Rate constants and activation parameters for this postulated process are calculated. It is concluded that the isomeric rearrangements of the cryptand (if of the same nature of the ones occurring during metal cations complexation) do not appear to be the rate-determining steps for the complexation reaction. Introduction Synthetic macrobicyclic ligands such as the cryptands, and monocyclic ligands, as crown ethers, have received considerable attention in recent years because of their remarkable affinity for metal cati0ns.l In fact, they are used as models of the corresponding natural cyclic ligands which are ubiquitous in nature as antibiotics (valinomycin, eniantin), heme, and chlorophyl. However, in order to understand the mechanism of metal complexation reactions, one must first understand the dynamics of the conformational change of the ligand. Whereas this study has been carried on for valinomycin in hexane-methanolz and crown ethers in water,3no study to the authors’ knowledge has been performed for the cryptands. This investigation was embarked on with the above aim in mind. Water was chosen as the first solvent because of its relevance to biological systems. Ultrasonic relaxation techniques of multiple nature (because of the broad spectrum to be covered) proved to be the effective technique for the present study. Experimental Section Materials. Cryptand 222 was obtained from Merck (Kryptofix 222, article no. 10647, E. Merck, Darmstadt) and was used without further purification. Water was twice distilled and used shortly thereafter. Solutions were prepared by weighing the cryptand directly into volumetric flasks. Equipment. The ultrasonic resonator, cell, and electronic assembly have been previously d e ~ c r i b e d . ~The temperature was measured directly in a metal block surrounding the cell and was maintained to within f0.01 “C. The frequency range covered was 1-30 MHz. The pulse interferometric cell and equipment have been described el~ewhere.~ The frequency range covered was 1-300 MHz. Results Figure l a shows the excess sound absorption coefficient per wavelength (a,,,X) vs. the frequency f (MHz) for 0.2 (1) Gordon A. Melson, Ed., “Coordination Chemistry of Macrocyclic Compounds”, Plenum Press, New York, 1979. (2) E. Grell, T. Funck, and F. Eggers in “Membranes”, G. Eiseman, Ed., Vol. 111, Marcel Dekker, New York, 1975. (3) G. W. Liesegang and E. M. Eyring in “Synthetic Multidentate Macrocyclic Compounds”, Academic Press, New York, 1978, and literature references contained therein. (4) A. Bonsen, W. Knoche, W. Berger, K. Giese, and S. Petrucci, Ber. Bunsenges. Phys. Chem., 82, 678 (1978). ( 5 ) (a) S. Petrucci, J . Phys. Chem., 71,1174 (1967); (b) G. S. Darbari, M. R. Richelson, and S. Petrucci, J. Chem. Phys., 53,859 (1970); (c) S. Onishi, H. Farber, and S. Petrucci, J . Phys. Chem., 84, 2922 (1980).
M cryptand 222 in water a t t = 25 “C. The circles correspond to data collected by the resonator technique according to the relation p = a,,X
“)
= A ( Af --
f
fo
where Af and Afo are the width a t half-power (-3 dB) of the resonance peaks, f and f o are the resonance frequencies a t peak maxima for the solution and the solvent, respectively (in practice f = fo). The squares in Figure l a are the values collected by the pulse technique according to the relation
where u is the sound velocity measured by an interferometric technique,6 a is the absorption coefficient (Np cm-l), B is the background absorption a t f >> f R I , f R I It should be noted that the pulse technique, being an a b s o k e method, gives the absorption coefficient a of the solution whereas the resonator technique, being a relative method, of the solution. By correcting the gives the value of Xa, a value by the quantity B f , the two methods give the series of point depicted in Figure 1. The solid line corresponds to the fitted function, the sum of two Debye relaxations:
where f R I and f R I l are the two relaxation frequencies, pml and pml the maximum sound absorption for wavelength at f = fRi. Table I reports the collected parameters according to eq I11 together with the values of B and the velocity u. Figure l b shows another representative plot of p vs. f for c = 0.15 M a t 45 “C. Calculations The relaxation frequencies did not appear to show any appreciable trend with concentration in the range studied. Even a t 0.1 M (although the precision was not adequate, the effect being too small), the ultrasonic spectra could be interpreted by similar values, within experimental error of the fR’s (--f5%). One can therefore infer that firstorder or pseduo-first-order processes best represent the data. Calculations based on the measured hydrolysis constants7 show only a minute proportion of the cryptand (6) Reference 5b.
0022-3654/81/2085-2287$01.25/00 1981 American Chemical Society
2288
The Journal of Physical Chemistry, Vol. 85,No. 15, 1981
Schneider et ai.
TABLE I: Relaxation Parameters and Sound Velocities for Cryptand 222 in Water"
t,"C
c, M
105k1
105Pn
25 25 25 25 25 35 45 55
0.40 0.30 0.20 0.17, 0.15 0.15 0.15 0.15
200 140 90 85 70 70 85 115
200 150 95 80 70 80 110 140
f ~M H~z , f ~M H~ z 75 6 75 6.5 70 7 70 7 70 7 80 9 90 12 100 18
10*7~, , cm-' sz 30 27 24.5 24 24 18.5 14.5 14
10-5u, cm/s
1.565 1.533 1.528 1.538 1.553 1.561 1.589
a c is the concentration (in mole/liter), and PI and p11 are the maximum excess sound absorptions per wavelenght at the relaxation frequencies f R I and f~~~(Mz), respectively. B is the background sound absorption of the solution at f >> f ~ f~~~ according to eq 11. u is the sound velocity (cm s-').
~
0 2 0 M (222) in woter, 25OC 140-
&A
120-1
l40 14 l3 0
L 31 32
3J 3 3.4
(1031~)
100
f(MHr)
-23
(b)
1
1
2
5
10
20
f(MHz.1
50
I
200 500
100
Figure 1. p vs. f (MHz) for cryptand 222 in water at (a) c = 0.2 M and t = 25 ' C and (b) c = 0.15 M and t = 45 OC.
to be in the protonated form. In order to investigate the possibility of a reaction between the cryptand and water to give the protonated species, we adjusted the pH of the 0.2 M solution from pH 12.40 to pH 11.20 by adding HC104, the presumption being of finding changes in the ultrasonic spectrum. No such changes are noticeable, the two spectra overlapping within experimental error. It is therefore concluded that the observed processes do not correspond to a solute-solvent reaction but rather to a molecular property of the cryptand. Because of the first-order nature of the processes it is suggested that this is due to the conformational changes out-out + out-in + in-in These symbolisms, of common use, correspond to the position of the lone pairs of the nitrogen atoms with respect to the bicyclic ring. The above reaction may be schematized as follows:
Figure 2. (a) Eyring plot of the quantity in (?-'/T) vs. (1/T) for the "fast" relaxation process according to eq VII. (b) Lamb plot of the quantity In (pr&T) vs. (11r ) for the "fast" relaxation process according to eq x.
be made is that itl, kl> kz,kzsince the two relaxation frequencies are separated by a factor of ten. From eq IV and the corresponding rate equations (see Appendix I, available as supplementary material), the two relaxation times (with the condition kl,kl > k2,kZ) may be given by q-l(fast) = kl
+
where
= k - J k l . For the fast process, one may write ?I-1 = kl k-1 = 27rfR1 = 4.4 x 108 s-1 (VI)
+
Further
and d In ( T - ~ / Z ' ) d(l/T)
where, naturally, no inference of the nature of Al or A3 may be drawn from the present data. In other words, Al could be either out-out or in-in. The only assignment that can (7) B. G.Cox, D. Knop, and H. Schneider, J. Am. Chem. SOC.,100, 6002 (1978).
-- d In ( k - l / T )
d In (+/Z') d(l/T)
d(l/Z')
AH-l*
R
+
K1d In K1 1 + Ki d ( l / T )
K1 aH1 (VIII) 1+K1 R
Figure 2a shows the data for In (?fl/T) vs. 1/T. Linear regression and statistical analysis applied to the data of Figure 2a gives the following: slope = -842 f 21, intercept
,
The Journal of Physical Chemistry, Vol. 85, No. 15, 198 1
Conformation of Cryptand 222 in Water
2289
l A ,
= 17.04 f 0.07, r2 = 0.999 (coefficient of determination).
Therefore
kn (Ti‘/T ) 12
Furthermore, for the fast process one may write for the excess sound absorption per wavelength at the relaxation frequency f = fR1
30
1- , -111
31
32
33
34
33
34
(1031~)
with 0,= 1/(pu2). This expression, using K1 = A2/A1 and Kz = A3/A2, may be rewritten as R
”=
(AVd)2 -E-AiAz -
R
% RT A l + A 2
20,
( A V d Z Ki R T K l + lA1
with AV,, the isoentropic volume change. However, from the mass law and mass conservation rule, one has
Al =
1
+ K 1 + K1K2
30
31
32
(1 0 3 1 )~
Figure 3. (a) Eyring plot of the quantity In ( T - ’ / T ) vs. ( l / T ) for the “slow” relaxation process according to eq XII. (b) Lamb plot of the quantity In T ) vs. (1/ T ) for the “slow” relaxation process according to eq XIV.
Equation X may be rewritten as
following: slope = d In ( q c l / T ) / d ( l / n = -2726 f 247, intercept = 21.00 f 0.79, rz = 0.988. For the slow process one may write for the maximum excess sound absorption per wavelength (at f = fRIr)
which gives (see Appendix 11),by neglecting the small temperature dependence of AV,, and of c(AV,, = AV,, -(8/pcp)AH1, with 8 the expansivity and cp the specific heat) d In (PIPST) dU/T)
1-162-
C
AHl[ I--- K1 R 1 + K1
1 + K1+ K1K2KlKz
I
+!!E!R
where
1 + KKIKz 1 +K1K2
(i m) 1
K1 1 + K 1 + KlKz
=
-l
+
(XI)
Figure 2b shows the plot of In (pI&T) vs. 1 / T for c = 0.15 M. Linear regression gives the following: slope = -1774 f 398; intercept = -1Sa6f l.3; r2 = 0.93. The solid line in Figure 2b has been drawn by using these parameters. Therefore, d In (bI0,T)/d(l/T) = -1774 f 398. From the intercept, AV,, = f33., f 11 cm3/mol. For the slow step and eq V one can write
(AVsl and AV,, are the volume changes associated with the single steps of eq IV; AV,, and AV,, are the corresponding parameters for the normal coordinates.) Then (see Appendix IV)
Further (see Appendix 111) d In (711-’/T) d(l/T)
-
d In
(PIIPST)
d(l’T)
1+1
+ K, (XIII)
Figure 3a depicts the plot of In (q-l/T) vs. (1/7‘). By applying linear regression to these data one obtains the
-
K1 Ml 2AV51 1 (1 K1)2 R Ab’,, 1 KlAVs1 2 K1 AH1 AH1 m R R 1+K, R
+ +
+
2290
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981 Maximum excess sound absorption per wavelength vs concentration for 2 - 2 - 2 cryptand in water at 25OC
Schneider et al.
mol, AS-l* = (-13.4 f 0.1) eu, and kl = 4.4 X lo8 s-'. Similarly for the slow process and eq XII, one can write
I
200 1
- 21.0 f 0.8 Then one calculates A K 2 * = (5.4f On5) kcal/mol, AS-; = (-5.5 f 1.,J eu, and k2= 4.4 X lo7 s-l.
C (rnole/drn3) Flgure 4. pW vs. the total concentration c for cryptand 222 in water at 25 "C: (A) pI = fast process; (0) pI, = slow process.
Figure 3b shows the data for In (pII&T) vs. 1/ T. By applying linear regression to these data, one obtains the following: slope = -2486 f 228, intercept = -17.2 f 7, r2 = 0.987. Further information may be obtained from the concentration dependence of pI and pn at t = 25 "C. Both eq X and XIV predict linear dependence of the p's with the total concentration. Figure 4 shows the correlation between pI,pII, and the total concentration. The values of pIand hn seem to align themselves on the same function with c, within experimental error. Linear regression, applied to these data, forcing the intercept through zero (50% statistical weight to the origin) gave a slope = 5.050 and r2 = 0.993. The solid line shown in Figure 4 is the result of this calculation. From the above expressions it is clear that, without knowledge of the equilibrium constants Kl and K2,it would be impossible to arrive at definite results. Specifically, eq VI, IX, XI, XII, XIII, XV, X, and XIV give eight numerical results based on the experimental data. On the other hand, the parameters necessary to characterize the system contained in these equations are kl, k+ k2, k+ AVl, AV2, AHl, AH2,AILl*, Since AV1 has been evaluated, this leaves eight results for nine parameters. In order to extract more information on the problem, we performed a computer simulation of the process based on the above eight equations and results, fixing the parameters K1 and K2 to arbitrary values. For K1 = K2 between 1 and 100 the computer gives negative values for A€Ll*. Changing the ratio between the Ks, namely, for Kl = 100, Kz = 50, K1 = 50, K2 = 100 leaves AK1* negative. For K1 = K 2 = 200 the activation parameters become positive but AV, becomes too large to be physically reasonable. On the other hand, for K1 = K 2 = 0.1 and K1 = K z = 0.2 all the activation parameters become positive and reasonable values of the AVs result. Changing the ratio between the Ks (namely, for K1 = 0.1 and K 2 = 0.01 or vice versa) alters somewhat the AVs. It would seem therefore that both K1 and K2 must be of the order of 0.1-0.2 or smaller, in order to obtain acceptable parameters. The above translates into saying that T i 1 N k-l and T~I-' k-2. But then, for the fast process and eq VII, one can write
and d In (T
