J. Phys. Chem. 1987, 91, 3854-3862
3854
Thus, the Monte Carlo techniques may find wide use in describing van der Waals complexes, energy-transfer "complexes", and transition-state properties a t and above the critical energies for reaction, especially when the reaction is studied over wide energy and temperature ranges.
Conclusions A general Monte Carlo technique for high-dimensional integrals is described and applied to calculations of sums of states for nonseparable anhannonic molecules. The method can be applied to sampling domains with complicated boundaries, and it has an efficiency that can equal unity in realistic applications. When applied to the spectroscopically determined properties of real molecules, the results show that the calculational technique gives accurate results and relative standard deviations comparable with
those obtained by a modern stratified sampling method, which is much more complicated to implement. The sums of states for the nonseparable vibrations of H20 and CHzO are significantly different from those for the separable vibrations. Harmonic vibrations are probably acceptable for empirically fitting RRKM models to experimental data, but theoretical calculations of rate constants and lifetimes must properly account for the nonseparable characteristics of the potential energy surface. The Monte Carlo technique described here is suitable for such calculations.
Acknowledgment. Conversations with William L. Hase and William R. Martin are gratefully acknowledged. This work was funded, in part, by the Department of Energy, Office of Basic Energy Sciences, and by the Army Research Office.
Mechanism of Complexation of Na+ with N-Pivot-Lariat 15-Crown-5 Ethers in Methanol at 25 O C Luis Echegoyen, George W. Gokel, Min Sook Kim, Department of Chemistry, University of Miami, Coral Gables, Florida 331 24
Edward M. Eyring,* Department of Chemistry, University of Utah, Salt Lake City, Utah 841 12
and Sergio Petrucci Department of Chemistry, Polytechnic University, Long Island Center, Farmingdale, New York 1 1 735 (Received: August 22, 1986; In Final Form: January 22, 1987)
Ultrasonic relaxation absorption spectra for NaC104 + methoxyethoxyethylmonoazo 15-crown-5ether (RN 15C5) in methanol at 25 OC, in the frequency range 0.5-400 MHz and concentration range 0.1-0.3 M, are reported. The data are interpreted by the two-step scheme: Na+ + RNI 5C5 F?. Na+, RN15C5 a (NafRN15C5), where the intermediate represents a complex species with Na+ residing outside the crown ether ring. The role of the side chain has also been studied by ultrasonic relaxation techniques with the crown ether monoazomethyl 15C5 (MeNI 5C5) (having only a -CH3 appendage) + NaCIO4 in methanol. Two concentration-dependent, Debye relaxation processes are observed with this crown ether. This suggests that the role of the side chain is only marginal in determining the presence of the upper frequency relaxation process. An isomeric single Debye process whose relaxation frequency is independent of concentration is observed for both crown ethers dissolved in methanol (at comparable frequencies with those reported) when Na+ is also present. 15C5 containing only carbon and oxygen atoms in the ring shows no relaxation process in methanol at comparable concentrations and at 25 OC. This suggests that the upper relaxation process is associated with the inversion of the lone electron pair of the nitrogen (rather than with the rearrangement of the side chain). A molecular analogy between RN15C5 crown ether and 221 cryptand is drawn. As a further test of the supposition that the upper relaxation of both RN15C5 and MeN15C5 when reacting with Na' is due to nitrogen inversion, the isomeric relaxation of 221 cryptand in methanol has been studied. Two Debye relaxations appear, probably associated with the inversions of both the nitrogens of 221 cryptands, in analogy with a previous interpretation of the ultrasonic relaxation of 222 cryptand in water by Schneider et al. With Na+ reacting with 221 a double Debye relaxation also appears, consistent with the above hypothesis. The role of the side chain of RNI5C5, when compared with MeN15C5 reacting with Na+ in methanol, appears mainly to be that of enhancing the stability of the final complex, with Na+ embedded in the cavity of the crown ether possibly coordinated also to the side chain. The influence of the cation on the two relaxations observed in methanolic solutions of NH4C104plus MeN15C5 is also reported. Surprisingly, AgClO., + MeN15C5 shows only a single relaxation at a frequency comparable to that attributed to the isomeric process, although the relaxation amplitude (maximum excess sound absorption per wavelength) is different from that of the isomeric process.
Introduction The kinetic study of the complexation of alkali metal cations such as Na+ with crown ethers containing side chains with polar groups is relevant to the biochemistry of cation transport through membranes. Such model systems may provide useful analogues of the surface of cells containing cyclic antibiotics (such as valinomycin) binding to Na+ and . 'K The side chain may be competing as a binding site for the cations, or it may be particpating in the cation capture in the step preceeding the encapsulation of 0022-3654/87/2091-3854$01 .50/0
the cation by the ligand cavity. The present paper reports an investigation of the mechanism of complexation of Na+ with the N-pivot-lariat crown ether methoxyethoxyethylmonoazo 15C5 ( R N 15C5). Methanol was selected as the solvent because of a previous'V2 equilibrium study by NMR. We have also investigated the methylmonoazo 15C5 (1) Echegoyen, L.; Kaifer, A.; Durst, H.; Shultz, R. A,; Dishong, D. M.; Goli, D. M.; Gokel, G. W. J . Am. Chem. SOC.1984, 106, 5100.
0 1987 American Chemical Society
Complexation of 15C5 Ethers with Methanol
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3855
+
NaCIO, 0.188M R- N 1 5 C 5 0.18h.l In MeOH : t = ? j o C
-u 0 3
60 100
200 f(MHr)--
500
t
u =1.133\ 105cn1S-’ B =34a 1 0 - ’ ~ c l n - ’ s ~
100
0 0.1
0.2
0.5
1
2
5 f(MHr)-
10
20
50
100
200
500
Figure 1. Excess sound absorption per wavelength p vs. frequencyffor NaCIO4 (0.19 M) + RN15C5 (methoxyethoxyethylmonoazo 15C5) (0.18 M) in MeOH at t = 25 O C . The solid line is the sum of two Debye relaxation processes. Inset shows a/f vs.fshowing the tail of the relaxation processes for the same system. ether (MeN15C5) with Na+ in methanol as well as the isomeric relaxation of the two crown ethers dissolved in methanol. A molecular analogy with 221 cryptand induced us to study the isomeric relaxation of the 221 ligand in methanol. This gave rise to the hypothesis that the first step of the complexation of both N-pivot-lariat crown ethers with Na+ is associated with the inversion of the lone pair of electrons of the nitrogen of the ring, rather than with the interaction of the side chain with the metal cation. As a further check on the proposed mechanism, ultrasonic spectra of NH4C104 MeN15C5 in methanol at 25 OC have been obtained and reported. Dramatic evidence of the influence of the cation on the relaxation frequency of the fast process is provided. This proves the influence of the cation upon the fast relaxation process, which is not due to the ligand alone. A single sequence of measurements reporting the ultrasonic spectrum of AgC104 + MeNl5C5 shows an absence of the slower relaxation process. Possible reasons for this phenomenon ate discussed below.
+
Experimental Section Equipment and procedures for the ultrasonic work have been reported previously3and in the references therein. The only change has been to automate the data capturing by a Tektronix 2465 oscilloscope possessing cursors to record the voltage of the first ultrasonic pulse-echo. These voltages were transferred to an 85B Hewlett Packard computer via a GPIB bus in the form of In (voltage) for step positions of the Mitutoyb Series 164 digital micrometer3 corresponding to increasing distances between the piezoelectric crystals in a dual-crystal interferometric sample cell. The set positions of the digital micrometers were also recorded via a BCD interface by the same computer. The digitized data at each frequency are subjected to linear regression by the same data-capturing program giving the attenuation coefficient ct from decibels vs. distances linear regression. The precision obtained in the attenuation constant a remains at about 1% in the frequency (2) Shultz, R. A.; White, B. D.; Dishong, D. M.; Arnold, K.A,; Gokel,G. W. J . Am. Chem. SOC.1985, 107, 6659. (3) Delsignore, M.; Farber, H.; Petrucci, S. J. Phys. Chem. 1986,90,66.
range 50-300 MHz4 and decays to about 2% in the frequency ranges 300-500 and 10-30 MHz. The RN15C5 and MeN15C5 were synthesized at the Utliversity of Miami. Purity tests were made by comparing highresolution N M R spectra with literature spectra. NaC104 (ACS reagent) was redried in vacuo (- 1 Torr) at t CT 70 OC in volumetric flasks. The NH4C104and AgClO, (G. F. Smith Co., Columbus, O H ) were redried in vacuo at room temperature for 24 h. The 221 cryptand was a Merck product which was used as received. Methanol (Fisher, ACS) was distilled over an aluminum amalgam prepared in situ by adding HgCI2 to freshly machined A1 turnings. The solutions were prepared by weighing NaClO,, NH4C104, or AgC104 directly in the volumetric flask, adding the crown ether by weight, and bringing to volume by adding freshly distilled methanol. The addition of the crown ether through a volume dispenser took no more than 20-60 s which corresponds to the time of contact with the open atmosphere. Addition of methanol and, after ultrasonic stirring of t h t solution to homogenize the liquid, final dropwise addition of methanol to the volume mark did not allow more than another 30 s of contact with the open atmosphere. Solutions were used within 1 day of their preparation.
Results Figure 1 is a representative plot of the ultrasonic relaxation spectrum of the system NaC10, + RN15C5 in methanol at 25 OC. The spectrum is depicted in the form of the excess sound absorption per wavelength p = ( a - Bp)u/fvs. the frequencyf. In this expression, cy is the sound attenuation coefficient expressed A, frequencies above the relaxation in N p cm-I, B = (ct/f’)b,,,for region, u denotes the sound velocity, h the wavelength, and tHus h = u / f The solid lime corresponds to the sum of two Debye single relaxation processes centered a t the frequencies fI and ji,,respectively (dashed lines). The solid line corresponds to the fudction
(4) Petrucci, S. J . P h p . Chem. 1967, 71, 1174. Petrucci, S.; Battistini, M . J . Phys. Chem. 1967, 71, 1181.
3856 The Journal of Physical Chemistry, Vol. 91, No. 14, 1987
Echegoyen et al.
TABLE I: Calculated Ultrasonic Parameters pI, fI,pH, fII,E , and Sound Velocity u for NaCIO, NH., An) in MeOH"
CNPCIO~,M
cRNI5C5,
0.286 0.180
0.282 0.188 0.140 0.08,
0.140
0.093
CNiCIO,,
C M ~ N I SMC S ~ 0.360 0.292 0.187 o.109
0.358 0.292 0.182 0.10~
M,
0. I 5 4
0.301 0.204 0.10,
C M ~ N I SMC ~ ,
0.206
A, MHz
0.212
Pll
110 100 100 70
x 105
MHz
fil,
30 20 15 PI1 x
IOS
AI, MHz
225 190 140 100
fi, MHz
x 105
110
X
MHz
fir,
15
x 105 340 260 130
A, MHz
PII x 105
12 10 8
410 315 225
x 105 108
fr, MHz
x 105
B
X
3.6 3.2 2.0 1.5
30
Pll
B
4.5 4 3 2
45
45 40 30 30
85
0.297 0.20, 0.10,
cAgC104,
x 105
x 105
CM~NISCS. M
cNH4C104*
h, MHz
70 80 70 30
Cm. M
C N . ~ LM . 0.134
x 105 205 135 85 50
+ RN15CS and MC104 + MeNlSCS (M = Na,
fir,
MHz
u
lot7,cm-' sz 42 38
u
x
x
B x ioi7,cm-l sz 33
u
B
u
1 .O
0.9
B
60
lOI7,cm-l s2 40.5 36.5 34.5
cm-l
X
s2
cm s-I 1.146 1.158 1.129 1.111
32.5
X
cm s-l 1.152 1.132 1.137 1.125
353
1.1
ji1, MHz
lOI7, cm-l s2 37 34 34 33$
x
cm s-l 1.127
x
cm s-I 1.148 1.150 1.121
u X
36.5
cm s-I 1.127
'Same parameters for NaCIQ4 + 221 cryptand in MeOH; f = 25 OC. TABLE II: Calculated Ultrasonic Parameters fit, f,, E , and Sound Velocity u for RN15C5, MeN15C5, and 221 Cryptand in MeOH at t = 25 'C
witb No Salt Present macrocycle ~macrocyclel M RN 15C5 MeN 15C5 22 1
x 105
0.20 0.20 0.15 30(
55 75 90
A, MHz 80 30 60
PI1 x
10'
AI, MHz
45
20
B
cm-I s2 33.5 33 35
X lOI7,
u X
cm
s-I
1.127 1.116 1.123
+
NsC10, 0.29,M Monoazomethyl 15C5 0.29,M In MeOH : t =25OC p, =80~10-~
I, =40MHz pII = N O X
t
n
0
10'~
Ill =3.2MHz
20(
P
x
n =38 x 10- "cm-
' s2
u =1.158x 105cms-'
i
f(MHr)Figure 2. Excess sound absorption per wavelength fi vs. frequencyffor NaClO, (0.29 M) + MeN15C5 (monoazomethyl 1 5 0 ) (0.29 M) in MeOH at t = 25 O C . The solid line is the sum of the two Debye relaxation processes.
where pI and pIIare the maxima in sound absorption per wavelength for the two relaxation p r o c y e s centered a t the relaxation frequenciesf; andfil. Table I reports all the calculated parameters pIrfirpIl,hI, B, and the sound velocity u for the concentrations investigated at t = 25 O C . Figure 2 is a representative plot of p vs. f for NaC104 MeN15C5 in methanol, a system investigated to probe the effect of the side chain of RN15C5, as discussed below. Although MeN15C5 has no side chain except for a -CH,appendage, two relaxations are readily discernible as depicted by the two dashed
+
lines of Figure 2. Table I reports all the calculated parameters the sound velocity u for the concentrations investigated at t = 25 "C. Figures 3 and 4 are the ultrasonic spectra of 0.20 M RNI 5C5 and 0.20 M MeN15C5 dissolved in methanol at 25 'C with no sodium ion present. These spectra are diagnostic for the molecular mechanism of the complexation of Na+ R N 15C5 as discussed below. The calculated parameters for the observed single Debye relaxation, namely pI,fi, B, and the sound velocity u are reported in Table 11. pI,fi, pI1,hl, 8, and
+
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3851
Complexation of 15C5 Ethers with Methanol
300-
O.2Oh.I In MeOH ; t =25'C
t
zx
p m = 5 5 ~1 0 - ~
I, =80MHz
200-
B =33.5~10-~~cn1-~~*
u =1.127x 105cms-'
i
100-
0 0.1
0.2
0.5
1
5 flMHzf-
2
10
I
I
I
20
50
100
I .
200
I
500
Figure 3. Excess sound absorption per wavelength fi vs. frequencyffor RN15C5 (0.20 M) in MeOH at t = 25 OC. The solid line represents a single Debye relaxation process.
0.20M In MeOH ; t =25'C
p m = 7 5 ~1 0 - ~
I
t
100
0 0.1
0.2
0.5
1
2
5 10 f(MHz)--c
20
50
100
200
500
Figure 4. Excess sound absorption per wavelength p vs. frequencyffor MeN15C5 (0.20 M) in MeOH at t = 25 OC. The solid line represents a single Debye relaxation process.
Because of the molecular analogy between RN15C5 and the cryptand 221, an ultrasonic spectrum of the latter compound in methanol has been obtained at c = 0.1 5 M and is reported (Figure 5 ) . Two Debye relaxations describe the spectrum (solid line) better than a single Debye description (dashed-pointed line) of the same spectrum. Figure 6 reports the ultrasonic spectrum of NaC104 221 cryptand (0.15 M) in methanol that is also describable by two relaxation processes. The calculated parameters are collected in Table I for the cryptate and in Table I1 for the cryptand. Figure 7 reports the ultrasonic spectrum of NH,CI04 MeN15C5 (0.20 M) in methanol at 25 OC. Two Debye relaxation
+
+
curves adequateIy describe the relaxation process. Whereas fir is of the same order of magnitude as that for NaC104 MeNl5C5,fr shows a remarkable cation dependence, the value offr for NH4C104 + MeN15C5 being about four times smaller than for the sodium ion system. Even more surprising is the finding for AgC104 + MeN15C5 (0.20 M) in MeOH at 25 OC. Here the lower relaxation process is missing (Figure 8), a single Debye curve being sufficient to describe the ultrasonic relaxation process. Table I reports the calculated parameters pIrfi,pII,fiI, B, and the sound velocity u for the concentrations of NH4C104 MeNl5C5 and of AgC104 MeNl5C5 in MeOH that have been investigated.
+
+
+
3858 The Journal of Physical Chemistry, Vol. 91, No. 14, 1987
Echegoyen et al.
221 C r y p t a n d 0.15M In MeOH : t =25OC
- - - Slngle Debye relaxatlon
300-
p m = 1 ? 8 x 1 0 - ~.4 : = 4 ~ . ~ \ 1; 0 - ' ~ c m - ' s * I, =50MHz u =( 1.123
0.004) S l0'cm 5
B = 35 x io- I7cm-
t
z 200-
S?
Debye process
-Double
p , =00x10-~ ; A , =?O.i\
X
L O I'
f l =OOhlHz
d
p l l = 1 5 Y lo-' ; A, =40.1
\
10
I,, =20MHZ
'
u =(I 1?7~000I)\l0~'Ctll. B =35u 10-~'cm-'s*
100
0 0.1
0.2
1
0.5
2
5 f(MHr)-
10
50
20
100
500
200
Figure 5. Excess sound absorption per wavelength p vs. frequencyffor 221 cryptand (0.15 M) in MeOH at t = 25 "C. The solid line is the sum of two Debye relaxation processes. The dashed-pointed line corresponds to a single Debye process.
Calculations and Discussion Complexation of Na+ with Lariat Crown Ethers. We first propose that the process of the complexation of Na+ and R N l 5 C 5 be described by the reaction ~ c h e m e : ~ . ~ kl
ki
k-t
k-i
Na+ + RNI5C5 eNa+, RN15C5 e(Na+RNISCS) (2)
where the intermediate symbolizes a complex species, with Na+ residing outside the crown ether ring, and the final product a complex species, with Na+ embedded in the ring cavity. The above scheme for two loosely coupled steps leads to the relations: ~ 1 - l= 7II-I
(2wfi) = k-l
= ( 2 ~ f i I )= k-2
+ kl(2ac)
(3)
+ kzKl(2~~)
(4)
with KI = kl/k-l, the equilibrium constant of the first step, K2 = kz/k+ the equilibrium constant of the second step, and Kz = XI(1 Kz) the overall equilibrium constant. Kz has been reportedZ to be K, = 3.47 X lo4 M-' for Na+ R N l 5 C 5 in methanol a t 25 O C . Then, the degree of dissociation a is readily calculated as K, = ( 1 - o)/uzc. Figure 9 reports eq 3 and 4 applied to the present data. Linear
+
+
regression gives k l , k-', and k2KIhence kz. Then from the calculated k*, (the intercept of 711' vs. 2ac is about zero), one obtains k..2. These results are reported in Table 111. Reitctjon scheme 2 does not specify, however, the role of the side chain, if any, io the complexation process that is customarily called the Eigen-Winkler mechanism.' In order to investigate this possible role we have eliminated most of the tail by using MeN 15C5. Surprisingly, two relaxations persist, thus relegating the side chain of RN15C5 to a subordinate role in the complexation. Under the hypothesis of the applicability of the same reaction scheme 2, Figure 10 has been constructed. Fi8ure 10 reports the 71-land si{' vs. ( 2 4 data for NaCIO, + MeNl5C5, ( 5 ) Chen, C.; Wallace, W.; Eyring, E. M.;Petrucci, S.J . Phys. Chem. 1984, 88, 2541. ( 6 ) Wallace, W.; Chen, C.; Eyring, E. M.; Petrucci, S. J . Phys. Chem. 1985, 89, 1357. (7) Eigen, M.;Winkler, R. In Neuroscience: Second Study Program, Schmitt, F. O., Ed.; Rockefeller University: New York, 1970.
TABLE 111: Forward and Reverse Rate Constants and Equilibrium Constant for tbe Two Steps for Complexation of Na+ with RN15C5 and of Na+ and NH,' with MeNlSC5 at 25 OC in MeOH according to Reaction Scheme 2 RN15C5 macrocycle MeN 15C5 macrocycle NaC104 Electrolyte k , = 9.0 X 1OIo M-' s-l k , = 9., x 109 M-1 s-1 k-, = 2., x 108 s-1 k-, = 5.9 x 107 s-1 K , = 429 M-l K , = I54 M-I k2 = i., x 107 s - ~ k2 = 5.9 x 106 s-1 k-, = x io5 S-I k-, = 3 + X IO5 s-l K2 = 80 K2 = 14.9 K~ = 3.47 x 104 M - ~ R K , = 2455 M-'" NH4CI04Electrolyte k l = 2.3 x 109 M-1 s-I k-, I., x 107 S-1 K , = 192 M-I k2 = 8.4 X IO6 s-I k-, = l . , X IO6 K , = I., K z = 1660 M-' a "Shultz, R. A. et al. J . Am. Chem. SOC.1985, 107, 6659.
TABLE I V Distribution of the Concentrations of the Species in the Equilibrium Na' Cl
+ XNl5C5 s Na', Cl
XNl5C5 e (Na'XN15C5)
c2
c3
C, M CI, M C2, M CP,M Macrocycle RN15C5 (X = R = CH30CH2CH20CH2CH2) 0.282 0.00284 0.00346 0.216 0.180 0.00226 0.00219 0.176 0.00 199 0.00170 0.136 0.140 0.087 0.0015 0.00106 0.084
0.358 0.292 0.182 0.107
Macrocycle MeN15C5 (X = Me = CH,) 0.0119 0.0218 0.324 0.0107 0.0176 0.264 0.0084 0.0109 0.163 0.0064 0.0063 1 0.0943
The reported',* value K, = 2455 M-I has been used to calculate u. Table I11 reports the calculated values of k , , k-,, k2, and k+ Table IV reports the distribution of the concentrations of cI ( = Na', or crown ether), c2 ( = Na+,RNI5C5 or Na+,MeNISCS),
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3859
Complexation of 15C5 Ethers with Methanol
I
B =33x i ~ - ~ ’ c m - ~ s *
100
u =1.1?7x 105cms-‘
0 0.:
2
1
0.5
0.2
5
10
50
20
t(MHz)-
100
500
200
+
Figure 6. Excess sound absorption per wavelength p vs. frequencyffor NaClO, (0.13 M) 221 cryptand (0.15 M) in MeOH at t = 25 OC. The solid line corresponds to the sum of two Debye relaxation processes. The dashed line corresponds to a single Debye process.
I/
,-\
\
,,003
\
\
!I
\
/
\
/
\
LD
2
.*-5
.-.A*..
\
/ \
200
0 =36.5x10-”crn-’s2
/ \
/
\ /
4
I \ /
’
\
\
/
\
\
01
0.5
I
I
I
1
2
5
I
I
lo f(MHz)
, --
--
-
-1-
J
-
Figure 7. Excess sound absorption per wavelength p vs. frequencyffor NH4C104(0.20 M) + MeN15C5 (0.20 M) in MeOH at t = 25 OC. The solid line corresponds to the sum of two Debye relaxation processes. The dashed line corresponds to a single Debye process.
and cj (= (Na+RNlSCS) or (Na+MeNlSC5)), calculated with the relations: c = c, c2 cj
+ +
K1 = c 2 / c 1 2
It also appears that the kl for Na+ + RNl5C5 is of the order of magnitude of a diffusion-controlled rate constant. The Smoluchowky equationEfor the rate constant expressing the molecular encounter between neutral particles is
(5)
= c3/c2 The conclusion drawn from Tables I11 and IV is that the role of the side in RN1scsappears to be that Of enhancing the stability of the complex by greatly increasing c j over c2 and cl. Kl three times larger and K 2 five times larger for RN15C5 reacting with Na+ in with to MeN15C5‘ kl is one order of magnitude larger for R N 15CS with respect to MeNISCS. The longer side chain is not producing the-upper relaxation since both ligands show an upper relaxation process.
kD
K2
E
8rLkT - 3.8
30009 where in methanol
X
1Olo M-l s-I
= 298.2 = o.oos445 This figure for kD is of the same order of magnitude as the experimental value kl = 9 X 1OloM-’ s-’. If one were to include a dipole ion potential interaction ui,d = -(ep e/&) in the diffusion expression leading to kD (as done by Debye9 for the corresponding ion-ion interaction at
(8) Von Smoluchowsky, M. 2.Phys. Chem. 1917, 92, 129.
3860 The Journal of Physical Chemistry, Vol. 91, No. 14, 1987
Echegoyen et al.
AgC104 0.21M+MeN15C5 0.21M in MeOH, t=25’C
- Single Debye 20(
- - - MacrocycleMeN15C5 0.20M MeOH a t t=25’C pm= 6 5 ~ 1 6 ~
p L m1=0 e X 1 0 - ~ f,
I
In
0 I-
in
f r = 5OMH.Z
=6OMHZ
B =36.5~10-’~cm-’s~
B = 3 3 x 1 0 - 1 7 c m- 1 s2
u = 1 . 1 2 7 ~ 1 0cm~ s-’
u = 1 . 1 3 8 ~ 1 0cm~ s-’
X
a
t
1 O(
0
2
10
5
20
50
-
100
200
500
f(MHz) Figure 8. Excess sound absorption per wavelength p vs. frequencyffor AgCIOp (0.20 M) + MeN15C5 (0.20 M) in MeOH at t = 25 OC. The solid line corresponds to a single Debye process. The dashed line reproduces the isomeric relaxation process for MeN15C5 (0.20 M) in MeOH. 7 -
7;’= k-,
+ kl(2uC)
6-
t
I
5-
r
2- 4 ‘E
r
‘c”
3-
c3
“Fast,”@ and ’ slow’, @ relaxation processes according to the two steps scheme:
‘“7
Na++
‘Fast,“@ and slow”, Q relaxation processes according to the two steps scheme:
O
+ Na+
d
I
0 0
1
2
3 4 (2uC)xl@,M-
5
6
7
Figure 9. T
