TRANSFER DIFFUSION
3303
Transfer Diffusion. 11. Kinetics of Electron Exchange Reaction between Ferrocene and Ferricinium Ion in Alcohols by I. Ruff,* V. J. Friedrich, K. Demeter, and K. Csillag Institute of Inorganic and Analytical Chemistry, L. Edtvds University, Budapest, Hungary
(Received January 18, 19Yl)
Publication costs borne completely by The Journal of Physical Chemistry
The rate of the electron exchange reaction between ferrocene and ferricinium ion has been determined by direct measurement of the diffusion coefficient of the ferricinium ions, increased due to transfer diffusion, in methanol, ethanol, and 1-propanol as solvents. The rate proved to be close to the diffusion limit near room temperature. The data are in good agreement with the isotopic exchange measurements at around -70" in methanol.
1. Introduction I n the previous part of this series, transfer diffusion has been theoretically discussed in detai1.l The physical background of this phenomenon is that the movement of a species due to its concentration gradient does not take place only by its migration, but in addition to this an apparent translation occurs owing to an exchange reaction, if the other reactant of this exchange is present. I n the case under discussion, this means, e.g., for ferricinium ion, that it gains the path of the iron-to-iron distance which should not be really moved along, when it collides with a ferrocene molecule and they exchange the electron. After the exchange, the newly formed ferricinium ion continues its translation from the point at which the ferrocene is situated. The apparent diffusion coefficient DAX'involves in this way the true diffusion coefficient DAXand a term corresponding to the electronic jump
/ where IC is the second-order rate constant of the exchange reaction, 6 is the distance between the center of the reactants in the activated complex, CA and CAX are the concentration of the reactants a t a distance x (supposing linear diffusion). The equation is formulated in a general way for any exchange reaction
+
+
AX A A AX (2) where X is the pointlike particle exchanged. I n the present case A should stand for ferrocene, AX for ferricinium, and X for the electron deficiency (hole) on the ferricinium ion, since the process has been followed by the diffusion measurement of the ferricinium ion. The determination of k can thus rely on the difference between DAX' and D A X ,if the second term in eq 1 ex-
ceeds the errors in measuring these two quantities in the presence and absence of ferrocene, respectively, for the saturation of alcohols with ferrocene does not allow higher concentration than 0.05 M , and the true diffusion coeficient of ferricinium proved to be about loM5 cm2 sec-l with a standard deviation of about 3% in the best case; the lower limit of observing any difference between the two diffusion coefficients gives
k 5 1.5 X lo9 M-l sec-l
(3)
in methanol. I n ethanol and propanol it is somewhat lower due to the lower true diffusion coeficients: about 6 X lo8 and 4 X 108, respectively. There was still some hope to get useful results, since in methanol the extrapolation of Stranks' isotopic exchange measurements2 a t -75, -70, and -65" to room temperature supported values high enough to fulfill eq 3. Concerning the general aspects of the knowledge of chemical exchange processes as fast as these figures, it is almost evident how useful a kinetic method can be giving comparable data for reactions which have been known as immeasurably fast ones so far. Thus the purpose of the present paper is to show the applicability of the transfer diffusion in the case of this particular reaction.
2. Experimental Section Materials. Ferrocene was purchased from the KochLicht Laboratories. It was purified by resublimation at 101". All the other chemicals were of analytical grade. Stock Solutions. HC104 (1 114) solutions have been obtained by diluting concentrated perchloric acid by the solvents used; 100 ml of the ferricinium solutions usually contained 5 ml from this stock solution, so the water content was sufficiently low. (1) I. Ruff and V. J. Friedrich, J . Phys. Chem., 75, 3297 (1971) (2) D. R. Stranks, Discuss. Faraday Soc., 29, 73 (1960).
The Journal of Physical Chemistry, VoL Y 6 , N o . 21v 1971
I. RUFF,V. J. FRIEDRICH, K. DEMETER, AND I(.CSILLAG
3304
A
s. E
Ql V
C
C
U
2
0
1
2
3
4
5
6
7
___3
c, mM
Figure 2. Calibration curve for ferricinium.
6
Figure 1. Apparatus for photographic measurement of diffusion coefficients.
The ferrocene solution was 0.05 M in methanol and 1-propanol and 0.04 M in ethanol. Ferricinium perchlorate solution was always freshly prepared by oxidation of 5 ml of ferrocene solution and 2.5 ml of 1 M perchloric acid by PbOz. After filtration this was diluted to 50 ml. Apparatus. Diffusion coeficient has been measured in a glass apparatus shown in Figure 1. Photographic evaluation has been used. Filter 3 (Schott GmbH. Type G4) permitted the sufficiently slow flow of the solution from tube 4 to tube 2 (flow rate 5-10 ml/hr). All three of the tubes could be thermostated. After filling tube 4 with the solution of higher density which was usually the ferricinium solution when measuring DAX or DAX’, while it was the ferrocene solution for the determination of DA,the solution was pressed through filter 3 by a rubber ball applied to the end of tube 4 that eliminatcd the bubbles under the filters. The solution above the filters mas then removed and replaced by the one into which the diffusion of the lower solution was wanted to proceed. After a period of thermostating, diffusion was started by opening stopcock 6. The communication of the air was enabled by the hole and slotted grindings 1 which could be closed by turning the stoppers. When the interface of the two solutions reached the middle of the illuminated tube, the stopcock was closed. The mercury lamp of a Pulfrichtype photometer served behind the tubes 2 to give the two illuminated circles 5, while the camera (Zeiss, Type The Journal of Physical Chemistry, Vol, 76, No, 21, 1971
C/
mM
.
5
Figure 3. Calibration curve for ferrocene.
Exacta Varex) was fixed in front of them. ORWO NP 15 film was used. If the blue solution of ferricinium was to be photographed, a red filter, eliminating the yellow color of ferrocene (also present in the case of the measurements of D’) , was applied. When ferrocene diffusion was measured, a violet filter gave better contrasts. The advantage of this method is that the diffusion can be followed by snaps taken at different time intervals, and it gives the possibility to start the diffusion from a very sharp interface; usually no convection could be seen, which was proved by the fact that the diffusion coefficient was really independent of the time of diffusion. Evaluation. The calibration curve (transmittance of the film us. concentration) had to be measured for both tubes, since there was some difference in the intensity of the light. For ferricinium and ferrocene solutions they are shown in Figures 2 and 3, respectively. Transmittance was determined by a Zeiss microphotometer by which it could be measured in each 0.1 mm of the film, which corresponded to 3.3 mm in the diffusion column.
3305
TRANSFER DIFFUSION
Figure 4. Calculated concentration distribution along the diffusion tube.
Figure 6. Dependence of Dt on the time.
Figure 5. Distribution of ferricinium in methanol at 25’ in different times of diffusion (I:03, 2 : 00, 4:02, and 6:OO hr in the order of decreasing slopes).
The slit width was 0.04 mm, so the transmittance could be measured within practically pointlike distances. Using the calibration curves, the concentration could be determined as a function of the height of the column. The plot c/co, where cg is the initial concentration of the species, us. the linear coordinate x was fitted t o a family of calculated curves (Figure 4), each curve corresponding to a certain value of Dt. These Dt values were then plotted us. the time t that should be theoretically a
linear function of t and cross the origin. If this intercept was really obtained, it proved the interface to have been sharp enough at t = 0. I n some runs the intercept on the time axis xas at negative values that meant a convection resulting in a diffusionlike concentration distribution when starting the diffusion. If the linearity still remained, diffusion coefficients were calculated by taking into account this additional apparent time as if diffusion had started earlier. I n Figures 5-10 some typical plots are shown to demonstrate these cases (Figures 6, 8, and 10 belong t o Figures 5 , 7 , and 9, respectively). Calculation of DhX’ differed somewhat from that described above. Considering that it may depend on x owing to it involves the ratio of the concentration gradient (see eq l),the graphical evaluation gave only a starting value to an iteration procedure as follows. The Journal of Physical Chemistry, Vol. 76, No. 91,1971
I. RUFF,V. J. FRIEDRICH, K. DEMETER, AND I(.CSILLAG
3306
1.0Q9.-
o&(II.-
U6.-
050.4--
03-
4
02-
-1.0
0
-0.5
w, c m
0
Figure 7. Distribution of ferricinium in ethanol a t 35" in different times of diffusion (I :00, 3: 30, and 5 : 00 hr in the order of decreasing slopes).
'I
0.u
- 0.5
c
0
0.5
X,
cm
Figure 9. Distribution of ferricinum in 1-propanol ab 35" in different times of diffusion (0: 32, 1 : 30, 2: 32, and 5 :30 hr in order of decreasing slopes).
1
Zt ) .
q
0.10-
0
"1' ' os
i
3
2
i
s+*
Figure 8. Dependence of Dt on the time.
If the condition
+
+
Figure 10. Dependence of Dt on t'he time.
is satisfied (where C A O and CAXO are the initial concentration of the reactants) which includes the assumption D A = D A X , the ratio of the gradients in equal to -1, Le. CA
CAX
= co =
CAO
CAX'
This can be taken as a first approach for calculating Using this value and its 75 and l25%, one can compute three theoretical curves for c / c o as a function of x. In doing this, the concentration gradients were calculated in the first iteration step as if no transfer
ICs2.
The Journal of Physical Chemistry, Vol. 76, N o . d l , 1071
diffusion occurred; in this way a better approach of DAX'at a certain x could be obtained, then the gradients were corrected by the new value of DAX' and the procedure was repeated until the same value of DAX'was twice obtained within 0.1 relative per cent. These curves were fitted again to the experimental points of the c/c" vs. x plot and the real lc62 was chosen by the best fit. The iteration was usually very short, within two steps the correct value was reached, and thus no appreciable influence of the ratio of concentration gradients could be obtained.
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TRANSFER DIFFUSION 3. Results and Discussion For ferricinium perchlorate the diffusion coefficients measured are summarized in Table I. It is seen that the errors range from 3 to 15% which can be considered as a satisfactory precision, since in the presence of ferrocene the increase in the diffusion coeficient is in general three times this error or more. Table I : Diffusion Coefficients of Ferricinium Perchlorate in the Presence and Absence of Ferrocene a t Different Temperatures Upper soln [HCIO4], [Fe(III)I, [Fe(II)l, [Fe(II)I, m A4 mM mM miM --Lower
t, OC
12.5 20.0
25.0
35.0
12.5 20.0 25.0 35.0
12.5 20.0 25.0 35.0
50 50 100 50 50 100 50 50 100 50 50 10 50 50
5 5 10 5 5 10 10 5 10 10 5 4.4 5 5
50 50 50 50 50 50 50 50
4 4 4 4 4 4 4 4
50 50 50 50 50 50 50 50
5 5 n
5 5 5 5 5
soln-
I n Methanol 0 0 45 50
9.17 0.87 12.0 i 1 . 4
0
0 40 0 0 0 35 40 42.5 45.6 0 45
I n Ethanol 0 36 0 36 0 36 0 36
0 O
a X 106, cmz sec-1
I
45
10.3 1.0.5 12.8 1 . 0 . 2 12.6 i 0 . 1
0
45 47.51 47.51 50 0 50
0 40 0 40 0 40 0 40
In 1-Propanol 0 0 45 50 0 0 45 50 0 0 45 50 0 0 45 50
which would meet the requirements have not been studied. The only test one can do is to measure the viscosity of the ferrocene solutions. If viscosity remains the same when adding ferrocene, the increase in the diffusion constant is by all means due to transfer diffusion. I n some viscosity measurements at 2 5 ” , exactly this was observed: 0.550, 0.655, and 1.95 cP was measured for pure methanol, ethanol, and l-propanol, while the values for the solutions were 0.558, 0.652, and 1.89, respectively. The temperature dependence of the true diffusion coefficient was calculated by least-squares method and resulted in +4.55 A 0.06, -0.25 A 0.5, and +8.22 f 0.08 kcal/mol for the energy of activation in methanol, ethanol, and 1-propanol, respectively. In Table 11, the diffusion coefficients of ferrocene are shown. The values of the energy of activation are 6.05 f 0.07, 0.88 =k 0.04, and 2.81 f 0.04 kcal/mol, respectively. The unusual behavior of the temperature dependence in ethanol is significant; its reason, however, is out of interest in the present work. (It should still be noticed that the solubility of ferrocene is somewhat less in ethanol which also shows an unusual behavior.)
16.8 i 0 . 5 16.4 k 0 . 6 16.0 i O . 9 15.4 i k 1 . 7 20.1 f 1.0
*
4.47 0.soa 5.24 i 0.40 4.66 i 0.08 6.98 f 0.58 4.28 i 0.50 7.33 i 1 . 0 4 . 3 3 =t0.56 5.91 i 0 . 2 2 0.94 i 0.20 1.24 i 0.15 1.63 =!= 0.02 2.58 f 0.21 2.08 =!= 0.06 3.64 f 0.09 2.73 f 0.02 3.89 1.0.09
The errors printed in italics do not arise from the scattering of the diffusion coefficients corresponding t o different diffusion times, but to the uncertainty of the intercept of the line Dt us. t. a
Table I1 : Diffusion Coefficients of Ferrocene t,
om%seo-1
15.0 20.0 25.0 35.0
In Methanol 50 50 50 50
15.3 f 0 . 7 16.7 f 2 . 4 23.5 i 2.0 29.5 i 3 . 5
15.0 20.0 35.0
In Ethanol 40 40 40
11.0 f 0.8 10.4 f 0.7 12.2 i 0.8
15.0 20.0 25.0 35.0
In 1-Propanol 50 50 50 50
6 . 4 8 i 0.19 7.45 i 0.38 8.61 f 0.41 9.00 i 0.41
From these data both the diffusion limit and the second-order rate constant of the reaction can be calculated. For the first quantity Smoluchowski’s equation3 was used ki’
To exclude the possibility that the increase in the diffusion constant in the presence of ferrocene is due to some other effect of changing the solvent structure, it would be preferred to repeat the measurements by using a system identical with that of ferrocene and ferricinium but slow enough t o give no transfer diffusion effect. Unfortunately such a system is not available, since the exchange processes of other metallocenes
bA x lo6,
[FeUI)I, mM
oc
=
+
~ T N A ~ ( D A ”D ~ x ” ) / 1 0 0 0
(4)
where N A is the Avogadro number, and (5)
DA” is a similar apparent diffusion coefficient as DAX” (3) M .V. Smoluchowski, 2. Phys. Cham. ( L e i p z i g ) , 92, 129 (1917).
The Journal of Physical Chemistry, Vol. 76, N o . $1, 1971
3308
I. RUFF, V. J. FRTEDRICH, K. DEMETER,AND K. CSILLAG
in eq 5 , but with the labels exchanged (cf. eq 38 in ref 1). In Table I11 the values of the diffusion limit are compared to the second-order rate constants measured. According to that the errors should always be additional quantities in both cases; while either the sum or the difference of the diffusion coefficients appear in kl‘ or k , respectively, the relative error is higher for the latter one. Thus, it can be concluded that the reaction takes place with a rate equal or very close to the diffusion limit.
Table I11 : Comparison of the Diffusion Limit and the Rate Constant Measured for the Ferrocene-Ferricinium Exchange kl’ t,
oc
10-8,
M - 1 seo-1
kd
x
M-1 sec-1
[HI +, mM
12.5 20 25 25 25 35 12.5 20 25 35
I n Ethanol 8 . 9 i 0.8b 5rk 3 8 . 4 =t 1 . 0 15i 4 10.1 i 1 . 1 c 19 It 10 8 . 8 i0 . 6 1oi 5
- 70
- 65
12.5 20 25 35
In 1-Propanol $3.3 4 . 8 rt 0.2b 4.8 4 1.2 5 . 4 i0 . 3 7 . 9 i0 . 8 6 . 6 i0 . 3 6.9 4 0.3 5.9 ==! 0 . 6
50 50 50 50
50 50 50
50
a See ref 2. b Extrapolated values. c Interpolated values. Calculated with 6 = 7.08 (see ref 2).
It seems t o be important to note that, on one hand, low true diffusion coefficients favor the application of the transfer diffusion method (see eq 1); on the other hand, the lower the diffusion coeficients the lower the diffusion limit. Thus, the range in k that could be studied a t all can be quite small due to a methodical lower limit and an objective upper one. This range in the cases under discussion is about one order of magnitude below the diff usion-controlled rate. Though the extrapolation of the diffusion coefficients to low temperature is somewhat uncertain, Stranks’ data for the rate constant in methanol are significantly less than the corresponding diffusion limit. I n this way the enthalpy and entropy of activation can also be calculated with satisfactory precision using No yes’ equation4 for the resultant rate L, when both diffusion and activation control is in operation The Journal of Physical Chemistry, Vol. 76,N O . $1, 1971
+
kl’/k2)
(6)
where kz is the rate constant of the activation controlled reaction. Substituting the temperature dependence of ICz according to Eyring, one has
(7) where k and h are the Boltzmann and Planck constants, respectively. Otherwise, when IC2 is written in the form ICz = 2 exp(-AF*/RT) where 2 is the number of collisions in unit time that can be taken as equal to the diffusion limit, the following equation can be obtained AF*
109,
In Methanol 0.1940.04b 0.000087i0.000022~ ? 0.25 st 0.04b 0.00017 4 0.00004a 1 0.35 =t 0.04b 0.00035 i 0.00018” ? 1 2 . 5 i 1.4b 14 i 11 50 15.8 i 1 . 3 13rk 4 50 2 0 . 3 i1 . 3 1 8 3 ~3 10 2 0 . 3 f 1.3 20h 3 50 20.3 i. 1 . 3 23i 3 100 19 i 14 27.1 i 1 . 4 50
- 75
d
x
k = Ll’/(l
- = R In T
fi - 1)
The term a t the left-hand side of eq 7 and 8 was calculated as a linear function of 1/T by a least-squares program. The activation parameters obtained are AH* = 14.8 =k 1.4 and AHx = 8.8 f 1.6 kcal/mol, AS* = 44 f 6 and ASx = 33 =k 9 eu in methanol. I t is to be noted that these values are very sensitive to the low temperature data of Stranks, while those measured by us do not influence their absolute value too much, but their errors. This is due t o that the Arrhenius plot cannot sweep in the large range Strank’s data would allow, but it must cross the range given by our points. Since the temperature range covered is quite large, the activation parameters can be given with a much smaller scattering than one would expect from the uncertainty of the rate constants. This is why no enthalpy and entropy of activation can be given for the reaction in ethanol and l-propanol which solvents were not studied at low temperatures. The thermodynamic parameters of the reaction seem, however, unreasonable, since one would expect values near zero for both the enthalpy and entropy of activation. This is due to that, on one hand, probably the structure of the complexes is the same in the oxidized and reduced form and, on the other hand, ferrocene is an uncharged molecule; thus neither Franck-Condon restriction nor Coulombic repulsion would appear. Some recent electrochemical measurements5 at low temperature support this expectation being in contradication with Stranks’ data. The main conclusion-in addition t o the proof of the existence of the transfer diffusion-is that the electron transfer reaction takes place through an activated complex in which the reactants are as close to each other as possible. This follows from the agreement of the diffusion limit with the rate constants measured near room temperature, for the diffusion limit is proportional to 6 while the transfer diffusion term in eq 1 de(4) R. M . Noyes, Progr. React. Kinet., l , 129 (1961). (6) I. Ruff, M. Zimonyi, and G. Farsang, to be published.
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CHARGE DENSITY ON THE PHOSPHORYL OXYGEN pends on The use of a smaller 6, if reasonable a t all, would result in that the diffusion limit is smaller than the rate observed. In this way, very important information about the mechanism of the reaction, namely the distance of the reactants in the activated complex, can be tested.
Acknowledgment. One of the authors (V. J. IF.) is very grateful to the Hungarian Academy of Sciences for the Graduate Fellowship awarded to him. The authors are very indebted to Dr. B. Rozsondai, who called their attention to the photographic method of the measurement of the diffusion coefficient.
The Charge Density on the Phosphoryl Oxygen in a Series of Phosphate Esters;
Tributyl Phosphate, a Monocyclic Phosphate, and a
Bicyclic Phosphate Ester1 by A. L. Mixon and W. R. Gilkerson* Department of Chemistry, University of South Carolina, Columbia, South Carolina
202'08
(Received April 1 I 1071)
Publication costs borne completely by The Journal of Physical Chemistry
The effects of adding a series of phosphate esters on the conductances of dilute solutions of piperidinium, N-methylpiperidinium, and N-ethylpiperidinium picrates in chlorobenzene at 25" have been measured. Cation-ligand association constants, KL, have been calculated for the N-methyl- and N-ethylpiperidinium cations with tributyl phosphate, n-octyl trimethylene phosphate, and l-oxo-4-ethyl-2,6,7-trioxa-l-phosphabicyclo [2.2.2]octane. The ratio of the cation-ligand association constant for the N-ethyl cation, &(Et) , to that for the N-methyl derivative, KL(Me),has been used as a probe of the electron density on the phosphoryl oxygen atoms in the esters, These results are compared with phosphoryl oxygen stretching frequencies, YPO, and with calculated (Huckel-MO) values of charge densities.
Introduction Bicyclic phosphate esters have been found2 to be poor extractants, compared to acyclic esters, for lanthanide metal ions. The same report contained the observation that these bicyclic esters had abnormally high phosphoryl oxygen stretching frequencies. Burger3 reported a linear relation between extractant ability and the phosphoryl oxygen stretching frequency, V P O : the lower the frequency, the better the extractant. Wagner4 carried out LCAO-MO calculations for a number of acyclic phosphoryl compounds and found an excellent correlation between the calculated phosphoryl PO a-bond orders and experimental values of phosphoryl oxygen stretching frequencies. The calculated values of the net charges on the phosphoryl oxygen also show excellent correlation with V P O . Presumably the extractant ability of a phosphoryl derivative is greater the greater the negative charge on the phosphoryl oxygen, and the larger the n-bond order of the phosphoryl bond, the smaller the net negative charge on oxygen and the larger the phosphoryl oxygen stretching frequency.
Recent Hucke 1-MO calculations by Collin6 indicate that the net negative charge on the phosphoryl oxygen decreases in the order ethylene phosphate anion > trimethylene phosphate anion > diethyl phosphate anion. Calculated values of the positive charge on phosphorus decrease more markedly in the same order. Boyde reported extended Huckel-XI0 calculations of both net charges on atoms and bond orders. Boyd's calculations indicate that net negative charge on the phosphoryl oxygen decreases in the order trimethyl phosphate > methyl ethylene phosphate. This is the reverse of the order given by Collin for the diester anions. The theoretical studies of both Collin and Boyd were concerned principally with the influence of the geo(1) This work has been subported in part by Grant GP 6949 from the National Science Foundation. (2) 8. G. Goodman and J. G. Verkade, Inorg. Chem., 5, 491 (1966). (3) (a) L. L. Burger, J . Phys. Chem., 62, 590 (1958); (b) J. L. Burdett and L. L. Burger, Can. J. Chem., 44, 111 (1966). (4) E. L. Wagner, J. Amer. Chem. Soc., 85, 161 (1963). ( 5 ) R. L. Collin, ibid., 88, 3281 (1966). (6) D. B. Boyd, ibid., 91, 1200 (1969).
The Journal of Physical Chemistry, Vol. 76,N o . 81, 1071
