DIELECTRIC RELAXATION DUETO CHEMICAL RATEPROCESSES
lock. The measured chemical shifts were identical. The temperature of the probe for the Kr determinations was 25.0 1.0" as measured from ethylene glycol calibrations. Equilibrium constants were calculated from eq 5 on the CDC 1604B computer of the Field Research Laboratory.
*
402 1
Acknowledgments. The authors wish to thank L. E. Nelson for operation of the HA-100 nmr spectrometer and C. H. Calvert for purifying the sulfide compounds. We are grateful to the staff of API Project 48 (Bartlesvile, Okla.) for the sample of 2,2,5,5-tetramethylthiacyclopentane and to Mobil Oil Corp. for permission to publish this work.
On Dielectric Relaxation Due to Chemical Rate Processes
by Gerhard Schwarz MasPlamk-Inatitut far physikaliache Chemie, G6ttingen, Germany,' and Institut f a r molekulare Biologie, Biochemie und Biophysik, StbckheirnlBraunachweig, Germany Accepted and Transmitted by The Faraday Society
(March 88, 1967)
The effect of chemical relaxation on the dielectric behavior of dipolar reaction systems is examined. When only small electric field densities are applied, the equilibrium of the over-all react,ion is not perturbed so that the static dielectric constant remains unchanged. Nevertheless, the chemical process may affect the dielectric relaxation by providing an alternate means of orienting dipoles. In principle, this should be measurable in cases where chemical relaxation proceeds at least with about the same rate as the rotation of dipoles. If the reaction rate is much faster than rotational diffusion, a distinct dielectric relaxation effect occurs which reflects directly the chemical relaxation process. Owing to the unfavorably large rotational diffusion coefficients of small molecules in solution, pertinent reaction systems can be expected, first of all, among those systems which involve macromolecular particles. Potential applications of the phenomenon aiming at the determination of chemical rate data are discussed, including even cases which originally could not be studied because of dipole rotation being too fast.
I. Introduction Dielectric polarization of an isotropic and homogeneous medium is described by the expression
P
=
M/V
=
EO(€
- l)E
(1)
relating the electric dipole moment per unit volume, P, to the electric field density, E, which induces it (M = over-all dipole moment, T' = volume, co = 8.854 X 10-14 f/cm). The quantity E represents the (relative) dielectric constant of the system. Owing to the finite rate of formation, the actual value of the dipole moment M will lag behind its equilibrium value, if the field
is changed fast enough. This is of particular significance for periodic fields of suffciently high frequencies. I n such cases a phase shift between M and E will occur, resulting in an energy absorption (dielectric loss). Also the amplitude of M will be changed. Using complex notation, this dielectric dispersion is adequately described by writing E = EOeiut(Eo = amplitude of the field, i = dw = angular frequency, t = time) and introducing a complex dielectric constant e =
E* = E'
- ie"
(2)
(1) Send inquiries t o the author a t this address.
Volume 71, Number 12 November 1967
4022
The (real) dielectric constant, e’, as well as the dielectric loss factor, e“, become functions of w within the dispersion range (for frequencies outside this region e’ is independent of w , while e” vanishes). Both quantities are accessible experimentally by means of electrical impedance measurements. The capacitive part yields e‘, the conductive one e”. Dielectric polarization may be the result of various effects. First of all, individual molecules can be polarized owing to slight deformations of their atomic and electronic charge distribution. This is generally brought about in extremely short times, corresponding to dielectric dispersion a t optical frequencies. Another important mechanism becomes effective if permanent molecular dipoles exist in the system. Then, an applied electric field tends to cause preferential orientation of the dipole axes-which are originally distributed at random-thus giving rise to an over-all nonvanishing dipole moment (orientational polarization). The rate of this process is ordinarily determined by the rotational diffusion of the molecules perpendicularly to their dipole axes. This results in dielectric relaxation at sufficiently high frequencies. The corresponding dispersion is observed at. microwave frequencies (=lo4 &IC) as far as small molecules in liquids are concerned. Since the time which is required to rotate a molecule increases rapidly with its length, orientational polarization in macromolecular systems may display relaxation at fairly low frequencies, e.g., at radiofrequencies as in the case of polypeptides2 or polynucleotides3 in solution. I n macromolecular systems considerable dielectric polarization and relaxation may occur also owing to polarization of the ionic atmosphere of polyions* or colloidal particles5 as well as to the effect of heterogeneities in the dielectric properties of the medium6 (Maxwell-Wagner effect). The respective relaxation frequencies are usually also rather low (kilocycle-megacycle range) ; they may decrease even below the kilocycle range for sufficiently large polarizable particles.’ Apart from the mentioned physical effects there is a potential chemical mechanism of dielectric polarization and relaxation. I t can be anticipated for systems where dielectric properties are changed during the course of a chemical reaction. The chemical equilibrium of such a process is shifted by an applied electric field as can be shown thermodynamically. Thus a field tends to change the dielectric polarization of the system via the initiated cheinical reaction. In this case the rate of dielectric polarization is, of course, determined by the respective chemical relaxation process. As an important consequence we conclude that an investigation of the corresponding dielectric dispersion can, in principle, be utilized to measure chemical relaxation times. The Journal of Physical Chemistry
GERHARD SCHWARZ
These quantities may be interpreted then in terms of the underlying chemical kinetics.* The fundamental potentiality of chemically induced dielectric relaxation and its inherent significance for the study of fast reactions in solution has been pointed out recently by Bergmann, Eigen, and De M a e ~ e r . ~ These authors directed attention also to a restrictive peculiarity of the phenomenon, namely, that any chemical contribution to the dielectric constant vanishes for E -+ 0. Thus it cannot be measured by means of the conventional impedance-measuring techniques which employ fairly low field densities. The effect should be measurable, however, in the presence of a suffciently high static electric field. By imposing such a condition upon a special system (200 kv/cm on the dimerization of 6-hexanolactam (e-caprolactam)) it has actually been observed experimentally and interpreted in terms of kinetic properties.s On fundamental grounds and in view of the comparatively difficult and inconvenient experimental problems involved in the application of high field densities, the question may arise whether a chemical reaction could nevertheless induce dielectric effects without a strong static field being present. This is indeed possible as is to be shown in this article. Our result does not contradict the above-mentioned conclusion provided we express it in the more precise form: “Small field densities do not perturb a chemical equilibrium sufficiently to produce a chemical contribution to the static dielectric constant.” It will turn out that actually not the dielectric polarization but the dielechric relaxation may be affected by a chemical reaction even for E + 0. In the above-mentioned experiment, strong fields were predicted to be absolutely necessary for any chemically induced dielectric effect. As a matter of fact, this rests upon the implicit assumption that the relaxation time of dipole rotation, T ~ is, small compared with the chemical relaxation time, T c h . Such an assumpt,ion is very well (2) (a) H. Watanabe, K. Yoshioka, and A. Wads, Biopolymers, 2, 91 (1964); (b) S. Takashima, ibid., 1, 171 (1963). (3) S. Takashima, J. Ph.ys. Chem., 70, 1372 (1966). (4) (a) G . Schwarz, 2. Physik. Chem. (Frankfurt), 19, 286 (1959): (b) M. Mandel and A. Jenard, Trans. Faraday SOC.,59, 2158, 2170 (1963). (5) (a) H. P. Schwan, G. Schwarz, J. Maczuk, and H. Pauly, J. Phys. Chem., 66, 2626 (1962); (b) G. Schwarz, ibid., 66, 2636 (1962). (6) (a) C. T. O’Konski, ibid., 64, 605 (1960): (b) H. Pauly and H. P.Schwan, 2. ,V&urforsch., 14b, 125 (1959). (7) H. P. Schwan, Advan. Biol. &led. Phys., 5 , 147 (1957). (8) (a) M. Eigen and L. De Maeyer in “Technique of Organic Chemistry,” S. L. Friess, E. S. Lewis, and A. Weissberger, Ed., 2nd ed, Vol. VIII, Part 2, Interscience Publishers, Inc., New York, N. Y., 1963, p 895; (b) G. Schwarz, Rev. Mod. Phys., in press. (9) K. Bergmann, M. Eigen, and L. De Maeyer, Ber. Bunsengee. Phyeik. Chem., 67, 819 (1963).
DIELECTRIC RELAXATION DUE TO CHEMICAL RATEPROCESSES
justified in that special case as well as for other small molecular dipoles which can freely rotate in a liquid ( ~ > ~ 10-lo h sec >> T~ = lo-” sec). That explains, of course, why actually no chemical effect has been observed so far at low fields. Nevertheless, there might be cases where the reverse relation T~ >> T& holds true. Under that condition a low-field dielectric dispersion caused by a chemical reaction can, in principle, occur. This may apply especially to macromolcular reaction systems (among them many of biological significance). There the rotational diffusion could well proceed more slowly than a chemical relaxation process. Furthermore, by manipulating systems this behavior might be achieved deliberately in order to utilize the resulting dielectric relaxation effect for kinetic measurements. In the following the problem is treated in more detail from a quantitative point of view. At first we shall establish the thermodynamic relations describing the effect of an electric field on a chemical equilibrium. Then a model reaction will be discussed which is especially suited to demonst,rate clearly the fundamental aspects of the phenomenon under consideration. This will clarify under which circumstances a chemical mechanism of dielectric relaxation at low fields can be expected. Finally a reaction process of more practical inberest, is to be investigated with regard to potential experimental applications.
11. Thermodynamic Foundation
4023
vidual reaction partners. The condition of chemical equilibrium is given by A = 0. Conventionally p i is expressed in terms of activity coefficients,f f , and concentrations, ci
+ RT In ft + RT In ci
(7) The standard chemical potentials p? depend on T , p , E (and on the nature of the soIvent), but not on {. Using (7) eq 6 may be transformed to p i = pi0
A/RT
= In
K*
-
cvtIn ci
(8)
i
with In K* = In K
K
= expi
-
zvtIn f f
(9)
i
-CvfpP/RT]
where K is the thermodynamic equilibrium constant. According to (8) we have at equilibrium In I(* =
CvfIn Ei
(11) (equilibrium quantities are always to be denoted by a bar). Since we are interested in the change of the equilibrium concentrations by an applied field, the effect of E on K* (= “apparent equilibrium constant”) will be investigated now. On account of the fact that the order of differentiation is immaterial when computing- second derivatives of G, eq 5 leads a t once to
The general chemical reaction process
+
~l’R1’ vZ’RZ’
+
.. e n’%”
n”R2” -l. . . (3)
may proceed in a solution which is assumed to be a closed and dielectrically isotropic system. Any changes in the numbers of moles, ni,of the reaction partners R’ and R” are determined then by
The quantity AM represents the molar change of the over-all dipole moment M which is produced by reaction 3 (Proceeding from the left to the right) while keeping the intensive variables T,p , and E constant. Inserting A from (8) into (12) yields
(4)
(13)
6ni = vi6{
where vi equals the respective stoichiometric coefficient, v”, for any R” species, but is taken as the negative v’ value for the reactants R’. The quantity 4 is introduced as the extent of reaction. The Gibbs free energy, G, of the system is to be considered a function of T (absolute temperature), p (pressure), E , and [. Its total differential can be expressed by the well-known relationlo dG = -SdT -k V d p - MdE ( S = entropy). The affinity
- Ad‘
‘s)
is related to the chemical potentials pf of all the indi-
provided the concentrations are measured in molalities (nf/lOOO g of solvent). The use of molarities (ni/lOOO ml of solution) requires, strictly speaking, to take into account the effect of E on V (electrostriction). This is done by adding the term -(l/V)(bV/bE)T,,,ECvi on
the right side of (13). However, even for
ts 1.
we may generally neglect it in those cases we are interested in, Le., reactions with an appreciable A M a t E + 0. (IO) Cf.! e.g., A. H. Wilson, “Thermodynamics and Statistical Mechanics,” Cambridge University Press, London, 1957.
Volume 7 1 , d-umber f.8 November 1967
4024
GERHARD SCHWARZ
I n order to describe the perturbation of equilibrium caused by the field E we differentiate K * for A = 0 instead of keeping 5 constant
(y*) (3) T,p,E
bE
The second term on the right side vanishes because A = 0, while the first one may be expressed in terms of M by means of (14)
T,p,A-O
By using the relation
This latter relation is readily established when comT S - pV using the puting second derivatives of G exact differential
+
d(G
as well as eq 8, 9, 12, and 13 we finally obtain
b In K*
+ TS - pV) = TdS
AM
(7 = ) R T
- pdV
- MdE - AdE (21)
(Equation 21 is immediately obtained from eq 5 . ) Now, a procedure which is analogous to that used for deriving (16) and (17) may be employed to evaluate the final form of (18), namely
fT,p
T,P
with
s,v
= fs.v
-(-)
1 bM RT at
S,V.E
(22)
with
b In K* Since (b In K * / ~ ( ) T , ~ , E = - ( b Z v i In f i / b E ) T , p , E , one sees immediately that f T , p = 1 if the activity coefficients are not changed during the reaction. This is frequently the case in practical work (e.g., if the f i are essentially determined by an excess of a nonreacting electrolyte). I n principle, eq 16 may not apply to the actual conditions of the reaction system. Chemical rate processes which produce measurable contributions to dielectric relaxation would be very fast. Hence they can be expected to proceed a t constant S and V rather than a t constant T and p. Let us therefore evaluate the pertinent quantity
b In K* ( b 6 ) S . V
-1
Obviously this result differs from (16) and (17) only by the fact that all of the derivatives with respect to 5 are taken a t constant S and V instead of taking them at constant T and p . Both kinds of derivatives can, of course, be related to each other. For instance, we have
b In K* =
( b e ) S , V , A - O
(18)
The variations of T and p due to t’he reaction (under the condition S , V = constant) can be expressed by measurable thermodynamic quantities. Using wellknown methods of evaluationll (compare also ref sa) leads to
where K* is considered a function of S, V , E , and 5. Because of (8) we may write
(11) Cf.,e.g., H. Margenau and G. M. Murphy, “The Mathematics of Physics and Chemistry,” D. Van Nostrand Co., Inc., Princeton, N. J.. 1943.
3 In K *
(y) (”) S,V,E
The Journal of Physical Chemiatry
bE
~,V,A-O
DIELECTRIC RELAXATION DUETO CHEMICAL RATEPROCESSES
4025
let us consider a special model reaction which is as uncomplicated as possible but nevertheless discloses clearly the basic aspects of our problem. The unimolecular reaction process Ai (26)
(AH and AV == molar change of enthalpy and volume, respectively, due to the reaction at constant T, p , E; p = density; CY = thermal expansion coefficient, H = compressibility, cp = specific heat, all taken a t constant values of the respective intensive variables; c V = specific heat a t constant volume; a , H , cp, and cv do not include contributions of the chemical reaction.) Thus-under adiabatic and isochoric conditions-a field E may affect a chemical equilibrium even for AM = 0. This will occur if M (or the dielectric constant of the system, respectively) depends on T or p , while AH or AV does not vanish. However, these effects can usually be neglected for systems with a AM of appreciable magnitude. Application of (24) to Zvi In ci and In K*, respectively, instead of M yields practically no difference ~ for cases where AH, AV, between f T , p and f s , except and the concentrations of the reaction partners are extremely large. VC’e may finally conclude that for many reaction processes of interest with a AM of considerable magnitude and no change of activity coefficients the simplc relation b In K*
AM
(27)
-
IC012 A
Az
(28)
k”*
turns out to be well suited for this purpose. The quantities kn12and kQzl denote the rate constants for the forward and reverse reactions, respectively. They are related to the equilibrium concentrations EO1 and COZ and by the wellthe “apparent equilibrium constant” K*o known relation
(29) We assume the rotational diffusion coefficients of both species AI and Az to be equal (= D,) ; furthermore Az may have a permanent molecular dipole moment p , while Ai has none. Actually such reactions might be encountered-to some approximation-even in practical work, e.g., among rotational isomerizations or proton transfer in zwitterions. I n the molecules of A2 an axis is defined by the direction of the dipole p. This axis is to be conserved for the transition Az + AI. Thus also an axis for the molecules of A1 is introduced. There will be a distribution of the axes with respect to the direction of an applied field E. The concentrations of these molecules which have axes pointing at the time t into the solid angle dfl = 27r sin e de (e being the angle between E and a molecular axis) can be expressed as dc1 = yl(B, t)dQ, dcz = y?(B,t)dQ
(30)
holds true as far as practically meaningful accuracy is concerned. Because of the general relation (1) we may expect that in ordinary reactions bM/bt is proportional to E , no matter what the thermodynamic conditions are. Hence b In K*/b,$-.t 0 for E --.t 0; i.e., the chemical equilibrium will not be perturbed at very low field densities. Consequently no chemically induced dielectric polarization due to additionally produced dipoles can occur. The situation will be different in this respect at sufficient high field densities. However, as is to be shown in the next section, the chemical process contributes, in principle, to the dielectric properties of the reaction system also a t low fields. This does not contradict the preceding conclusion because only the dielectric relaxation is affected, while the static dielectric constant remains unaltered.
The distribution functions y1 and y2 (=concentrations per solid angle) may change on-ing to the chemical reaction process (28), but also on account of rotational diffusion and-in the case of Az-because of the momentum of force exerted by the field. This is described by the two partial differential equations
111. A Simple Model Process
It should be emphasized that-owing to the perturbation of equilibrium by the field-the rate constants klz
For the sake of simplifying the mathematical effort,
- = v12 bt
+ D,V2y2 - div j ,
The effect of the reaction rate is represented by chem
kizyi
chem
-
kziyz = kzi(K*yi
- yz)
(32)
Volume 7 1 , Number 13 Nonember 1967
GERHARD SCHWARZ
4026
and ktl as well as the corresponding K*depend on E and 8. For E = 0 they are, of course, identical with kolz, ~ O Z I , or PO, respectively. The term DrV2y,(i = 1, 2) in (31a, b) results from rotational diffusion according to Fick’s law. Application of the Laplace operator V 2 yields in our case vzy, =
-- {sine%}
(33)
sin 0 de
Finally the angular flux of axes directed by the field E can be introduced as
Dr
j , = -y2--pEr
kT
sin
e
pEr
kT
1 b -{ y 2 sin2 e] sine be
(35)
(36) with
corresponding to equilibrium a t E = 0. Any terms of second or higher order in E , will be neglected. Now, inserting (36) into the rate term v12 according to (32) leads to pEr
= -{ k0iz@i
kT
= F I ( ~cos ) 8, @,E, = F z ( t ) cos e
- koa@z] + k02172-
(with FI and F2 being independent of e), we finally transform (31a, b) to a system of two ordinary and linear differential equations for F1 and Fz
dFi- -{ dt
koiz
+ 2DrjK +
kOziFz
- koz1y2Er
(41a)
b I n F dE,
There are two cases of degeneracy. One of them concerns Dr = 0, i.e., no rotational diffusion. It is described by only one relaxation time, namely, the chemical relaxation time pertaining t,o process 28, Tch, which is obtained according to 1
-=
k012
To h
d In I(*- N A ~ C O e S-~- p COS e RT kT bE,
+
(42)
k021
On the other hand, if there is no chemical rate process (ko12 = kO21 = 0) but a finite D,, a relaxation time due to rotational diffusion
(38)
(43) is found (of course, two rotational relaxation times would exist in the more general case of different D, values for A1 and A,). I n any event, however, the degeneracy will be destroyed if both the chemical as well as the rotational effect must be taken into account. I n this case the characteristic equation of (41a, b) yields, in principle, two different’ eigenvalues. Their negative reciprocals represent two relaxation times, 7 1 and rZ. As is readily determined, we have 11 = T ~ while for 7 2 the relation (44) holds true. The final solution in terms of y functions is obtained by means of standard procedures. For a periodic field of angular frequency w , the function yz takes the form
}
Suppose the conditions of eq 27 apply here; then (39)
(NA = Avogadro% number) since the molar change of the dipole moment due to the reaction 28 equals The Journal of Physical Chemistry
(40)
1
Fortunately it will not be necessary to find the general solution of (31a, b). Since pE,/kT de = e o A e * ~ (47)
4027
of the molecular dipoles, no matter whether there is a chemical reaction or not. CaseII. T r = Tch, Le., rotational diffusion and chemical relaxation proceed with comparable rates. TJnder these circumstances T~ and T Z are different. However, since the difference is not large enough to distinguish them clearly from the measured dispersion curves (this requires rr 2 107&), merely a flattening of the curves will occur. Thus a direct determination of the individual relaxation times is not possible. Nevertheless their effect on the dielectric constant and the dielectric loss might be measurable provided sufficiently sensitive experimental techniques are available. For instance, at the high-frequency limit of the dispersion range we obtain
- _
lim WAC" -- Axm -1 - l-+ (
After evaluation it follows A t * = cp(w)Ae0 = 1
A€'
A€:
+
iWTr +
1
A€*'
+
iwT2
(48)
with
€oA€'
T*
+ +
+
(n = refractive index of Az) provided the static dielectric constant of the system, E, is essentially determined by the solvent and its dispersion occurs at frequencies >> 1/r2, Otherwise gr should be smaller (gr + 1 in the limit), although this is generally difficult to ascertain quantitatively. Splitting Ae* into its real and imaginary parts enables us to determine the increments of the dielectric constant, A d , and the respective dielectric loss, A d ' (according to (2)). Apparently, the dielectric relaxation behavior depends very much on the order of magnitude ) from each other. Let by which rr and r 2 ( 5T ~ differ us consider the three possible cases. Case I . r , > ~ ) Tch >> T ~ ( * ) ; i.e., the rotational dispersion ranges of the macromolecules (at lower frequencies) and the small molecules (at higher frequencies) must be sufficiently far off the intermediate range of chemical relaxation. Macromolecules with a T,'") below sec are readily available. Sufficiently rigid binding of small reacting groups to such large carriers might well be encountered in certain systems of interest or could be, in principle, achieved deliberately by appropriate procedures. Thus the method would be adequate for dipole reactions with relaxation frequencies in the megacycle range (ie., Tch = 10-6-10-9 see). I n order to form an idea of the magnitude of the effect, eq 72 may be evaluated numerically for a special case with = 0.5, MD = 3 D., co = 1 M , and gr = 1. The result is Aecho = 0.15. For solvents with a dielectric constant e 5 10 we find for the corresponding loss angle Ae,h"/e' = 10-*-10-'. This is far above the limit of measurability, at least as far as (nonaqueous) solutions with little electrolytic conductivity are concerned. I n the case of cooperative reactions even considerably larger effects can be expected (cf. ref 13). ~
Acknowledgments. Stimulating discussions with Professor 31. Eigen and Dr. L. de Maeyer are gratefully acknowledged. The author is also indebted to Mrs. M. Barnes for proofreading the manuscript.
