April, 1959
VELOCITY FIELD IN ELECTROLYTIC SOLUTIONS
expressions.40 In all probability both factors made a significant contribution. Doty and Steiner22 have found further that in systems displaying a non-random distribution of scattering centers, external interference should result in negative dissymmetries in light scattering. Calculations, however, have shown that in the case of STA, the last would be significant only in the very low concentration range, at which the excess turbidity is very small. Attempts a t such measurements have met with failure so far, as the experimental error mas found to be greater than the expected difference in scattering at angles of 45 and 135O. Light Scattering from Other Heteropoly Acids.Kerker, Lee and C h o ~ have * ~ reported light scattering data for the 9 and 12 phosphotungstic acids which seem to exhibit the same extreme dependence of H ( C Z / T )on C2. In the absence of salt, the apparent molecular weight of the 12 acid was found to be 35% lower than the formula weight, while for the 9 acid it was 25% lower than the monomer weight. In the case of the 12 acid, the failure of H ( C 2 / 7 ) to decrease sharply with Ce or even to deviate markedly from linearity is probably (40) It is interesting to point out that caloulation of k from a light soattering expression involving the Verwey-Overbeek potential“ also results in a discrepancy. In this oase, however, the intercept yields a value of k much larger than that obtained from the slope. (41) E. J. W. Verwey and J. T. a. Overbeek. “Theory of the Stability of Lyophobic Colloida,” Elsevier Publishing Co., Inc., Ameterdam, 1948. (42) M. Kerker, D. Lce and A. Chou, J . Am. Chem. Boc., 80, 1539 (1958).
633
due to the fact that the measurements were not carried to concentrations sufficiently low to detect the strong curvature. As Fig. 2 indicates, at a concentration of 20 g./l., which is comparable to the lowest concentration employed by Kerker, Lee and Chou, H(C2/r) for STA is still linear in C2. In the case of the 9 acid, the slight curvature that Kerker, Lee and Chou observed a t low values of Czis consistent with our results and the expectations from the theory.22 Quantitative comparisons of our results with those obtained for the phosphotungstic acids is impossible since the latter measurements do not extend to sufficiently low concentrations for the use of expressions such as eq. 5. Conclusion The above-described light scattering measurements on silicotungstic acid can be considered as good experimental evidence for the non-random arrangement in solutions of macro-ions under conditions of low screening. Although no rigorous quantitative treatment of the experimental data is available a t the present time, the theory of Doty and Steiner can be used as a first approach for the analysis of such systems. It yields good agreement with experimental data within the limits of the assumptions in its derivation. A more exact theoretical analysis of the situation would seem desirable, and work to that effect has been initiated. Acknowledgment.-The authors wish to thank Professor S. J. Singer for suggesting this problem and Dr. B. A. Brice for his encouragement in the course of these studies.
THE VELOCITY FIELD IN ELECTROLYTIC SOLUTIONS BYRAYMOND M. FUOSS~ Contributionfrom I’Istituto d i Chimica Fisica dell’ Universitd di Roma, Rome, Italy Received October 84,1968
The velocity field produced in an electrolytic solution by an external electrical field has been studied further. Both the hydrodynamic term in the relaxation field and the Onsager electrophoresisterm can be derived from the Navier-Stokes equation, specialized to the conductance problem. It is shown that the hydrodynamic radius which ap ears in the analysis must be identified with a, the center-to-center distance at contact between ions of opposite charge. the ions are represented as spheres of diameter a, then the hydrodynamic radius R is twice the electrostatic radius; this result follows from the facts that no charee except that of the reference ion can penetrate a sphere of radius a around the latter and that information concerning size can only be detected conductimetrically from the consequences of interionic contacts.
8
In a recent2 derivation of the limiting curvature of the conductance function, it was necessary to calculate the velocity field which an ion sets up in the surrounding solvent when an external electrical field causes the ion to move. If the ions are assumed to have non-zero volume, the simplest model which can be used is that of spheres-in-continuum. For this model, an external field will cause the 11ion,7 to move with uniform velocity; “solvent” will be pushed away in the direction of motion in front of the moving ion and will pour in to fill the space vacated by it as it moves. In a plane through the center of the sphere, perpendicular to the field direction, no radial displacement of the solvent will (1) On sabbatical leave from Yale University. aeoond aemester 1957-1958. Grateful acknowledgment ie made for a Fulbright grant. (2) R. M. Fuoss and L. Onaager, Tam Joumw,, 81, 688 (1957).
occur, while solvent just ahead or behind the ion on the line of the diameter parallel to the field will move parallel to the field with the ionic velocity. In other directions, the velocity of the solvent will be a vector quantity with components in both radial and field directions. A nearby ion will therefore be moving, not in a stationary medium, but in a medium which is in motion. The problem therefore may be stated: given, the above model, what is the velocity of the solvent at a point P(r, e) as a function of distance T from the reference ion and the angle 8 between P , the reference ion and the field direction? This problem was considered by Debye and Huckel* in their first treatment of the conductance problem; their solution was stated in terms of the (3) P. Debye snd E. HPokel, Pliyrik. Z., X I . 305 (1923).
RAYMOND M. Fuoss
634
Stokes radius R which entered through the boundary condition that the relative velocity of the solvent be zero at the surface of the ion. By means of a different approach to the problem, Fuoss and Onsager2 succeeded in eliminating the Stokes radius from all but a small term in the relaxation field; the electrostatic diameter a appeared everywhere else in their conductance equation as the distance parameter which necessarily must be introduced when the next approximation beyond point changes is used to represent the ions. In order to avoid two arbitrary distance parameters, a and R , Fuoss and Onsager replaced R by a/2, arguing that the term in which it appeared was at most only one-twelfth of the radial component of total velocity and that the whole velocity term AXv in the relaxation field was itself small compared to the other higher terms, and therefore ,that any error made in replacing the hydrodynamic radius by the electrostatic radius would be insignificant. This arbitrary solution of the problem of two parameters by simply ignoring it is aesthetically unappealing, however satisfactory it may be from the practical point of view of numerical calculation. The previous derivation of the conductance equation also contains another unsatisfying element: the electrophoresis term (which is by far the largest of the interionic force terms), as derived originally was simply grafted on to by Onsager and FUOSS,~ the relaxation term in order to obtain the final result. A more direct approach would have been to derive both of the hydrodynamic terms (electrophoresis and AX") directly from the velocity field. It is the purpose of this paper to present a more thorough study of the velocity field. As expected, both hydrodynamic terms are derivable from the NavierStokes equation. An unexpected by-product is the fact that, as far as conductance is concerned, the hydrodynamic radius turns out to be equal to a, the center-to-center distance of anion and cation at contact if we require that the velocity of the reference ion at vanishing concentration be given by Stokes law
Vol. 63 ug
= (Xei/Cr)(
- r/2 + E/r)
(4)
The symbols are defined as follows: g is viscosity, X is field strength, e is elementary charge, K is the Debye-Huckel parameter, u is an auxiliary vector from which the total velocity v in the liquid at a distance T from the reference ion can be computed by (l), and UA and US are the contributions to u from the atmosphere and charge, respectively, of the referen e ion. The constants B and E were found2 to h e the values
A
B = (1 f Ka
+ K'Ua/2 + K8Us/6)/Kz(1f Ka)
(5)
and E
-R"6
(6)
The constant B was evaluated by requiring that the first derivative of UA be continuous at T = a, Le., at the surface of the sphere which never may contain any charge except that of the reference ion. The constant E was evaluated by the condition that the velocity of the liquid relative to the ion must vanish a t T = R; in terms of the spherein-continuum model, this simply means that the liquid wets the sphere perfectly. The necessity for the appearance of two distance parameters, one of electrostatic nature and one of hydrodynamic nature, is thus inherent in the problem. As we shall see later, the limiting form of the velocity equation for vanishing concentration requires that a equal R. Substitution of (4) in (1) evaluates vs
A
and this result, combined with (3), gives for the total velocity
At r = a, (8) reduces to
vo = X e / 6 q R
Reflection shows that this result is consistent with the model and with the conductance phenomenon: the only way that we can tell experimentally that ions are not point charges is by observing the consequences of interionic collisions, and it obviously takes two ions to make a collision. Hence only an average distance can appear. In order to save space, we shall take as given equations 5.5, 5.15, 5.19 and 5.21 of ref. 2, which are ,respectively vv =
UA
+
V(V.U)
- AU
(1)
The x-component of (9) can be rearranged to give The concentration dependent term of (10) is immediately recognized as the electrophoresis term in the velocity AV,(U)
=
- XeK/6m(l + KU)
(11)
A t concentration equal to zero, (10) reduces to
= (Xei/4~)[r/2 B/r-
which must be the velocity which the field imparts to an isolated ion because (12) is the velocity in the liquid at T = a and must simultaneously be the velocity of the ion of diameter a contained in the sphere of radius a. If we assume that this velocity is given by the Stokes equation @*(a)
and (4)
L. Onaager and R. M.Fuosrr, Tam JOURNAL,$6, 2689 (1032).
Xe/6wR
(13)
then equating (12)and (13) leads to a relation be-
a
SOLUTIONS VELOCITY FIELD IN ELECTROLYTIC
April, 1959 tween R and a R3
+ 3aBR - 4a3 = 0
(14)
which can only be satisfied if R = a
vr
are equal if and only if a1 and as are equal; that is, the condition that a be a concentration-independent parameter leads to the result that the model must be one for which
( 15)
Finally, substitution of R = a in the radial term of ( 9 ) gives (a) = 0
(16)
635
a = al = a2
(24)
As a matter of fact, the concentration dependence of u, as defined by (20), actually is slight for most practical cases. If we let a1 = 2a2,O < I < 1; then
ie., the radial component of v(a) vanishes, as it should. The result (15) might be interpreted as follows. As long as an ion is free (not in contact with another ion), electrostatics can tell us nothing about its size. For the idealized model of a sphere in a continuum, hydrodynamics can give us a relation between size and single ion conductance; since, however, Walden's product is known to vary even for a given electrolyte in mixtures of two solvent^,^ the radius calculated in this way can only be an approximation. In the conductance problem, thiqdifficulty is irrelevant for symmetrical electrolytes, because individual ionic radii are not needed; only the center-to-center distance at contact appears in the equations. Hence it is not surprising that, as far as the velocity term in the relaxation field is concerned, the average value a alone appears. Stated alternatively, it makes no difference what the individual radii are before (or after) contact, and at contact only the sum can appear. But the sum (al a2)/2is the center-tocenter distance. The parameter a entered the theory by explicit use of the Debye-Huckel potential
+
+
Vi(?)= (e-"Tej/Dr)[exp(Kaj)/(l xaj)]
(17)
for ions of non-zero size in the equation of continuity. Actually ai in (17) was replaced by a in the derivation. Let us recall the definition of a! in terms of al and a2. Formal theory gives for the activity coefficient of an ion of speciesj
+ Kaf)
- In fi
= @'c1/i/2(1
+ Ka)-' + ( 1 f .a%)-']
= (6'C'/2/2)[(1
(19)
must be replaced by 2 / ( 1 + ~ a ) , whereby a is perforce defined, and so defined, is a function of concentration. Algebraic rearrangement of the expression 2/(1
+
Ka)
= [(l
+
xa)-1
+ (1 +
KQd-11
(20)
gives
al + as - 2a = K(aal + a h - 2 a 1 4 (21) If we want a value of a which is independent of concentration, (21) must reduce to the identity 0 = 0, which will be true only if simultaneously al
and a(al
+
@
=
2a
+ ad = 2a1a2
(22) (23)
Eq. 22 states that u is the arithmetic mean am of al and u2, and substitution of (22) in (23) gives the result that a must also be the geometric mean (alaz)'/* of the two diameters. These two means (6)
H.Sadek and R. M.Fuoas, J . Am. Chsm. Soc., 79, 801 (1960);
76, 6897, 6902, 6906 (1964).
-
vr = ( X e cos 1 9 / 2 ~ ) [ p ~ex(a-t)
(1
where p3
=1
+
f Ka)] (26)
KT)]/K*Ta(l
+ + ~*a'/3
(27)
KU
ThicJ differs from the previous equation (5.26) by the appearance of K2a2/3in place of in the constant term in the square bracket in (26). Integration of the differential equation then gives explicitly AXdX = [ a W / 6 w ( l KaI4(o1 odlh -
+ + + od (13 + 3d2)/48Pz + F ( ~ a ) / 2 +
~ ~ a / 3 6 1 n l p ~ (Kl ~ ) ( W I
where
(18)
But since single ion activities cannot be measured, the quantity in square brackets in (19)
- lnf,,
and the function multiplying a, in (25) is not at all sensitive to K. For K = 0, a always equals a,, of course; for the extreme case z = 1/10 = al/a2, ala, = 0.965 for ~a~ = 0.1 and for the absurdly high value Kaz = 1.0, a l a m = 0.762. For the cases of real interest where approximately a2>al> 0.3 aa and K U < ~ 0.2, the difference between a and the arithmetic mean of ul and a2is experimentally undetectable and hence not a suitable subject for discussion. Finally, we consider the consequences of using (15) instead of the original guess R = a / 2 in AX,. Using R = a in (8), the radial component of the velocity, which is needed for the evaluation of AX,, becomes
r7
(28) (29)
and F(Ka) is the transcendental function which appeared in the previous analysis. Approximating F(Ka) as before, the velocity term in the relaxation field is found to reduce to Ax,/x =
- [r*ab/6.ml(o1+ os)][0.2543 + 0.25 In Ka + 1/6b1 - ~H/6q(w1+ (30) ~
2
)
As before, this hydrodynamic term is now combined with the electrophoresis term, because (30) shows that they have similar coefficients. The mobility of an ion of species 1 is given by UI
dleiloi
- leil~/6d+ l ~ a )( ]1 -k f l / x )
(31)
where u = 1/299.79 is the factor which converts electrostatic units of field strength to volts/cm. If Ah denotes the equivalent conductance, corrected for electrophoresis and the hydrodynamic term (30), &e., ignoring temporarily all of the relaxation field except the term AX,, (31) and a similar expression for species 2 leads to Ah
no - ( u F e r / 3 w ) [ ( l + Ka)-'
+ H/2]
(32)
yhere F is the faraday equivalent and u F e u / 3 ~ c 1 /= : ~9
(33)
where @ is the Onsager electrophoresia coefficient.
NOTES
636 Hence =
-~
~ 1 1 ,
Approximating (1+KU)-’ can be simplified to Ah = do
+ K a ) - ~ + ~ / 2 1 (34) by (1-KU + ~ ~ a (34) ~ ) ,
- BC’/,(l - Ka + H / 2 ) - &‘/%‘a2
Vol. 63
shows that it usually can be neglected, provided ~a < 0.2. Replacing H by its explicit value, we have finally =
- Bc,,,
+
Bc’/*~a[ll/l2- ( b / S ) ( l n rta
(351
+ 1.0172)]
(36)
When combined with the other relaxation terms, The last term of (35) gives the major part of the (36) will then give the final conductance equation.6 former term J2c’’’; comparison with experiment (a) R. M. F ~ibid.,~ so, 3163 ~ (1958). ~ ,
NOTES THE ACTION OF REACTOR RADIATION ON SATURATED FLUOROCARBONS BYJ. H. SIMONS AND ELLISON H. TAYLOR University of Florida, Q a h m d l e , Florida, and Chemist?# Diviaon,
Oak Ridge National Laboratory,1 Oak R&e, Received September 1.5, 1068
Tennessee
The extraordinary stability of fluorocarbons to heat and to most reagents led to early hopes of a similar high stability to ionizing radiation. Experimenh upon fluorocarbon polymers,2 however, indicated a greater than average sensitivity to radiation. Interest in these materials for applications involving radiation therefore waned, despite the fact that non-polymeric saturated fluorocarbons would be expected to act differently. Other radiation studies of fluorocarbons have been motivated by interest in radiation polymerization of some unsaturates, and have shown that these can be polymerized when irradiated in glass.’ On the hypothesis that the observed radiation instability of the polymers and the polymerizable monomers might result from impurities (residual H in impure fluorocarbons), reaction with the container (glass), or from innate instability of particular compounds (polytetrafluoroethylene), the present experiments were initiated. Fluorocarbons and fluorocarbon derivatives of low hydrogen content had become available, and it was therefore thought possible finally to establish the degree of radiation stability of this class of compound. ~
Experimental Materials.-GFl~less than lo-*% hydrolyzable F; b.p. 82’; optical density 0.51 at 2700 b.; 0.23 at 2460 A.; 0.67 a t 2206 b. C8F16Q-a mixture of cyclic fluorocarbon oxides of this comDosition but undetermined structure; b.P. 101’ (constant over whole sample); optical density 0.5i at 2536 ‘il. (C,Fp)aN-purified to remove H-contahing compounds; b.p. 177’. Methods.-Each sample waa further purified by evacuation while frozen, melting, refreezing, and further evacuation, followed by distillation through PaOsi?to the irradietion vessel. This was a 4.inch length of 1-mch aluminum (2 S) tube with a flat bottom and a * #-inch aluminum tube a t the top, fabricated by heliarc weding. It was attached (1) Operated by Union Carbide Corporation for the U. 8. Atomic Energy Commiseion. (2) 0. Sisman and C. D. Bopp, Physical Properties of Irradiated Plastics, USAEC Unclassified Report, ORNL-928 (June, 1951). (8) D. 8. Ballantine, A. Glines, P. Colombo and B. Manowits. “Further Studied of the Effect of Gamma Radiation on Vinyl Polymer Bye-,” BNL-294 (March, 1954).
to the vacuum system through a Kovar-to-PFex seal. After being filled, the vessel was sealed off while frozen, first in the glass and then by pinching and welding the aluminum. The samples were irradiated for 4 weeks in the Oak Ridge Graphite Reactor a t a neutron flux of 5.5 x 1011 cm.-2 sec.-l and a t a maximum temperature of 110’. The total energy absorption a t this position should have been 3.6 X lo-‘ cal./g., sec., for a ample composed of graphite, based upon calorimetric measurements by Richardson and Boyle‘ corrected for the flux ratio between their position of mesaurement and the resent location. About 82% of the energy (in graphite7 arises from absorption of y-rays and the remainder from stopping of fast neutrons.4 The samples were opened by attaching them to a vacuum system, cooling in liquid air, and then drilling a small hole in the top of the entrance tube by hand with a steel twist drill. The first sample opened was completely removed from the am o d e by vaporization. Because of the formation of hi h-toiling substances this became a len thy procedure. ‘%heother samples were similarly opened lut,.after the low-boiling material was removed, air was adrmtted, the ampoule removed from the system, and the content@ poured into the distillation apparatus.
Results There was no evidence of corrosion, the inner wall of each capsule being clean and bright after irradiation. There was no large amount of gas in any case, and in particular no F2 or CF4. Each irradiated sample was clear and colorless. The results of distillation and other examination of the samples are given in Table I, and the distillation curves (5 to 10 theoretical plates) in Fig. 1. Discussion The irradiation effected chemical changes in these three materials. Thermal changes cannot have contributed since the compounds are stable to 50O0. The yield, G, can be estimated from the distillation analyses and the energy absorption in graphite, assumed equal to that in the fluorocarbons. This assumption is probably accurate to f 20% except for (C4F&N, in which the reaction N14(n,p)C14increases the energy absorption appreciably. For CTFls, with 35% of the molecules transformed for a total energy absorption of 2.2 X e.v./g., G = 2 to 3 molecules transformed per 100 e.v. absorbed. This value is based only on the total energy absorbed, since such reactor experiments give no way for separating the effects of the different kinds of radiation. If the relatively (4) D. M. Richardson, A. 0. Allen and J. W. Boyle, “Calorimetric Meaeurement of Radiation Energy Dissipated by Various Materish Placed in the Oak Ridge Pile,” USAEC Unclassified Report, O R N L 129 (December, 1848).
.
+
4
