4316
NOTES
Table 11: Conductances of the A M F C-103 Membrane in Different Ionic Forms A, Ionic form
lOG, ohm-1 om-1
ohm-1 oms equiv-1
Li +
1.20
1.33
Na + K+ Rb +
1.45 1.77
c5+
CaZ + Mg2+ La3 +
1.51
1.93 1.95
2.03 2.17 2.60
0.48
0.49
0.43 0.09
0.55
0.09
alkali metal ions is Li+ < Na+ < E(+ < R b + < Cs+, the normal sequence observed in their aqueous solutions. The X values obtained for mono-, bi-, and trivalent ions are in close agreement with the findings of George and Courant.’ The values of X vary for monovalent ions from 1.33 to 2.6, for bivalent ions from 0.5 to 0.55, while the value for the trivalent lanthanum is as low as 0.09. Because the membrane is held in equilibrium with sufficiently dilute solutions (0.01 N ) , there are very few coions in the membrane phase, and so in all the cases considered (Table 11), the current-carrying species are the cations. The charge on the ion has a very big effect on the mobility of the ions in the membrane. This is in contrast to what is observed in aqueous solutions, in which the limiting ionic conductances are not related to charge type as most of the conductance values fall in the range from 38.7 for lithium to 77.8 for rubidium. As the charge on the counterion is increased, the membrane conductance decreases. This may be ascribed to increased ion association with the membrane exchange groups.
Acknowledgments. The work was supported by a grant from the Office of Saline Water, U. S. Department of the Interior.
Thermal Conductivity and Diffusion Parameter : Critical-Point Behavior by L. Seigel Department of Physics, Columbia Unicersity, N e w York, N e w York 10027 (Received February 26, 1968)
I. Thermal Conductivity : Critical-Point Behavior Classical measurements of the thermal conductivity, A, near the critical point are beset by numerous technical difficulties. Recently, it has become possible to circumvent these difficulties by using the half-width Rayleigh scattered light as a probe for A (this requires The Journal of Physical Chemistry
independent determination of the thermal diff usivity and the isothermal compressibility). In this section, we make two observations concerning the Rayleigh “line” which suggest a more direct optical approach for determining the critical-point behavior of the thermal conductivity. In 1934, Landau and Placzek3 predicted that the spectrum of light scattered from entropy fluctuations in a fluid would be Lorentzian in shape with a halfwidth, WL-P, given by
where 2 is the momentum-transfer vector, p is the density, and C, is the specific heat at constant pressure More recently, Fix(the thermal diff usivity =h/CPp). man and others4J proposed that the effects of longrange correlation would modify the Landau-Placzek result so that
Here
K,
the inverse correlation length, is given by /
1
\I/,
(3) where ICB is the Boltzmann constant, T i s the temperature, @T is the isothermal compressibility, and R is a short-range parameter, typically the order of the intermolecular distance. Our first observation is that the Fixman term, O L - P K ~ /isK ~proportional , to the thermal conductivity. To see this, we employ the wellknown thermodynamic relation
(4) In the critical region, C, is more weakly divergent than C, and hence may, we assume, be neglected in eq 4. If factors (eq 3 and 4) are combined and if it is assumed that R and (dP/bT)v are nearly independent of temperature, our first observation is readily verified. Currently a number of research groups are engaged in a search for the Fixman term; if successful, the temperature dependence of the thermal conductivity can be extracted from these studies. A parallel attack on the measurement of A may be (1) This work was supported in part by the Joint Services Electronics Program under Contract DA-28-043-AMC-O0099(E) and in part by the U.S. Army Research Office (Durham) under Contract DA31-124-ARO-D-296. (2) (a) J. V. Sengers, Ph.D. Thesis, van der Waals Laboratory, Amsterdam, The Netherlands, 1962; (b) J. V. Sengers and A. Michels, Papers S y m p . Thermophys. Properties, Rnd, Princeton, N . J . , i962, 434 (1962). (3) L. Landau and G. Placzek, Physik. 2.Sowjetunion, 5,172 (1934). (4) M. Fixman, J . Chem. Phys., 33, 1363 (1960). ( 5 ) W. D. Botch, Ph.D. Dissertation, University of Oregon, 1963, p 63.
NOTES
4317
made by examining the intensity of the scattered spectrum. The spectral profile, I(k,w),may be written (5) where I @ ) , the integrated intensity, is proportional to the Ornstein-Zernike factor, 1/(K2 K ~ ) . Our second observation is that the numerator of eq 5, I ( z ) w ~ / ,is, proportional to the proposed Fixman term. From our first observation it follows that I(R)wI/,must be proportional to A. Thus when an accounting is taken of the temperature dependence of the amplitude as well as the half-width of the Rayleigh “line,” it is possible to determine, in a single experiment, the exponents characterizing the divergence of the thermal conductivity and the isothermal compressibility along the critical isochore. (Alternately, the same end may be accomplished through independent measurement of I ( Z ) and
+
Wl,,.)
11. Diffusion Parameter, a : Critical-Point Behavior The density of the diffusion flux, i, of one component in a binary mixture in which there is no temperature gradient may be described bye i
=
(6)
-agradp
where p is the chemical potential and a is a diffusion parameter. Little experimental attention has been paid to the possible temperature dependence of a near the critical mixing point. We shall presently show that the ideas of section I, when applied to a binary mixture, suggest a straightforward optical approach for determining the critical point behavior of a. From elementary thermodynamics, it is simple to show tfhat the mass-transfer coefficient, D = ( a / p ) ( d p / d ~ ) pmay , ~ be written
(7)
I n fact, such a study of the effects of long-range correlation in a critical binary liquid mixture, isobutyric acid-water, has been reported recently by Chue8 His work indicates that D + ( T - Tc)0.67.Although Chu attributes the behavior of the mass-transfer coefficient as being totally due to the dependence of ( ~ T / ~ c ) on T temperature, his data suggest that K~ + (2’ - TC)’.O2 with the result that a + ( T - T,)-0.35. I n a similar study of the binary mixture aniline-cyclohexane, Berge and Volochineg found
D
+
( T - T,)’J.55
(10)
Again, we see the suggestion of either a divergent a or an extremely nonclassical value of y . Unfortunately in this case, we cannot differentiate between these two possibilities because a Fixman term was not observed. Further work is needed to clarify the behavior of a.10 Finally, to complete the analogy with the simple fluid, we note that, in the absence of a Fixman correction, the exponents characterizing the divergence of a and ( b n l d c )may ~ be determined by measuring the temperature dependence of the amplitude along with the half-width of the Rayleigh line.
Acknowledgment. The author wishes to thank Professor H. Z. Cummins for helpful discussions. (6) L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” AddisonWesley Publishing Co., Inc., Reading, Mass., 1959. ‘(7) P. Debye, Phys. Rev. Lett., 14, 783 (1965). (8) B. Chu, J . Chem. Phys., 47, 3816 (1967). (9) P. Berge and B. Volochine, submitted for publication in Phg8. Lett. (10) Subsequent to submitting this note, relevant work by Chu and Swift has been brought to the author’s attention: B. Chu, F. J. Schoenes, and W. 1’. Kao, J . Amer. Chem. Soc., 9 0 , 3042 (1968); J. Swift, to be submitted for publication.
HzO-D~OIsotope Effect in Partial Molal Volumes of Alkali Metal and Tetraalkylammonium Salts
where 7~ is the osmotic pressure and c is the concentrati0n.I The half-width of Rayleigh scattered light may be expressed as
by B. E. Conway and L. H. Lalibert6 Department of Chemistry, University of Ottawa, Ottawa, Canada (Received April 15, 1968)
where the inverse correlation length,
K,
is given by
The analogy with the case of the simple fluid becomes apparent by identifying T + P and c + p. In particular, we observe that the second term of eq 8 is proportional to the diffusion parameter, a. Thus any singularity in the behavior of a near the critical point may be uncovered by a temperature study of the Fixman term.
The comparison of thermodynamic properties of ionic solutions in light and heavy water might be expected to provide information concerning the degree to which these properties are sensitive to and react upon the rather subtle differences’-* in the properties of the two solvents. (1) G. A. Vidulich, D. F. Evans, and R. L. Kay, J . Phys. Chem., 71, 656 (1967). (2) A. A. Maryott and E. 12. Smith, National Bureau of Standards, Circular 514, U. S. Government Printing Office, Washington, D. C., 1951.
Volume 7g3 .\‘umber
12
November 1968
