RICHARD N. WORK
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Vol. 63
A CONVENIENT NEW FORM OF ONSAGER’S EQUATION FOR THE DIELECTRIC CONSTANT OF POLAR SOLUTIONS1 BY RICHARD N. WORK Department of Physics, The Pennsylvania State University, University Park, Pa. Received Auguat 16, 1968
Onsager’sequation for the dielestric constant of solutions of polar molecules in polar or non-polar solvents has been recast to express the relation between the dipole moments of the polar molecules present and cs - em the orientational contribution to the dielectric constant, where ell is the static dielectric constant and e m is the high fre uency dielectric constant. The new form of Onsager’sequation is convenient to use and is valid over the whole range of sJution concentrations, within the limitations of Onsager’s model. The effect of the free volume of the solution has been included. It is shown that serious errors can be introduced by the usual assumption that the volume actually occupied by the molecules is equal to the volume of the solution. The new equation is particularly useful in studies of short-range forces that lead to directional correlations between polar molecules, or in studies of the asphericity of molecules.
Introduction
the Mossotti internal field do not take into account
A new form of Onsager’sZ equation for the di- the dipole-dipole interaction force, and hence electric constant of solutions of polar molecules has been developed in connection with a study of the detailed structure of copolymers.* It is believed that the new equation presents sufficient advantage under wide enough circumstances to justify a separate discussion. The result obtained previously has been generalized to include cases where the volume fraction actually occupied by the molecules of the solution is less than unity, and it is pre,sented here with revised notation. There are several advantages in the use of Onsager’s equation for the determination of dipole moments of molecules in solution. First, the equation is derived rigorously from a model, which, though incomplete in certain respects, is clearly defined. Hence, differences which are observed in the values of the dipole moment of a given polar molecule, when measured in the vapor phase and in various solvents can be attributed to factors not included in the model, viz., molecular asphericity and directional correlations arising out of short-range intermolecular forces, instead of attributing them to “solventeffects,” a term which is not very meaningful. Secondly, Onsager’s equation is valid over the whole range of solution concentrations. This simplifies the experimental task of determining dipole moments by permitting measurements in solutions of sufficient concentration that the dipolar contribution to the dielectric constant be large enough to be measured easily. In addition, the dependence of the measured value of the dipole moment on concentration can be used to st8udy short-range forces4 in the liquid and molecular shape.5 I n contrast to this, equationse based on (1) The research reported here was started at the Plastics Laboratory, Princeton University, Princeton, New Jersey, under joint sponsorship of the Army and Navy under Signal Corps Contract No. DA36-03980-70154; ONR 356-375. Sponsorship has been continued at The Pennsylvania State University by the Atomic Energy Commission under Contract No. AT-(30-1)-1858. (2) L. Onsager, J . A m . Chem. SOC.,S6, 1486 (1936). (3) R . N. Work and Y.TrBhu, J . A p p l . Phys., 2’7, 1003 (1966). (4) J. G. Kirkwood, J . Chem. Phya., 7 , 911 (1939). (5) F. Buckley and A. A. Maryott, J . Reaearch Nall. Bur. Standards, 5.9, 229 (1954). (6) C. J. F. Battcher, “Theory of Electric Polarization.” Elsevier Publishing Co., Houston, Texas, 1952; C. P. Smyth, “Dielectric Strurture and Behavior,” McGraw-Hill Book Co., Inc., New York, N. Y . , 1955. Theee authors give comprehensive reviews of the many equations for the static dielectric constant of solution of polar molecules in Ibn-polar solvents.
they are valid only in the limiting case of dilute solutions of polar molecules in non-polar solvents. This leads to certain difficulties. First, the orientational contribution to the dielectric constant of the polar molecules in a dilute solution is small, making difficult the measurement of this quantity. This difficulty is partially alleviated by methods6 making use of several measurements in the region of higher concentration and extrapolating to infinite dilution. But, the extrapolation laws are uncertain because there is no clear way to separate the effects of dipole-dipole interactions, short range forces and molecular shape. Finally, the requirement that measurements be made only in dilute solutions eliminates the possibility of using the variation of the dipole moment with concentration to study molecular association. Onsager’s equation has not been widely used for the calculation of dipole moments from measurements on solutions because the calculations, involving Onsager’s eq. 35 or 36,2are rather cumbersome. I n addition, the question of how to take into account the free voIume of the solution has been left open. I n the following, a new form of Onsager’s equation is derived that is relatively easy to use and the question of the free volume is examined quantitatively. Derivation Onsager’s equation is derived from a model in which each molecule of the j t h kind is represented as a homogeneous and isotropic sphere of radius aj, having an index of refraction nj, corresponding to the atomic and electronic polarizabilities of the sphere, immersed in a medium of dielectric constant est and containing a rigid dipole pj = pjy(nj2 2)/3 at its center. Here, piv is the dipole moment of an isolated molecule. In the following it will be necessary to distinguish between the moment p j v and (pjv)On the latter being the value obtained by applying Onsager’s equation to measurements of the dielectric constant of s o htions. The new equation was developed to express Av in terms Of the orientational contribution to the dielectric constant es - e, rather than ea - 1. The average-square dipole moment is given by the expression Av = 22,
+
THEONSAGEREQUATION FOR DIELECTRIC CONSTANT OF POLAR SOLUTIONS
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549
Discussion (p2jv)Onr where zj is the mole fraction of component j. The static dielectric constant es is The discussion of eq. 6 given here is divided into determined by making measurements at frequencies three parts. First, suitable methods are described sufficiently low that the distribution of dipolar for evaluating the quantities in eq. 6 which are vectors is always in equilibrium with the applied not directly measurable. Second, the error that is alternating field; while the high frequency dielec- introduced by assuming that the sum of the molectric constant e m i, determined at a sufficiently high ular volumes is equal to the volume of the solution frequency that the distribution of dipolar vectors is considered quantitatively. Finally, some further does not change under the influence of the field. advantages in the use of the modified version of In deriving the new equation, we begin with Onsager’s equation are pointed out. Onsager’s2eq. 35. This gives the static dielectric Evaluation of elm and P.-In eq. 6, the quantities constant e, of a solution containingj = 1,2, , n vi, and hence V, and e j m are the only ones that are molecular species, each having a dipole moment as not directly measurable. Now, the molecular determined in the vapor phase of piv, and a con- radius uj and the high frequency dielectric constant centration of Nj molecules per unit voliime. Froh- q m of the molecular sphere can be treated as conlich’has re-expressed the equation as stants, since they depend only slightly on temperature. Hence, a single determination of the value of uj or ejm, for each pure liquid, a t a convenient temperature suffices to estahlish these quantities 2e. 1 47 + 2ea ___ (n*j - 1) - djNj] (1) for any mixture. Hence, the ratio of the density + n2j 3 the pure liquid at the temperature of interest where k is the Roltzmann constant and T is the tofo the density of the pure crystalline phase can be absolute temperature. We will use the symbol taken as an adequate approximation of Vi, the e j m to represent nj2,and (pjV)on to emphasize the fact that the conditions of Onsager’s model are not volume actually occupied by the molecules in the pure liquid at the given temperature. always fulfilled. The relation between Vj, e m and e j m can be estabRepresenting (4/3) au,Wj, the volume fraction lished by considering eq. 6. Since we are interested actually occupied by the jth kind of molecule by here only in the atomic and electronic contributions vj, and defining the apparent moment (pj)spp by to the dielectric constant, we can set e8 = e m and the relation obtain the relation
-
+ nV)2
we have, with some rearrangement of the terms in eq. 1
Whence, it follows that and hence for a pure liquid
where em is the high frequency dielectric constant of the solution. The quantity has no simple interpretatiOn in terms of a physical model. By making use of the identity 2es 2ea
+ +
em ejm
(ejm
-
1) =
em
-
1
+ 2es + ~
2ea
+ ajm
(ejm
- em) (4)
Eq. 3 can be rearranged to yield the expression
where P 3 Zvj is the volume fraction actually occupied by the molecules. Finally, using the relations Nj = NA (F/lM)Xj and Av. = 23j. (yj2)appwhere is the density of the solution, M is its average molecular weight, N Ais Avogadro’s number, and zj is the mole fraction of component j , we have, on rearranging eq. 5
(ejm
- 1) =
(em
-
1
1 - 3e,(l
- V j ) / ( Z e m + 1) (9)
Thus, using eq. 9, an estimate of T’j from density measurements, and a measurement of e m of the pnre liquid, ?ne can obtain an estimate of e j m . The relation V = zjVj, establishes the value of fr, Magnitude of the Free Volume Effect.-In order to simplify the calculations whenever possible, it is of interest to determine the conditions under which the effect of the free volume can be neglected. By making use of eq. 8 and 9, it is possible to recast eq. 6 into the form
where
Also the left-hand-side of eq. 10 can be written as
(7) H. Frohlich, “Theory of Dielectrics,” Oxford University Press, London, 1949, eq. 6.38,
RICHARD N. WORK
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and ,E
2.0
The significance of B and C’ is the same as in eq. 11 except that d m and e m are here identical. Values of C and C‘ are shown in Fig. 1 as a function of the free volume fraction Vf for different and = values of es and Em. Curves for es = e m give the limiting values of the factors C’ and C for the case8 of very large and very small dipole moments, respectively. The limit of C, as becomes large, is zero in all cases. I n Fig. 2 values are given of C” where 1 - C” 5 [ ( l - C)/(1 C’)2]’/z. The quantity C” represents the fractional contribution to (pC2jv)On obtained by taking into account the free volume of the pure liquid. The calculi tion of the factor C for solutions can often be simp’ified by neglecting the second term I I 1 .,E 1.5 Ed2.0 --- E,= 3.0 - in C (eq. 11). This term is important only if .30is not much larger than e m and the free volume fraction Vi is large. I n addition, the term C itself rapidly becomes negligible with respect to C’ as e, becomes large. The factor C represents the dependence on molecular size of the enhancement of the molecular polarization in the direction of the applied field due to the polarization of the surroundings. Its magnitude depends on the values of e8, e m and each e j m , and in the case of mixtures, C is the weighted sum of the contributions of all molecular species present. The factor C’j represents the enhancement of the dipole moment of the FREE VOLUME FRACTION, Fig. 2.-Values of the fractional error in the dipole j t h species through the polarization of the molecule moment introduced by the usual assumption that the vol- in the direction of the dipole vector by the dipole ume of the liquid equals the sum of the molecular volumes part of the internal field. That is, the dipole as a function of the actual free volume in the liquid. pj, by polarizing the surroundings, produces an additional electric field at the molecule which where further polarizes the molecular sphere of dielectric constant ejm, thus increasing the moment of the molecule. This factor depends only on es, e m and e j m of the particular molecule of the j t h kind. Therefore the correction C’j is calculated separately and for each species of molecule in the solution. It is true that the free volume error is not the only source of the differencebetween and (/.tjv)On. Here ejm is the macroscopic high-frequency di- In fact, deviations of molecular shape from the assumed spherical shape can lead to errors of up to electric constant of the pure liquid j. It will be seen later that in many, but by no 40%,4 and short range forces that are responsible means in all, cases the factor C and sometimes even for directional correlations between polar groups C‘ sre negligible leading to a considerable simpli- can also introduce considerable difference between fication of the computations. I n intermediate pjv and (/.tjv)On. Hence, unless the latter effects cases the second term in C (eq. 11) is negligible are known, there is still no assurance that the correction for free volume will lead to true values of leading to a lesser simplification. To investigate the magnitude of the terms C and the dipole moment. The use of the corrected value C’,it is convenient to consider a pure polar liquid. will however usually lead to a significantly better Then eq. 11 can be simplified by using eq. 8 and 9 value of the dipole moment. In studies of molecular shape or molecular association a knowledge of to express ejmin terms of Vj. The result is the correction i8 important. 3 ‘ 3 k T M %a e m (Es Further Advantages.-As mentioned previously (P2jv)On= ( Z - n ) G x G 3 c. the ratio of (p2jv)o, to (p’jy) can be used to study molecular shape factors and short-range intermolecular forces. These effects can be separated with now, the factors C and A , given by the ex- because they depend in different ways upon solupressions tion compositions. The shape factors depend only A(l - P)/P upon e8, and they are essentially constant for values C= of e. > 10. On the other hand. the effect of di1 - B ( l - P)P E#
+
+
I
~
t
April, 1959
THEORY OF
THE
DONNAN MEMBRANE EQUILIBRIUM
rectional correlations arising out of short-range intermolecular forces depends only upon the average separation of the interacting molecules, and hence upon their concentration. Thus, by appropriate variations of solution compositions, either effect can be made constant (or negligible in the case of interactions) while the other can be varied. The validity of Onsager’s equation for any solution composition makes its use highly desirable in such studies, and the relative simplicity of the new form makes ite application practicable. Since the new form of Onsager’s equation relates (pjv)On to ea - e m in a way that is nearly linear, Le., by a linear equation with nearly constant coefficients, the quantities p , es arid e m in the coefficients, need only be determined to precisions corresponding to that of es - em. This contrasts with the requirements of methodsa using increments of molar polarization where p and Xj must be deter-
551
mined to unusually high precisions because the increment is usually a small difference of two large numbers. In any case, the quantities e l m and p need be determined only to nominal precisions since C and C’ are usually small compared to one. Sometimes cB and em can both be determined from measurements of a single specimen in a single cell, e.g., bulk polymers. In this case, the difference (e, - em) can be determined to an accuracy comparable with that of e8 or e m leading to improved accuracy of the result in spite of the fact that (e, em) represents a small difference between two larger numbers. This is because the same cell constants apply in the calculation of (es - e m ) from a capacitance increment as in the calculation of es or em separately, and the capacitance increment can be calibrated to the same accuracy (about 1 part in 1000) as the total capacitance, providing it is of the order of lppf. or greater.
THEORY OF THE DONNAN MEMBRANE EQUILIBRIUM. 11. CALCULATION OF THE OSMOTIC PRESSURE AND OF THE SALT DISTRIBUTION IN A DONNAN SYSTEM WITH HIGHLY CHARGED COLLOID PARTICLES’ BY D. STIGTER~ AND TERRELL L. HILL Department of Chemistry, University of Oregon, Eugene, Oregon Received Auoust 18, 1968
As shown earlierlg-.6the McMillan-Mayer statistical theory enables one to express the osmotic pressure and the salt dietribution in a Donnan system in terms of expansions in powers of the colloid concentration. The (virial) coefficients in thess expansions are formulated with the help of “cluster integrals” depending on the potential of average force between the relevant particles. I n this paper the second and third virial coefficients in the above expansions are evaluated by numerical integration for the case of spherical, highly charged colloid partic1es;surrounded by a Gouy-Chapman type of ionic double layer. It is shown that the value of the virial coefficients depends mainly on the (Debye-Huckel) potential field in the outer art of the double layer. This justifies the uRe of an approximate Debye-Huckel type of potential of average force in evaguating the cluster integrals. The calculations are put in “corresponding states” form which generalizes the applicability of the results.
1. Introduction I n a recent paper,3 various approaches to the osmotic pressure, salt distribution and membrane potential in a Donnan system were compared, all charged species being treated as point charges. I n particular, the application of the statistical theory of McMillan and Mayer4 to the Donnan equilibrium was discussed, In another paper,s the McMillan-Mayer treatment was applied to the more general situation in which two species, though present on both sides of the membrane, are not in equilibrium across the membrane, while the other species are in equilibrium. Specialization of the general results led to the (equilibrum) salt distribution in a more conventional Donnan system (one non-equilibrium species). The present work deals with calculations of the (1) This work was supported by a research grant from the Heart Institute of the U. 8. Public Health Service. (2) Western Utilization Research and Development Div., Albany 10, Calif. (3) T. L. Hill, Faraday SOC.Diec., P i , 31 (1956). Hereafter denoted by I. (4) W. G. McMillan and J. E. Mayer, J . Chem. Phys., 18, 270 (1945). ( 5 ) T. L. Hill. J . Am. Chem, &e., 80, 2923 (1958).
osmotic pressure and of the salt distribution in a Donnan system, using the McMillan-Mayer approach and a model for highly charged colloid particles. Features of particular interest are : (1) unlike the treatment in Section 4 of I, we are able here, through the use of reference 5, to express both the osmotic pressure and the equilibrium salt distribution in terms of appropriate cluster integrals. Thus, the entire discussion is put on a firm statistical mechanical foundation. (2) A consequence of this situation is that the concept of “membrane potential” does not enter the discussion a t all. (3) The calculations below can be put in “corresponding states” form and hence are rather general in applicability (though restricted, of course, to the model chosen), We consider two solutions separated by a semipermeable membrane. The solvent and the ionic species 1, 2 . . . j have the same chemical potentials in both solutions, but colloid particles are present on one side of the membrane only (the “inside” solution). The concentrations (number densities) of small ions are denoted p l , p z . . . pi and p,*, pz* . . . pi* in the inside and in the outside solution, respect,ively. In the inside solution the
