J. Phys. Chem. 1986, 90, 132-136
132
Some consideration has been given to the relationship of f12 to the intradiffusion coefficient D,*. This has been a controversial area where, on the one hand, it has been assumed that D , , = R T / f 1 2(as compared to eq 27) and, on the other hand, various authors are critical of this assumption based on theoretical g r o ~ n d s . ~ ~ ~ ~ ~ ~ ~ Comparison of values offi2/fi2 from mutual diffusion (eq 22) and sedimentation (eq 10) where the superscript zero denotes from tracer intradiffusion, and infinite dilution, fiZ0/(fl2 +flal) D,/D; from N M R spin-echo techniques where D, is the selfdiffusion coefficient are shown in Figure 5 for dextrans T70 and FDR7783. For both dextrans there is good agreement between the reduced quantities obtained from mutual diffusion and from sedimentation; the values from tracer diffusion tend to be higher at higher concentration. Excellent agreement is found between Callaghan and Pinder’sj9 self-diffusion study of dextran T170 in D 2 0 by N M R and tracer diffusion of FDR7783 whereas some difference is seen in the self-diffusion measurements of dextran T70 by N M R at 75 OC and corrected to 25 0C25to those made by tracer diffusion (Figure sa). The conclusions drawn from these comparisons are thatfi2 from mutual diffusion and sedimentation are similar and thatf,.] is relatively small and negative in sign as compared to f12. A further outcome of this work is the possibility that if f i 2 represents a coefficient describing solute-solvent exchange across the boundary in a volume-fixed frame then the tracer diffusion 0 in eq 27) may measurement (being similar tof12 sincef,., only represent an incomplete diffusion process or exchange (Le. solute-solvent exchange) as governed by eq 25 which requires solute-solute exchange.
TABLE I: Comparison of Extrapolated Values of MI/fII (from Figure 4) to Infinite Dilution for both Sedimentation and Diffusion Measurements
dextran type T10 T20 T70 FDR7783 T500
Mnl
?!VI
0.6 1.65 3.95 12.03 30.30
1.73 1.24 1.76 1.32 1.70
lo4 g mol-’
M.
(Mllfi2)c,-Ol io-” g cm dyn-I s-’
sedimentation diffusion 3.78 5.20 7.32 12.20 14.53
2.47 4.65 5.25 10.30 7.0
fIzml fgTSa
1.53 1.12 1.39 1.18 2.07
f12s and fi2”’ are the frictional coefficients from sedimentation and mutual diffusion, respectively.
to whether the M , / f I 2values from the two techniques are the same as has been found for less polydisperse polymer fraction^.^^,^^ Certainly the approach to molecular weight independence of the M , / J 2term does point to the possibility that it reflects dynamic properties of the polymer segment in semidilute solution rather than the individual molecule. Therefore, we would predict that the diffusion of the polymer segment in the dextran gel (Le. Sephadex) (when elastic constraints are low) would be similar to that found in semidilute solution as its Q value is similar (Figure 3). These correlations have already been experimentally established for solute diffusion in semidilute solutions and gels of p~lyacrylamide,’~ polystyrene,’’ and gelatin.j*
-
(33) Edmond, E.; Farquhar, S.; Dunstone, J. R.; Ogston, A. G . Biochem. J. 1968, 108, 755. (34) Comper, W. D.; Preston, B. N. Adu. Polym. Sci. 1984, 55, 105. (35) NystrBm, B.; Roots, J. J. Macromol. Sci. Reu. Macromol. Chem. 1980, C19, 35. (36) Tanaka, T.; Fillmore, D. J . Chem. Phys. 1979, 70, 1214. (37) Munch, J. P.; Lemarechal, P.; Candau, S.J . Phys. (Paris) 1977,38, 1499. (38) Amis, E. J.; Janmey, P. A.; Ferry, J. D.; Yu, H. Macromolecules 1983, 16, 441.
Acknowledgment. This work was supported by grants from the Australian Research Grants Scheme. We thank Mr. G. Wilson for performing the ultracentrifuge runs. (39) Callaghan, P. T.; Pinder, D. N. Macromolecules 1983, 16, 968.
Statistical Mechanics of Bolaform Electrolytes Joseph E. Ledbetted and Donald A. McQuarrie* Department of Chemistry, University of California, Davis, Davis, California 9561 6 (Received: June 17, 1985; In Final Form: September 16, 1985)
Bolaform electrolytes are a simple type of polyelectrolyte in which charges are separated by a chain of atoms in the polyion. This paper applies the methods of statistical mechanics to obtain thermodynamic properties for bolaform electrolytes. In particular, the linearized Poisson-Boltzmann equation is solved for an ellipsoidal model of a bolion. The electrostatic potential is then used to calculate thermodynamic properties. A comparison with available experimental activity coefficient data shows good agreement with the theory at low concentrations.
Theory and Model Formulation Bolaform electrolytes are a peculiar type of polyvalent electrolyte in which the charged sites are separated by a chain of atoms within the ion. In this work we will mostly be interested in divalent bolions. Examples of such ions are disulfonates and diquaternary ammonium ions. These ionic compounds were named by Fuoss’ and it is known that dilute solutions of bolaform salts have markedly different thermodynamic properties from those of normal (simple) electrolytes such as calcium chloride or copper(I1) sulfonate. Very little theoretical work has been done on this problem since the original work by Rice and Nagasawa.2 In this ‘Based on thesis submitted in partial fulfillment of requirements for Ph.D.
degree.
0022-3654/86/2090-Ol32$01 S O / O
paper we will present a model for bolaform electrolytes and solve the Debye-Hiickel equation exactly for this ionic system. We will also compare our theory with experimental activity coefficient data on real aqueous bolaform salts. The ionic model that we shall use to represent the bolaform ions in solution consists of a uniform fluid with dielectric constant E. The bolaform ion is immersed in this fluid. The other ions in the system are imagined to be charges with zero size. We assume that the potential external to the ion of interest is satisfied by the linearized Poisson-Boltzmann equation (1) Fuoss, R. M.; Edelsen, D. J . A m . Chem. SOC.1951, 73, 269. (2) Rice, S. A.; Nagasawa, M. ‘Polyelectrolyte Solutions”; Academic Press: New York, 1961; Chapter 6.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 133
Statistical Mechanics of Bolaform Electrolytes
V2$o = where
K
K
~
$
~
(outside)
(1)
is the Debye screening parameter defined by K2 = 4a pi(zie)2 ekT i
are called the radial and angular coordinates, respectively. The solutions to eq 7 are m
S ( V ) = S e ( i h ~=)
C ’~,,(W)Pfl(~)
(8)
’ i”d,(ih,E)h,,(l)(iht)
(9)
n=O or I m
The summation is over all the ion types in solution, pi is the bulk or average ion concentration, and zie is the charge on the ion. We also assume that our model of the bolaform ion is such that within the ion the potential satisfies Laplace’s equation except at the location of the two charges at .?I and Z2,
V2$i = zle6(Z - 5,)
+ z2e6(x’- Z2)
(inside)
(3)
Here 6 represents the Dirac delta function. Consequently, the problem of solving for the potential and thus the local charge distribution reduces to solving eq 1 and 3 along with the electrostatic continuity boundary conditions on the surface of the bolaform ion $i(surface) = $,(surface) %V$,(surface) = fi.V$o(surface)
(4)
R(E) = Rp(ih,E)= n=O or 1
The prime on the summations indicates that n assumes even integer values or odd integer values, dependent on whether E is even or odd, respectively. The subscript E denotes a particular solution to eq 7 corresponding with a particular eigenvalue Be. These functions (eigenfunctions of eq 7) involve expansion coefficients d,(ih,E) which parametrically depend on ih (i is the imaginary unit) and C. The spherical Hankel function of the first kind, hil)(ihE),gives the correct limiting form of the potential for large distances from the ion. The complete solution to the Helmholtz equation in these prolate spheroidal coordinates is a summation over a product of the eigenfunctions Se and Re, m
The first boundary condition is the requirement of continuity of the potential across the interface, and the second reflects the continuity of the normal component of the electric displacement vector due to the absence of a surface charge on the surface on the bolaform ion. These boundary conditions assume that the dielectric constant inside the bolion is the same as the dielectric constant in the fluid. Thus our model does not involve “image” charges. We must now choose the geometry for the bolaform ion so that a solution of these equations can be obtained. A versatile geometry and yet one in which the above linear differential equations are analytically tractable is an ellipsoid of revolution with one charge located at each focus. We consider the ion to be axially symmetric and define two coordinates
E e (ri + r2)/d,
7 E (ri
- r2)/d
where ri is the distance from the ith focus. The ranges of these coordinates are -1 < 7 < +1 and 1 < 6 < 0 3 . They are related to the rectangular coordinates by the transformation
The coordinates 5 and 17 are called the radial and angular coordinates, respectively. Note that d is the separation between the charges. We recognize eq 1 and 3 as the Helmholtz and Laplace equations, which are separable in this coordinate system. The solution of Laplace’s equation is
where $s is the potential due to the point charges and is given simply by Coulomb’s law. The quantity Pnrepresents the Legendre polynomial of order n. Outside the ion we have the Helmholtz equation with K # 0, and the substitution of $o(E,q) = S ( s ) R ( ( )into eq 1 yields the ordinary differential equations d dR -(E2 - 1) - [B h y ] R = 0 (7) 4 dt d dS -(1 - 72) - [ B h2+]S = 0 ds do where B is a separation constant and the parameter h2 K2d2/4. The solutions to these equations are restricted to only certain values of B (eigenvalues). Similar equations have been well studied3 and are known as spheroidal wave equations. The variables 5 and 11
+
+ +
(3) Flammer, C. “Spheroidal Wave Functions”; Stanford University Press; Stanford, CA, 1951.
$o(s,E)= C CtSt(ih,s)Rt(ih,t) e=o
(outside)
(10)
This equation, along with the solution within the ellipsoid
(11)
and the boundary conditions on the surface of the ellipsoid (E = 50) +i(E
=
EO)
= +o(E=Eo)
(12)
*V$i,c=lo = [fi.V$01,=,,
forms the solution to the potential at all points in space. Any number of terms in eq 10 and 11 can be computed numerically to yield essentially an exact solution to the problem. The functions Se(s) and Re([)can be accurately evaluated once the coefficients dn(ih,E) are determined, and then the boundary conditions (12) can be applied for N values of 7,resulting in a linear matrix problem of order 2N. These equations can then be solved for the boundary condition coefficients ( A , and C,) and the method can be iterated by using larger N a n d different values of 7 until convergence is obtained. An accurate test of the resulting solution is to test the boundary conditions for agreement. These resulting coefficients can be used in eq 10 and 11 to obtain the reduced potential at all points in the system:
The primed coefficients (A’ and C’)are reduced coefficients. The calculations were carried out on a CDC 6600 computer. This was necessary because the program that evaluates the expansion coefficients dn(ih,E) requires the large exponential range available with this computer. The number of coefficients required for a minimum of five-digit accuracy in the surface potential depends on the input parameters h and Eo (shape of ellipsoid). For values of practical importance (0.001 < h < 10 and 2 < to< loo), no more than 10 terms are needed to obtain the desired accuracy. The parameter Eo gives the actual surface of the ion and is the inverse of the eccentricity of the ellipse. The major axis of the ellipse is dEo and the minor axis is d(Eo2- l)I/*. A value of to = 2 gives an axial ratio of (major axis/minor axis) = 2/3II2. Actual values of tomust be decided on either experimental evidence or a detailed theoretical model of the ion of interest. Figures 1 and 2 show contour plots for the reduced potential for a few different parameters.
134
The Journal of Physical Chemistry, Vol. 90, No. 1, '986
charging a central ion keeping all other ions fully charged, or one could imagine charging all the ions simultaneously. The first process is called the Giintelberg charging process, and the second is called the Debye charging process. If we consider charging all the ions simultaneously (the Debye process), then the increase in the (Helmholtz) free energy due to the ions in solution is just a sum over all the ions in the system
t4
t2
y/d
1
0
Jcharged uncharged
-2
-4
-4
-2
0
tz
+4
X/d
Figure 1. Contour diagram of the potential near a bolion for to= 2 and h = 0.1. Reduced potential contours shown are for @ = 3, 1, 0.5, 0.3, and 0.25. The location of the charges are shown by crosses (+).
r4
Ledbetter and McQuarrie
t
I
1
dA = A - A0
AeI =
dXICj(rj,h)(ezj)
(16)
J
The integral over the charging parameter X involves the potential at thejth ion due to all other ions ( + j ) and this potential depends in a complicated manner on the parameter X because the Debye parameter K ( X ) is a function of the charging parameter. Fortunately, this potential qj can be determined as a simple function of K for spherical geometries, but more complicated geometries suggest the use of the other charging process. For the Giintelberg charging process, let us consider a particular ion. Now we imagine a reversible charging of each charge within this particular ion by an amount dX, thus resulting in the work of charging this particular ion
where $s(?j;X) represents the canonically averaged electrostatic potential at the location of t h e j t h charge in this particular ( p ) ion due to all other ions in the system. In other words, one removes the @ ion by subtracting the unscreened Coulombic potential due to each of the internal charges from the average potential
where +(7,;X) solves the Poisson-Boltzmann equation with a charge zgeX on the ion p. To relate the work of charging to thermodynamic properties, we realize that charging the ion /? results in an electrical contribution to the chemical potential
If one then assumes that any nonideality is due only to this electrical contribution, the activity coefficient of the p ion is defined by the relation
k T In y B = p B - pideal
I 'p ;
(20)
Because we have assumed a very dilute solution of the ion 0, the (Helmholtz) free energy is approximately's8
where N, is the number of p ions in the solution, and Ao is the energy in the absence of the 0 ions. Differentiating this free energy with respect to the counterion charge qc = z,e yields the canonically averaged potential at the counterion
where $c is the potential at the counterion location due to all other counterions (Le., in the absence of p ions). The activity coefficient can now be computed by charging the counterion according to eq 19 (4) Olivares, W.; McQuarrie, D. A. J . Chem. Phys. 1976, 65, 3604. (5) McQuarrie, D. A. "Statistical Mechanics"; Harper and Row: New York, 1976; Chapter 15. (6) Davidson, N. "Statistical Mechanics"; McGraw-Hill: New York, 1962; Chapter 21.
(7) Scatchard, G.; Kirkwood, J. G. Z.Phys. 1932, 33, 297. (8) Kirkwood, J. G. "Proteins"; Gordon and Breach: New York, 1967; Chapters I1 and 111. (9) Rock, P. A. "Thermodynamics"; University Science Books: Mill Valley, CA, 1983.
The Journal of Physical Chemistry, Vol. 90, No. 1. 1986 135
Statistical Mechanics of Bolaform Electrolytes k T In yc = pL,cl =
I’
dX(ez,)$,(X)
We now specialize our formalism to the ellipsoidal model for a bolaform ion, but it should be clear how this method could aoply to any geometrical ion model. For this model, we have two charges in the ion 0, one at each foci of the ellipsoid, z,e and z2e. The potential +8(?,) required in eq 19 is the “inside” potential from the Poisson-Boltzmann equation after subtracting the potential due to the charges at the foci. Substituting the exact solution to the linear Poisson-Boltzmann equation in this geometry into (19) yields the electrochemical potential of the bolion (zl = z2 = z8/2 3
0,3
+U
c
0,2
z)
where A,,, is the sum over the even reduced coefficients. In the limit of large toand small h = ~ d / 2 ,the first coefficient A. dominates this sum and we see that the bolion chemical potential thus becomes proportional to this coefficient. The electrochemical potential of the counterion is computed by using eq 19 with $,(A) from eq 22 with qc = Xz,e
0,l
I
I
0,l
Specializing to a solution containing bolions and counterions only, we then obtain the activity coefficient of the counterion from eq 24 -In y.,- = BC, bolaform electrolyte
2tkT
(25) One can also obtain the activity coefficient for the bolions (from eq 23) in a similar fashion, but this leads to negative chemical potentials for the bolion. Instead, we use the extended DebyeHiickel theory for the bolion activity coefficient -In y B =
(z842K 2ckT(1 + K R )
[extended Debye-Huckel]
where we determine R by equating the volume of an equivalent sphere with the ellipsoid volume d R = $to3- t0)1/3 We are now ready to calculate the mean activity coefficient. For an electrolyte with the formula BC,, the mean activity coefficient is defined by Ya
= (Yc“Yg)l’n+l
For reference we present the mean activity coefficient expression from our analysis for a BC, type bolaform electrolyte
-In
(z,e)2n2K = 2ckT(n 1)
+
[t
1 -k
1 aA,,, -
]
(4 dh)
(26)
The earliest measurements of thermodynamic properties of bolaform electrolytes were done by Lapanje et al.1° where a comparison of the ellipsoidal and rodlike models was done. These investigators used an electrochemical cell at 25 O C : Hg(l),HgzClz(s)JKCl(satd)lsample solution)AgCl(s),Ag(s) to measure the activity of chloride ions in solutions of ethylenediamine, diethylenetriamine, triethylenetetramine, and tetraethylenepentamine. This bolaform series ostensibly forms a homologous series of doubly charged ions in which the end nitrogen (10) Lapanje, 1960, 83, 1590.
S.;Haebig, J.; Davis, T.; Rice, S.A. J . Am. Chem. SOC.
0.2
iMc Figure 3. Experimentally determined chloride ion activity coefficients (0)and the theoretical curve based on eq 25 vs. the ionic strength. This solid line is the theoretical curve.
atoms bear a positive charge each. The separation between the charges can be estimated from molecular models. The experimental data for ethylenediamine are shown in Figure 3. Also shown is the theoretical activity coefficient for chloride ions calculated from eq 25 (n = 2, z, = -1). The parameter Eo was assumed to be equal to 2 in this calculation. Lapanje’s choice of towas not substantiated by experiment or argument. The most recent thermodynamic measurements on bolaform electrolytes are by Bonner et al.11J2 The aqueous bolaform electrolytes studied by Bonner and co-workers were m-benzenedisulfonate (BDS), 1,Zethanedisulfonate (EDS), and 2,7anthraquinonesulfonate (ADS) salts. Each of these doubly charged anions can be represented by a rigid ellipsoid of revolution in our theory. The mean activity coefficients can be calculated by using eq 26 if toand d are known. We can obtain these parameters from the diffusion coefficients of the bolaform sodium ~ a l t s ’ ~and J ~accurate molecular models. Mean activity coefficient data for the sodium salts of ADS and EDS are shown in Figure 4. Our theoretical estimates are given by the solid line in this figure. Note that the Debye-Huckel limiting law (DHLL) is a dismal failure for this salt. The quantity Mc is the molarity of the counterion, Le., the sodium ion concentration. The theoretical curve is for both salts; our model predicts the data to lie on the same curve. The agreement with the experimental data is only qualitative. This system undoubtedly involves nonidealities that are not electrostatic in origin. The main feature is that our theory can qualitatively account for the limiting slope of the data and our theory is not a bad approximation considering that there are no “fitted” parameters in our model. Activity coefficients were,also measured for the copper(II), nickel(II), and zinc bolaform salts. In Figure 5 we see that data for divalent cationic salts of these bolaform ions are independent of the particular cation (up to about 0.5 M). Once again our theory predicts a single curve for these salts as shown in the figure. Our theory suggests that the systematic difference between ADS and BDS (EDS) activity coefficients must be due to effects other (11) 4290. (12) (13) (14)
Bonner, 0.D.; Rushing, C.;Torres, A. T. J . Phys. Chem. 1968, 72, Bonner, 0. D.; Kim, S. J . Phys. Chem. 1969, 73, 1367. Uedaira, H. Bull. Chem. SOC.Jpn. 1973, 46, 401. Atkinson, G.; Petrucci, S. J. Phys. Chem. 1963, 67, 1880
J . Phys. Chem. 1986, 90, 136-139
136 I I
/
1.5
/
I
I
II
DHLL
/
/
/ /
/
DHLL
/
3.0
I
I'
I
I I
8
/
8
I I
8
/
ADSEI
/
ADSNa2
I I
I
/
2.0
EDSR
-
p
BDSM
EDSNa?
0.5
1.0
/
0.2
0.4
0.6
0.2
iMc Figure 4. Mean activity coefficient vs. Mc'12for 1-2 bolaform salts. The
solid curve is for the rigid ellipsoid model. The dashed curve shows the Debye-Hiickel limiting law for a 1-2 salt. than charge separation or bolion size. Our theory does not include effects such as ion-solvent interactions which are probably important in these aqueous salt solutions. In summary, our theory is able to account for the qualitative behavior of activity coefficients for some disulfonate bolaform salts.
0,4
0.6
i"c Figure 5. Mean activity coefficient vs. MC1I2for 2-2 bolaform salts. The solid line is the rigid ellipsoid theoretical curve; the dashed line is the DHLL. M represents Cu, Ni, or Zn. The experimental data for the different metal ions are indistinguishableon this plot.
Accurate data on a unique series of bolions would be helpful to ascertain the utility of our theory. Our model is expected to provide the "electrostatic part" for a simple hybrid theory that can be used to calculate thermodynamic properties of real systems.
Phase Transitions in Adamantane Derivatives: Adamantanecarboxylic Acid Pierre D. Harvey, D. F. R. Gilson,* and I. S. Butler Department of Chemsitry, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received: June 24, 1985; In Final Form: September 18, 1985)
The phase transition in adamantanecarboxylicacid, CloH15COOH,has been investigated by Raman and infrared spectroscopy, differential scanning calorimetry, and proton spin-lattice relaxation. The transition was observed both by DSC and NMR methods at 251 K, AH = 2.25 kJ mol-'. There is little change in the vibrational spectra above and below the transition, indicating that the transition does not involve the order-disorder transition characteristic of other substituted adamantyl derivatives. The carboxylic acid exists as a dimer in both phases. The barriers to the adamantyl group rotation are 19.3 and 32.0 kJ mol-' in the high- and low-temperature phases, respectively.
Introduction Studies using a wide variety of experimental methods have shown that substitution at either the 1- or 2-positions of adamantane does not preclude the order-disorder transition typical of the parent compound, although the temperature and enthalpy and entropy changes may be quite different. Certain substituents such as the methyl, hydroxyl, carboxylic acid, and amino groups may also possess an internal rotation of the substituent group, which could lead to additional disorder. For example, in 1adamantanol, where intermolecular hydrogen bonding would be expected to influence the phase properties, the transition temperature is increased to 353 K and the entropy of transition to 41 J K-I mol-', compared with 208 K and 16.2 J K-' mol-', respectively, in adamantane itself.',* 0022-3654/86/2090-0136$01.50/0
There has been an extensive series of studies of the barriers to rotation in adamantyl derivatives by Amoureux and Bee and co-~orkers,~-' who have used NMR and incoherent quasi-elastic neutron scattering methods. For the high-temperature phase, the (1) J. P. Amoureux, M. Bee, C. Gors, V. Warin, and F. Beart, Cryst. Strucf. Commun. 8,449 (1979). (2) S . S . Chang and E. F. Westrum, J . Phys. Chem., 64, 1547 (1960). (3) M. Bee and J. P. Amoureux, Mol. Phys., 50, 585 (1983). (4) M. Bee and J. P. Amoureux, Mol. Phys., 48, 63 (1983). (5) J. Virlet, L. Quiroga, B. Boucher, J. P. Amoureux, and M. Castelain, Mol. Phys., 48, 1289 (1983). (6) J. P. Amoureux, M. Castelain, M . Bee, B. Arnaud, and M . L. Shouteeten, Mol. Phys., 42, 119 (1981). (7) M. Bee, J. P. Amoureux, and A. J. Dianoux, Mol. Phys., 41, 325 ( 1980).
0 1986 American Chemical Society
