J. E. LIND,JR.,H. A. A. ABDEL-REHIM, AND S. W. RUDICH
3610
~~~~~~
Structure of Organic Melts1
by John E. Lind, Jr., H. A. A. Abdel-Rehim,*and S. W. Rudich Department of Chemistry, CorneZl University, Ithaca, New York
(Received June 17v1966)
The Walden product of viscosity and electrical conductivity has been measured for a number of fused organic salts and compared with values computed from two theoretical models. The models are that for hard spheres and Rice's acoustical modification of h irkwood's theory. They both yield essentially the same expression for the Walden product, although the latter better accounts for the actual viscosity of the melts. This exrression for the Walden product is insensitive to energy terms and depends primarily upon the structure of the melt. The theory adequately predicts the Walden products a t 250" for salts with small spherical ions such as Pr4NBF4 and Pr4NPFe. As the alkyl chains are made longer, the measured Walden products become smaller than the computed values because of the sensitivity of the friction factor to the clogging of interstices in the melt by the long chains. For most salts the temperature dependence of the Walden product is small. The only exception to this is the salt Pr4NBPh4where the large temperature derivative most probably arises from like-ion interactions which do not appear to be significant for the other salts.
Introduction The viscosity and electrical conductivity of several organic salts have been measured as a function of temperature. Most of the transport data on liquid systems of any complexity have been analyzed in terms of theories in which a free volume or an energy of activation is empirically calculated from the data. These parameters could be called ensemble averages of a property of the bulk liquid. There is another group of theories which take as their basis the molecular properties such as size, mass, and pair potential. I n these latter theories the major part of the temperature dependence does not appear explicitly but implicitly through the radial distribution function. Thus the temperature dependence is not easily calculated because of our inadequate knowledge of the distribution function, and these theories have not been very useful except for liquid argon. However, for some systems the Walden product, is very insensitive to temperature and it is for these systems that comparison with these models is fruitful. Our systems were chosen with their ions as symmetrical as possible to make the comparison easier with these theories. Two theoretical extremes will be compared: that of the hard sphere and that of the acoustical modification of IGrkwood's theory. The Journal of Phvsical Chemistry
Both yield the same results for the Walden product although the latter better accounts for the measured values of the individual transport coefficients. The product depends primarily on the size of the ions and their molar volume and is insensitive to the intermolecular potential. Thus we will be able to focus attention on the structure of the melts with little consideration for the energy terms.
Experimental Section Eastman tetraalkylammonium halides were used in the preparation of the salts. The tetra-n-butylammonium iodide, Bu4NI, recrystallized twice from methyl alcohol, has a melting point of 146.0'. The Bu4XBrwas recrystallized from 9 :1 benzene-cyclohexane mixtures and melted at 119.5' under argon. The halides were dried for several hours under vacuum a t temperatures no greater than 60" since they deconipose readily. All others were dried between 50 and 100" overnight before (1) This study was aided by a grant from the Office of Saline Water, U. S. Department of the Interior. (2) Part of this paper is based on a thesis presented by H. A. A.
Abdel-Rehim, a U.A.R. Government Fellow, to the Graduate School of Cornel1 University in partial fulfillment of the requirements for the M.S. degree.
3611
STRUCTURE OF ORGANIC ~IELTS
melting points were taken under an argon atmosphere. The tetraalkylammonium tetrafluoroborates were prepared by the fluoroboric acid titration of an aqueous solution of the tetraalkylammonium hydroxide which had been prepared from the iodide by reaction with silver hydroxide. The resulting tetrabutylammonium tetrafluoroborate, Bu4XBF4, was recrystallized four times from 1:1 methanol-water mixtures and had a melting point of 162". The tetrapropylammonium tetrafluoroborate, Pr4NBF4,was recrystallized twice from water and had a melting point of 248", and the tetra-nhexylammonium fluoroborate was recrystallized from methanol and melted a t 91". The tetraalkylamnionium hexafluorophosphates were prepared by adding to a hot concentrated aqueous solution of the tetraalkylammonium iodide a 10% aqueous solution of potassium hexafluorophosphate. The desired salt precipitated and was recrystallized several times from methanol. For the Bu4NPFG salt, fluorine analyses yielded 29.62 and 29.31% in agreement with the theoretical yalue of 29.45%. The salt crystallized from methanol had a sharp melting point of 247", while the use of other solvents such as methanol-benzene, ethyl alcohol, and acetone resulted in low melting points with wide melting ranges. The Pr4NPFs was also recrystallized from methanol three times and had a melting point of 237.0'. The tetraalkylammonium tetraphenylborides were prepared by metathesis in water of the sodium tetraphenylboride, NaBPh4, and the tetraalkylammoriiuni iodide. The salts were recrystallized either from 3 :I acetone-water mixtures or preferably from 3 :1 methanol-acetone mixtures. In the latter mixtures excess acetone was boiled out until incipient precipitation occurred. The BuiNBPh4 salt melted at 236.5' and the Pr4NBPh4at 205.7'. All salts except the tetrafluoroborates had to be recrystallized immediately before they were used. They were then dried in the measuring apparatus under vacuum a t temperatures up to 100". For most salts extended periods of drying were avoided to prevent decomposition. In order to remove the last trace of solvent, the salts were quickly fused under argon. They were then sparged with argon and/or dried under vacuum again. In the case of the halides which are the least stable of the salts, the sparging of the melt with argon was effective in removing the volatile decomposition products which were presumably the tributylamine and the butyl halide. Any trace of solvent in the hexafluorophosphates caused etching of the glass so the apparatus was recalibrated after each measurement on these salts. The initial point of each run was always rechecked by the last point in order to detect decomposition. All measurements were made with the salt
under an atmosphere of argon, and each piece of apparatus was corrected for thermal expansion. The measurements of conductance were made with a shielded Wheatstone bridge of the Shedlovsky type.3~4 The resistance of the salt in the conductance cell was measured a t frequencies between 1 and 20 kHz, and the resistance was extrapolated to infinite frequency as either the reciprocal of the frequency or the reciprocal of the square root. Most of the measurements were made with a Pyrex cell with bright platinum electrodes whose constant was about 27 cm-l. The cell was calibrated both by a 0.1 denialj solution of KC1 in water and by comparison with an erlenmeyer cell whose constant is 2.0714. This cell was calibrated a t 250" by the LZF methode6 The densities were measured in a modified Lipkin bicapillary pycnometer' of either 1- or 2-ml capacity. The viscosity measurements were made in modified Cannon-Ubbelohde semimicro viscometers, sized 25, 50, 75, 100, 150, and 200. Most of the viscometers were calibrated by the Cannon Instrument Co. After modification they were intercompared with each other and with an unmodified viscometer. S o change of the viscometer constantsoccurred in the modification. The intercomparisonsof the viscometers of size 25,50, and 75, having constants of 0.002438, 0.003742, and 0.005684 cs/sec, respectively, were made with carbon tetrachloride, water, and nitrobenzene. There was agreement between the calibrations with CC14 and nitrobenzene to better than 0.1%. However for water, discrepancies of 1% were found. Since the flow times for water lie between those for the other two liquids, this phenomenon is not caused by dissipation of kinetic energy, but must be caused by the high surface tension of water in these small viscometer bulbs. The correction for the surface tension is a function of the surface tension divided by the density.8 The values of this quotient in cgs units are: CCL, 16.8; nitrobenzene, 36.4; and water, 72.6. Since this quotient for molten tetraisoamylammonium iodide9is only 24, no corrections need be made to the data for the fused organic salts. Tables I-V contain the data and the equations which have been fitted to the data. I n all of these equations
(3) T. Shedlovsky, J . Am. Chem. SOC.,52, 1739 (1930). (4) H. Eisenberg and R. M. Fuoss, ibid., 75, 2914 (1953). (5) G. Jones and B. C. Bradshaw, ibid., 5 5 , 1780 (1933). (6) J. E. Lind, Jr., J. J. Zwolenik, and R. M . Fuoss, ibid., 81, 1557 (1959). (7) M. R. Lipkin, J. A. Davison, W. T. Harvey, 2nd S. S. Kurta, Jr., I d . Eng. Chem., Anal. Ed., 16, 55 (1944). (8) G. Barr, Proc. Phys. Soc. (London), 58, 575 (1940). (9) P. Walden, Bull. Acad. Sci. St. Petemburg, 8 , 405 (1914).
Volume 70, Number 11
November 196'6
J. E. LIND,JR.,H. A. A. ABDEL-REHIM,AND S. W. RUDICH
3612
including the equation for the densities, the temperature is in degrees Kelvin. The limitation in the precision of the measurements was the decomposition of the salts. The tetrafluoroborates are the most stable, the halides the least. I n general, small amounts of solvent do not alter the slope of the Arrhenius plots but cause dis-
Table I: Conductance Data for Salts with Nonlinear Arrhenius Plots -HexrNBFd--
yPrrNBPh4A
t, o c
214.42 218.90 223.40 227.93 230.19 225.65 221.15 216.64 212.18
5.921 6.137 6.349 6.742 6.644 6.446 6.240 6.026 5.808
cHexrNBF4t , "C A
136.01 105.32 115.67 125.27 130.82 120.58 110.35 100.36 149.38 140.78 132.02 123.39
0.8318 0.2925 0.4282 0.5932 0.7077 0.5079 0.3533 0.2406 1.1984 0.9484 0.7318 0.5563
t,
oc
114.90 107.88 98.45 111.59 127.53 144.78 157.86 160.53 169.56 178.56 187.54 196.88 206.05 215.42 224.96 164.95 173.88 182.88 192.10 201.24 210.58 220.11 229.60
A
0.4164 0.3229 0.2231 0.3684 0.6356 1.0538 1.4836 1.5879 1.9585 2.3822 2.8595 2.4114 4.0093 4.6845 5.4323 1.7628 2.1545 2.6024 3.1187 3.6876 4.3304 5.0470 5.8217
-ButNBFrt , "C
A
162.35 173.46 184.68 196.15 167.93 207.86 219.74 190.43 208.18 265.96 255.81 243.96 231.52
4.735 5.865 7.144 8.585 5.287 10.195 11.959 7.862 10.237 19.947 18.057 15.923 13.864
placement toward higher conductance and lower viscosity. The B U N is the only salt on which Walden'O made a measurement, and his conductances are about 3% higher than ours. His salt had been crystallized from water rather than methanol, and thus it may not have been water free. The addition of a drop of water to 5 g of melt increased the specific conductance by 5%. These results are plotted in Figures 1 and 2.
Discussion I n an attempt to ascertain information about the structure of these melts, two models will be used which represent extremes in the spectrum of momentum transfer between molecules. I n the hard-sphere model, the transfer of a large amount of momentum can occur during a collision. I n Rice's acoustical modification11~12 of Kirkwood's model, small amounts of momentum are continuously transferred between molecules during The Journal of Physical Chemistry
Table 11: Viscosity Data for Salts with Nonlinear Arrhenius Plots -HexrNBFd--
7 B u h N B F r -
oc
10011, poises
102.88 111.07 119.49 123.50 107.05 115.32 154.06 128.15 136.50 145.12 149.58 140.78 162.76 171.69 180.71 189.81 167.11 176.19 185.27 194.44 203.81 213.14 222.51 229.63 199.22 208.41 217.76
106.16 76.02 55.25 47.17 88.98 64.16 18.43 40.61 30.90 23.65 20.84 27.03 14,627 11.751 9.589 7.894 13.101 10.560 8.603 7.136 5.872 4.968 4.295 3.804 6.445 5.421 4.601
t,
-
oc
1001, poises
162.31 173.44 184.72 189.17 166.72 179.05 168.82 207.75 219.73 231.70 196.33 169.15 243.95 219.73 265.99 207.85 256.16
9.954 7.844 6.304 5.803 9.015 7.027 8.660 4.258 3.557 3.017 5,121 8.560 2.587 3.557 2.005 4,252 2,233
t,
------Pr4NRPhr----1004, t, o c
poises
219.91 210.38 225.82 215.22 231.78 217.70 238.09 243.80 223.47 256.05 212.77
8.117 9.839 7,292 8.940 6.560 8.506 5.886 5.356 7.631 4.429 9.338
Brownian motion in the fields of the surrounding molecules and strong correlation with the surrounding medium is introduced by coupling the molecule to an acoustical wave. For the application of both models to fused salts the assumption is made that each ion is surrounded by and interacts with only ions of opposite charge. The Coulomb field ensures that the nearest neighbors are oppositely charged. Rice13 has shown that, while the Coulomb field determines the density of the salt, it does not contribute to the friction constant in the zeroth approximation and contributes less than 5% to the viscosity of KCI. Therefore, the equations derived for simple liquids can be applied to the ionic (10) P.Walden, H.Ulrich, and E. J. Birr, Z . Physik. Chem., 130,494 (1927); 131, 1, 21, 31 (1928); P.Walden and E. J. Birr, ibid., 160, 45,57, 161 (1932). (11) s. A. Rice, Mol. Phys., 4, 305 (1961). (12) B. Berne and S. A. Rice, J . Chem. Phys., 40, 1336 (1964). (13) S. A. Rice, Trans. Faraday Soc., 58, 499 (1962).
3613
STRUCTURE OF ORGANIC M E L T S
Table I11 : Fused Salt Densities
Salt
Temp range, "C
No. of
Density eq
Std dev
data
0.0001 0.0001
4 10
0.0001 0.0003
14 12
- 7.039 X 10-4T 1.4460 - 8.388 X 10-4T 1.1906 - 5.812 X 10-4T 1.3252 - 6.557 X 1.1435 - 4.945 X 10-4T
0.0002 0.0003 0.0003 0.0003 0.0012
10 18 4 14
- 5.772 X
0.0003
25
- 7.637 X - 6.286 X 1.2433 - 3.224 X
Pr4NBF4
252-259.5 259.5-274
d = 1.3116 d = 1.2397
Pr4NPF6 Pr4NBPh4
240-272 209-239
d = 10-4T d = 0,14613 3.70736 X 10-3T 4.29706 X 10-6T2
Bu4NBr BL14NI Bu~NBF~ BLI~NPF~ Bu4NBPh4
119-135 147-162 163-266 256-275 240-270
d d d d d
Hex4NBF4
102-218
d = 1.1296
10-4T 10-4T
+
-
= 1.287 = = = =
10-4T
9
Table IV: Equivalent Conductance Temp range, Salt
O C
Pr4NBF4 Pr4NPF6 Pr4NBPh4
249-282 240-272 212-233
No. of
Conductance eq
In A = 8.1154 - 2519.4/T In A = 8.9646 - 3068.9/T In A = 5.9266 9497.64/T
+
Std dev
data
-
0,001 0.0005 0.0005
16 15 9
-
0.07 0.005 0.001
7 13 13
0.003 0.001
5 12
0.002
35
2.79915 X 106/T2 Bu4NBr BU4NI Bu~NBF~
117-135 148-167 162-266
BLI~NPFB BudNBPh4
256-275 241-267
Hex4NBF4
98-230
In A = 8.64 - 3891/T In A = 11.573 - 4623.8/T In A = 3.8298 1811.00/T 1.2199 X 106/T2 In A = 8.5072 - 3019.2/T In A = 8.8060 - 3455.5/2'
+ +
In A = 3.3488 2042.6/T 1.4287 X 106/T2
-
Table V: Viscosity
Salt
Temp range, OC
Pr4NBF4 Pr4NPF6 Pr4NBPh4
249-274 244-269 210-256
BuaNI BLI~NBF~
162-266
Bu~NPFB Bu4NBPha
256-281 244-269
10% = 27.0 a t 148.8' In 10% = -0.59671 - 1661.37/T 1.27194 X 106/T2 In 10% = -5.7532 3602.95/T In 10% = -6.4455 4144.5/T
HeqNBF4
102-230
In 10% = -0.6263
Equation for viscosity
In 1% = In 1007 = In 10% = 1.65156
+
-5.0112 3038.41/T -5.1583 f 3251.4/T -0.4930 - 2072.35/T X 106/T2
+
+ + -
1987.1/T
+
+
Std dev
No. of data
0.0007 0,001 0.002
5 5 11
... 0.002
17
0.002 0.001
5 5
0.004
27
1.49496 X 1Oe/T2
Volume 70, Number 11 November 1966
J. E. LIND,JR.,H. A. A. ABDEL-REHIM, AND S. W. RUDICH
3614
40
20
1 .o 0.8
0.6 0.4
0.18
0.20
0.24
0.22 100/T.
0.26
number of salts. However, before these results can have any significance, we must show whether these theories can a t least predict the order of magnitude of the transport properties separately, so that the results for the Walden product are not happenstance. In the next paragraphs, estimates are made of the parameters which would have to be used in order that each of the theories yield the measured viscosity of Pr4NPF6 a t 250”. These parameters are then compared with other data to determine if the model is reasonable. This particular salt is used since, of the salts investigated, its ions are the most spherically symmetric ions which differ least in size and mass from one another. The viscosity and friction constants for hard spheres areL4 7 = (4/15)(2~~~~)’/2p2~4g(a)
Figure 1. Equivalent conductance of fused salts.
l
10 $ 8
2
1 0.18
I
0.20
0.22
0.24
0.26
lOO/T.
Figure 2. Viscosity of fused salts.
melt. For ions of unequal mass, the mass is replaced by twice the reduced mass. If the total particle density is used, the radial distribution function for the nearest neighbor and other odd-numbered shells would be essentially the same as that for a simple uncharged liquid but the function would become zero for the evennumbered coordination shells. Obviously these models for spherical molecules will not adequately represent all the salts being considered here. However, deviations from these models can be used to indicate the structure of these melts just as deviations of the compressibility factor from unity indicate the structure of real gases. We will show that both the hard-sphere model and the acoustical model predict the Walden product for a The Journal of Physical Chemistry
= (8/3)pg(a)~2(21.L~kT)1’2
(1) (2)
Both [ and 7 are zeroth order which means that the distribution function in momentum space is Alaxwellian about the local mean velocity and that there is local equilibrium in coordinate space. In the expression for the viscosity the kinetic term is neglected since its contribution is negligible at liquid densities. I n these expressions the major part of the temperature dependence must be placed in the volume dependence of the radial distribution function g(u) which is evaluated at the collision distance u. Davis and LuksI5 state that the hard-sphere theory with additional higher order terms, which contribute 20-30% to the viscosity, yields about 70% of the viscosity of liquid argon when they used an empirical distribution function which had fitted a square-well model to the data. Using an arbitrary radial distribution function of 6 and their collision diameter of 3.16 A, about half of themeasuredvalueof the viscosity of argon at 90°K is given by eq 1. This value of 6 for the distribution function is close to the largest value of the radial distribution function evaluated at the surface of the sphere which was obtained from the computer calculations of Alder and Wainwright.I6 A similar calculation for Pr4NPF6at 250’ with the collision diameter given in Table VI yields about one-fourth of the total viscosity. Thus a value of g ( u ) of 25 would be required to reproduce the measured values of the viscosity for the fused salt. Since 25 seems unrealistic, the additional contribution must (14) Cf.ref 8 or 15; H. C. Longuet-Higgins and J. A.Pople, J . Chem. Phys., 25, 884 (1956). (15) H. T. Davis and K. D. Luks, J . Phys. Chem., 69,869 (1965). (16) B. J. Alder and T. Wainwright, “Proceedings of the International Symposium on Transport Processes in Statistical Mechanics,’’ Interscience Publishers, Inc., New York, N. Y., 1958, p 97; J . Chem. Phys., 33, 1439 (1960).
STRUCTURE OF ORGANIC NELTS
3615
where
Walden Products and Reduced Viscosities at 250’
Table VI:
Salt
PraNBF4 Pr4NPFe Pr4NBPh4 Bu~NBF~ BQNPFB Bu4NBPh4 Hex4NBFd
A7
A7
(obsd)
(calcd)a-d
0.601 0.638 0,359 0.408 0,479 0,395 0.215
0.637 0.648 0.556 0.570 0.562 0.529 0.478
11= 1007
2.22 2.88 4.86 2.40 3.1lE 4.38 2.82‘
J0
+
x ~ [ z ~ u ” ( x )4u’(z)]g(z)dz
=
A
27.1 22.2 7 . 3ge 17.0 15.4e 9.03 7,64’
Calculated from eq 11 using the following ion sizes (in angstroms): BF4-,* 2.76; PFe-,’ 2.93; BPhh-,‘ 4.72; Pr4N+,d 4.40; BUN+,^ 4.79; H ~ X ~ N5.52 +,~ by using 15.7 ml/methyl group. * L. Pauling, “The Nature of the Chemical Bond,” Cornel1 University Press, Ithaca, N. Y., 1960; H. Bode and H. Clausen, 2. Anorg. Allgem. Chem., 265, 229 (1951); A. Bellanaca, Struct. Rept., 12, 190 (1949). W. R. Gilkerson and W.-Y. Wen J. L. Stewart, J . Phys. Chem., 65, 1465 (1961). and S. Saito, ibid., 68, 2639 (1964). e Extrapolated values.
arise from the greater surface for interaction occasioned by the chains of the cations if the model is to be retained. Otherwise, van der Waals forces and stronger correlation with the surrounding medium must be introduced. Consider the Rice-Kirkwood model where these latter factors are t,aken into account. The equations given below are to the same approximation as those for the hard-sphere m ~ d e l . ~ , ~They , ” do not account for the perturbation of the pair distribution function from the spherically symmetric form about the mean local fluid velocity. The Kirkwood model assumes that two ions move in Brownian motion relative to one another. In the time interval during which the pair remembers the history of its motions, they experience a displacement which is small compared to the distance of separation of the pair. In this approximation and with the assumption that the gradient of the mean local velocity in viscous flow is small and uniformly the same for all species, the equations for the Iiirkwood model given in ref 17 can be generalized to the ionic melt where anions and cations have different mass by substituting twice the reduced mass, 2 p , for the mass. I n the acoustical modification the force between two molecules is interpreted as the force acting on a mass of continuum fluid equal to that of one molecule. While not exact, we shall for convenience assume for a two-component fused salt that the force acts on a mass equal to twice the reduced mass. (3)
(4)
I2
=
im
[x%”(x)
+ 2 ~ ~ ’ (]g(z)dz 2)
=
P W
where x is the distance from the center of an ion scaled by the distance r to the minimum of the potential well. The intermolecular potential u(x)is not known for these salts. Even for molecules for which approximate potentials are known, there is great uncertainty in the values of these integrals because they are very sensitive to the second derivative of the repulsive potential. The Lennard-Jones potential is certainly not correct especially at the high values of e/kT that will be necessary. U(2) =
E(z-12 - 22-6)
(7) However, it will be used to obtain a rough estimate of the values of e/kT and g required for this theory to yield the measured viscosity of Pr&PF6. These results may then at least indicate if this theory can better account for the viscosity than the theory for the hard spheres. For simplicity the radial distribution function is approximated by a square peak of height g in the region where the potential energy is less than some specified value of kT. The value of g is taken as zero elsewhere. At large distances where g should be unity, the integrand is so small that this approximation is very good. For the assumed values of e / k T of 1, 5, and 10, the values of the product [ (e/kT)g]required for Pr4NPFe are 15, 33, and 36, respectively. Here the velocity of sound is assumed to have a value comparable to that of the inorganic salts of 1500 m/sec.18 These values are of the same magnitude as the 25 required for the hard-sphere model to fit the data. I n the Kirkwood model, however, this number is the product of two parameters rather than the single parameter of the hard-sphere theory. Thus the Iiirkwood theory can better approximate the higher viscosity especially if a steeper repulsive potential is used. For the Lennard-Jones potential, (q‘kT) will most likely be of the order of 5-10, For these values, the range of integration is so narrow that this average g is (17) S. A . Rice and J. G . Kirkwood, J . Chem. Phys.,31, 901 (1959). (18) J. O’M. Bockris and N . E. Richards, Proc. Rou. Sac. (London), A241, 44 (1967).
Volume 70, Number 11
,Vovember 1966
J. E. LIND,JR.,H. A. A. ABDEL-REHIM, AND S. W. RUDICH
3616
adequately represented by the maximum height of the first peak of the radial distribution function. At values of q‘kT near unity this maximum is about 3, and it may become somewhat greater at these large values of c / k T . Thus e/kT may lie in the range of 5-10, and although these values are high, they are not unreasonable. This parameter must account for the larger forces between an anion and cation where there are ion-induced dipole forces, where there is a small direct contribution of the Coulomb field, and where the chains of the cation can partially envelope the anion to increase the energy of interaction. Thus the acoustic model with its additional parameter seems to account sufficiently well for the viscosity of Pr4NPF6to serve as a model for comparison and the hard-sphere model is of less certain usefulness. Now that we know that at least one of these theories can readily yield the viscosity of the salts, the Walden product will be computed. Both theories yield essentially the same result for this product which indicates that the product is independent of the exact mechanism of momentum transfer. Thus should a theory be developed which introduces greater correlation into the hard-sphere model and which might thus permit a better approximation of the viscosity, one would not expect any serious alteration of the expression for the Walden product. Both the viscosity and friction constant are very sensitive functions of temperature, whereas the Walden product of conductance and viscosity with its small temperature dependence shall prove to be a quantity dependent upon the geometrical structure but not the depth of the energy well. Either transport property by itself is dependent on both factors.
h
= 2qkdT
3.718u02/v
The Journal of Physical Chemistry
j--x~h(x)g(x)dx
2
Aq = 3.718-
‘1
(10)
(h(x) - 2xu’)g(x)dz
If the term 2xu’ in the denominator is small, the denominator is the normalization constant for the function weighting x2 in the numerator. Now for any potential, 2xu’ goes through zero at x equals unity, while g(x) peaks near there. Simultaneously h ( x ) is large for values of z of unity or less as shown for the LennardJones potential in Figure 4. Thus 2xu’ can be neglected with the result that Aq S 3.718r02(x2)h,/7 Z 3.718r02/
(11)
where the angular brackets indicate the average over the function h(x)g(z). This average x 2 should be just slightly less than unity from the behavior of h ( z ) and g(x). The result is that the expression for the Walden product is essentially the same as that given by the hard-sphere theory, except the collision parameter is replaced by the distance to the bottom of the potential well. Note that this product depends only upon the ion sizes and the volume of the salt and depends indirectly on the depth of the well only through the radial distribution function if the potential is of the form cf(z). This latter dependence is slight since the distribution is strongly influenced by the Coulomb forces rather than
(8)
The factor of 2 accounts for the 2 moles of ions/mole of salt, k is the Boltzniann constant, and q is the charge on the ion. Here the Nernst-Einstein equation must be assumed to relate the conductance to the friction constant. For inorganic salts this would be a grossly inadequate assumption. However, the results of the analysis will show that eq 8 fits the data for simple tetrapropylammonium salts so well that deviations from the Kernst-Einstein equation are probably small. In these salts the van der Waals forces are larger and the Coulomb forces smaller than in the inorganic salts. The Walden product of the organic salts is nearly insensitive to temperature much more like dilute solutions in nonpolar liquids than like inorganic fused salts. The hard-sphere theory in the zeroth approximation yields A7 =
v
where u o is in angstroms and is the molar volume of the salt. Xote in Figure 3 that this equation gives the correct sign of the temperature dependence through the implicit dependence of the volume upon the temperature. The acoustical theory yields
(9)
$P 1 I
a
I
I
0.3
0.2
0.18
0.20
0.22 0.24 100/T.
0.26
0.28
Figure 3. The temperature dependence of the Walden product. Solid lines are the experimental results. Dashed lines represent the Walden product a t the melting point scaled by l / v .
3617
STRUCTURE OF ORGANIC MELTS
400
I
I
1
\
two possible causes for the deviation from this model. Either like-ion interactions are significant or the chains of the cations clog the interstices in the melt and prevent motion of the ions. Consider the first of t,hese possible causes: the effect upon the Walden product when one species of ion is much larger than the other and like-ion interactions can occur. Normally only one of the two ion species will suffer like-ion interactions and the Walden product becomes 117 =
k[Pap-l
+
(Pap
+
Pua)-'I[~ap
+
~ a a l
(12)
where CY and p are the two species of ions. Again the density of all ions is used and the radial distribution function for like-ion interactions is just that for a single component liquid but set equal to zero in the regions of the odd-numbered coordination shells. For the acoustical theory 0.8
1.0 1.2 Scaled pair distance, X.
hq = 3.718roap26(y,raa/rap)/P
1.4
Figure 4. Dependence of the functions of I I and 1 2 upon distance of separation of a pair of ions for the Lennard-Jones potential depicted on the graph.
by the van der Waals forces. Thus in the Walden product we have a property which is essentially independent of the detailed mechanism of momentum transfer. The experimental values of the extrapolated values of the Walden product at 250" are compared in Table VI with the computed values from eq 11 where r" has been computed from X-ray measurements and partial molar volumes in dilute solutions; (z2)is taken as unity; and P is the molar volume of the salt a t 250". The comparison is made at 250" because most of the salts are molten in this region; and also for low-melting salts with long alkyl chains, 250" is sufficiently above their melting points to avoid the region of high curvature of the Arrhenius plots of their transport properties. The temperature dependence is sufficiently small to make the temperat,ure at which the comparison is made relatively unimportant. Since most of the temperature dependence of the transport properties arises from the radial distribution function and there is no adequate simple theory describing it, comparison a t constant temperature is most reasonable. For the tetrapropylammonium salts of the fluoroanions and even for Bu4NPF6, the measured values of the Walden product are nearly the same as the computed values; while for all other salts the measured values are considerably less than the computed values. The argument so far is not convincing if the Walden products for the other salts cannot be accounted for by this model. There are
(13)
where ~ ( Y Y
rola/rap) =
The integrals are expressed either in CYCY or aP space with their corresponding potentials; in and c would correspond to an effective mass and speed of sound for the interaction given by their subscripts.
The latter statement is more uncertain for the radial distribution function for the second shell does not necessarily peak a t the minimum of the potential well in (YCY space because of the geometrical constraints set by the first shell. Now y will be positive and will generally be less than unity because the integral I ~ ( C Y C Y ) is much smaller t,han I2(cYp). The first of two factors which make the ratio of these integrals small is that much of the repulsive part of the like-ion interaction cannot be seen in the second coordination shell and this repulsive part is the major contribution to the integral. Secondly, the radial distribution peaks at a much lower value in this second shell than in the first shell which contains unlike-ion interactions. Obviously, if there are no like-ion interactions, y is zero and 6 is unity. The value of 6 then increases above unity, as the interaction parameter becomes positive and as the likeVolume 70,Number 11 November 1966
3618
J. E. LIND,JR.,H. A. A. ABDEL-REHIM, AND S. W. RUDICH
ion-size parameter raa increases above that for unlike ions re@. As an example, when the larger ions are twice the size of the smaller ones, the factor 6 increases about linearly with y and is 1.53 when y is 0.5. Thus like-ion interactions cannot account for the low experimental values of the Walden product but can only account for high values of the product. If we can find a reason for the low values of the product for salts such as Bu4KBPh4, the anomalous temperature dependence of the Walden product of PrJYBPh? from a high value of the product at low temperatures to one comparable with Bu4KBPh4 at high temperatures can be ascribed to like-ion interactions. This rapid decrease in the Walden product of Pr4KBPh4would then result from the rapid decrease in the interactions of the larger rigid anions in this salt as the structure of the melt expands with temperature. At a high temperature near the melting point of BuJSBPh4, the conductivity of Pr4S B P h 4is still slightly lower than that of the tetrabutylammonium salt, and the Walden product of the P r S + salt is approaching that of the Budsf salt. The Bu4KBPh, salt IS presumed to be without significant likeion interactions because there is no large variation of the Walden product with temperature. The BuJX+ ion is just sufficiently larger than P r 4 S + to prevent interaction between the phenyl rings of the anions. X o w consider the second possible cause of the low Walden products: that the cation chains clog the interstices of the melt. In order to identify this mechanism we must first determine whether the viscosity or the friction factor is causing the low values of the Walden product. S o t e in Table VI that the viscosity coefficients of the BF,- and PF6- salts form a well-ordered series by size or mass. The viscosity for the PFe- salt of a given cation lies about 30% above that for the BF4- salt, and for a given anion the yiscosity increases 8yc per methyl group in the type of chain attached to the cation. The order is not nearly so regular for the electrical conductivities where the distinctions among all anions of BuJf salts are blurred compared to the distinctions among the P r 3 + salts. If the viscous behavior is assumed to be normal because of these regular relations among the viscosity coefficients, low conductivities (ie,,high friction factors) must account for the low values of the Walden product. It is not unreasonable that clogging of interstices in the nielt would affect the conductivity more than the viscosity. The conduction requires ltinetic transfer of mass of adjacent ions in opposite directions, whereas the viscosity arises from the macroscopic deformation of the whole structure with momentum being transferred primarily by phonons rather than by kinetic transfer of mass. The kinetic transfer would be seriously reduced by the long The Journal of Physical Chemistry
alkyl chains, while there would be little effect upon the phonons. Thus the abnormally low values of the Walden product for the tetraphenylborides and especially HexSBF4 (Hex = hexyl) would be attributed to the fact that the interstices in the melt are clogged. The measured Walden product for Bu4NBF4is considerably below that for Bu4SPF6as well as below the value calculated from the theory. The BF4- may just be sufficiently smaller than the PF6- ion to allow appreciable clogging by the chains of the Bu4N+ion while the larger size of the latter anion drastically reduces such clogging. While the above explanation does not correspond directly to the model of rough hard spheres it should be noted that when rotational as well as translational momentum transfer can occur, the Walden product decreases slightly. For a dilute gas in the first approxi, * Walden ~ prodmation of Chapman and E n ~ k o g , ’ ~the uct decreases to 80% of that for smooth spheres in the extreme case where all the mass is placed on the surface of the spheres. As in the case of the fused salts, the primary effect is in the friction factor which increases about 40% compared to an increase of the viscosity of about 15%. Having now rationalized the data for the tetraalkylammonium salts by this approach, we will briefly consider other available data. The experimental values of the Walden products of the alkali halides21,22 are often as much as twice the value of the products calculated from this theory by using Pauling radii. The measured values decrease with increasing temperature and especially for salts with a large disparity in the sizes between the anion and cation the product decreases extremely rapidly. In the case of KaC1, the product decreases from three times the calculated value a t 800” to twice the calculated value near 925” and continues decreasing at higher temperatures. This behavior suggests contributions from like-ion interactions in the case of these small ions. In the case of the alkali nitrates,22the Walden products can be calculated from the diffusion coefficients. The products actually increase slightly with temperature except for Lis03 and from the theory yield ion-size parameters at 350” of 5.35, 4.46, and 4.53 A for Lin’O3, S a N 0 3 , and IZn’Os, respectively. The value for LiNOs may be high be(19) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” John Wjley and Sons, Inc., New York, N. Y., 1954, p 509. (20) D. W. Condiff, W. K. Lu, and L. S. Dahler, J . Chem. Phys., 4 2 , 3445 (1965). (21) A. Klemm, “Molten Salt Chemistry,” M. Blander, Ed., Interscience Publishers, Inc., New York, N. Y.,1964, pp 564-578. (22) B. R. Sundheim, “Fused Salts,” B. R. Sundheim, Ed., McGrawHill Book Co., Inc., New York, N. Y., 1964, p 224.
STRUCTURE OF ORGANICMELTS
cause of like-ion interactions for there is a small decrease of the Walden product with temperature in contrast to the other two nitrates. The ion-size parameters for the other two nitrates are only 0.5-1.0 A larger than the largest estimate of the sizes of the ions. Unfortunately, data are not available for salts of comparable size to the nitrates but with more symmetrical ions. Small tetraalkylammonium salts decompose before melting well over 300".
Conclusion We have shown that the acoustical model with re% sonable parameters will yield the viscosity of a salt such as Pr4NPF,. This theory, as well as that for hard spheres, yields the correct values of the Walden product a t 250" for Pr4NBF4and Pr4NPF6as well as approximating that for Bu4NPF6. Both theories require only one parameter, the ion size, in order to predict the Walden product. When ion sizes from partial molar volumes and X-ray measurements are used to predict the product for the other salts, the measured values are
3619
much smaller than the calculated values. The small experimental values are caused by the abnormally high friction constants which arise, not from the increased forces of interaction, but from the long chains which clog the structure and impede movement in the melt. Finally the rapid decrease of the Walden product of Pr4NBPh4 with temperature from abnormally high values of the product may be caused by the decrease of the anion-anion interactions in this salt as the structure expands. Although this type of analysis is restricted to salts where the Walden product is relatively insensitive to temperature, it does provide a description based upon molecular parameters which is sensitive to the structure of the melt rather than the energy of interaction between ions. The analysis is also essentially independent of the detailed mechanism of momentum transfer. Both the hard-sphere model with its discontinuous potential and lack of strong correlation and also the acoustical modification of Kirkwood's model with its slowly varying potential and high correlation with the surrounding medium predict the same results.
Volume '70,Number 11 November 1966
