T H E
J O U R N A L
O F
PHYSICAL CHEMISTRY
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U.S. Patent Ofice @ Copyright, 1989, by the American Chemical Society
VOLUME 73, NUMBER 7 JULY 1969
Friction Constants for Tetra-n-propylammoniumTetrafluornborate by S. W. Rudich and John E. Lind, Jr. Department of Chemical Engineering, Stanford University, Stanford, California 9.4306 (Received January 17, 1060)
The diffusion of tetra-n-propylphosphonium tetrafluoroborate into its ammonium homolog was measured by cm2/sec a capillary technique between 252 and 269’. The cation diffusion constant, which is 0.86 X at 252O, is an approximation to the self-diffusion coefficient of the cation. An estimate of the error in this approximation is made from the electrical conductances of the salts. The conductances of both the phosphonium and arsonium salts are reported. From the diffusion coefficient and the electrical conductance, the cation-cation and cation-anion friction constants are calculated. They are four to five times as large as the friction constants for KC1; however, the ratio of the friction constants for ions of the same relative size is about the same for Pr4NBF4 and KCl. Although KCl has nearly twice the coulomb energy, the coulomb potential does not appear to make significant contributions to the friction constants. The viscosity of Pr4NBF4, estimated from the friction constants by a hard-sphere model, is closer to the measured value than a similar calculation yields for KC1. However, the effects of the smaller coulomb potential in the organic salt cannot be separated from the effects caused by its having larger, polyatomic ions at twice the hard-sphere density of KC1.
Introduction The organic salts melt at sufficiently low temperatures to permit measurements of many properties with relative ease compared to similar measurements on the alkali halides. The organic anion-cation distance is a t least twice that of the alkali halide and thus the organic salt provides a melt with lower charge density which can then be contrasted with the alkali halides. We report here measurements of diffusion and electrical conductivity for tetra-n-propylammonium tetrafluoroborate, Pr4NBF4,and its phosphonium homolog Pr4PBF4. The purpose is to detect any differences of the data for the organic salt from those of KC1 which might arise from the lower coulomb energy of the organic salt. Conversely, the purpose is also to determine if the transport properties of these organic salts might be characterized by simple theories using spherically symmetrical potentials. The fluoroborates are the most stable of the organic salts, and the tetrapropyl salts are the ones with the smallest cation which melt sufficiently low to form stable liquids. As was shown previously in a paper on friction constants (FC),’ the self-diffusion coefficients as
well as the electrical conductance are required to calculate the viscosity. Of the two self-diffusion coefficients, the smaller coefficient will contribute more to the viscosity. Since the cation of Pr4NBF4 is the large ion, its smaller diffusion constant along with the conductance will dominate the calculation of the viscosity. This cation self-diffusion coefficient for Pr4NBF4 is estimated from the interdiffusion of a phosphonium cation “tracer” into the ammonium salt.
Experimental Section The preparation of the salts has been given before as well as their viscosity coefficients, densities,2 and one of their conductivities.8 The newly measured conductances of the phosphonium and arsonium salts, given in Table I, are plotted in Figure 1. The diffusion measurements were made by standard capillary techniques in which the tracer diffuses from a capillary into a well-stirred reservoir. Eleven diffusion (1) G.Morrison and J. E. Lind, Jr., J . Phys. Chem., 72, 3001 (1968). (2) 8.W.Rudich and J. E. Lind, Jr., J . Chem. Phys., in press. (3) J. E. Lind, Jr., H. A. A. Abdel-Rehim, and S. W. Rudich, J. Phys. Chem., 70, 3610 (1966).
2099
'2100
S. W. RUDICHAND JOHN E. LIND,JR.
I
I
I
230
I
I
1 1
1
250
240
260
TEMPERATURE, "C Figure 1. Equivalent conductance w a function of temperature.
capillaries were constructed from 0.089-mm i.d. F-hard stainless steel tubing with an average length of 1.015 f: 0.004 cm. They were held vertically in a 1.5-cm diameter circle with their open ends protruding about 0.1 cm from a stainless steel holder and with the other ends gold soldered to a stainless steel base plate to close off the bottom ends of the capillaries. The assembly could be rotated about the center of the circle of capillaries. The diffusion run consisted of filling the capillaries with the slightly more dense phosphonium salt and then rotating the holder with the capillaries submerged in the ammonium salt. The ratio y of the average concentration of phosphorus in the capillaries after the run of duration t to the initial concentration in them permitted computation of the diffusion coefficient D from the equation4
Dt/P = n(1 - y)'/4
(1)
where 1 is the length of the capillaries. Equation 1 is the first term in a series expansion and it is valid to 0.1% for values of y greater than 0.5. Table I : Electrical Conductivity --PrrAaBF4"-
--PrrPBF&----
T,OC
A
T,OC
A
244.30 254.01 263.56 273.75 249.13 255.97
27.60 30.04 32.56 35.15 25.80 31.22
215.51 224.65 234.39 244.01 254.00 263.75 220.32 232.17 244.25 256.42
23.62 25.51 25.05 30.48 32.96 35.45 24.71 27.55 30.49 33.56
PrhPBFr: In A = 7.5025 - 2320.2/T, &0.0012. Pr&BFI: In A = 5.2574 246.63/T 6.2800 X 1O6/TB, kO.0010. a
+
-
The Journal of Physical Chemistry
The phosphorus was analyzed by the methods of Kolmerter and Epstein6 following the conditions suggested by Quinlan and De S e m B The phosphonium salt was oxidized with persulfate, neutralized to the phenolphthalein end point with concentrated NaOH, and then treated to form the yellow molybdovanadophosphoric acid. The absorbances of the unknown and the NaH2P04standard solutions were measured at 4000 8 in a Beckman DB spectrophotometer. The standards were prepared in a similar fashion to the unknown and a t the same time. Readings were taken 15 min after completing the preparation. The concentration of the unknown was interpolated between the reference solution concentrations since Beer's law was not strictly followed. The precison was about 2%. Several sources of systematic error were considered and corrections were made. First, eq 1 applies when the concentration of tracer remains at zero in the reservoir. However, in these experiments the reservoir contained only 4.5 g of salt, so that the loss of phosphonium from the capillaries during the run and by thermal expansion just before immersion yielded a maximum concentration of about 0.5% for the longest runs. A small correction was made by assuming the reservoir had its mean concentration throughout the run. The correction to the diffusion coefficient was about 1%. Diffusion runs of 70-sec duration were performed to determine the true initial amount of tracer in the capillaries. The iterative calculation process estimated the diffusion constant from the long duration diffusion run and then calculated back to the initial concentration from the measured concentration at 70 sec of diffusion. This procedure automatically corrected for concentration effects brought about by immersion, expansion of tracer out of the capillary at the start and contraction of nontracer into them at the end of the run, as well as the sweeping of fluid out of the end of the capillary by the stirring. However, if the last effect is significant, the effective length of the capillary should be used instead of its actual length. The following dye experiments were performed to estimate the effective length of the capillaries. A 20% by weight aqueous sucrose solution was found to have about the same density and viscosity at 25" as the molten phosphonium salt. Phenolphthalein was added to this solution and made alkaline with NaOH, and it was placed in the capillaries. A 30% by weight glycerol solution was placed in the reservoir to simulate the ammonium salt. Runs of 5 sec a t both 21.7 and 43.: rpm showed by spectrophotometric analysis at 5730 A a loss of 1%. Runs of 45 and 60 sec showed losses of 1 and 2.4%) respectively. Although the diffusion coefficient of the dye is not known, if it were as large as 10-6 cm2/sec these latter figures (4) A. T. McKay, Proc. Phgs. Soc., 42, 547 (1930). (5) J. Kolmerter and J. Epstein, Bnal. Chem., 30, 1536 (1958). (6) K. P.Quinlan and M. A. De Sesa, ibid., 27, 1626 (1955).
FRICTION CONSTANTS FOR TETRA-~-PROPYLAMMONIUM TETRAFLUOROBORATE
2101
Table I1 : Diffusion of Pr4PBF4 into Pr4NBF.j Stirring rate, rpm
Time, 1 , 8ec
21.7 43.4 43.4 43.4
10,000 14,000 19,000 19,000
Averagea concentration a t t
r+-
43.4
T = 252'; 1.262 1.158 1.009 1.012
=
D++ x l o t s
7
cmz/aec
5 . 6 1 X 10-gg/sec 0.678 0.621 0.541 0.542
1 . 8 0 8 2 ~0.008
70
Corrected
g/seo
0.84 0.83 0.90 0.89
3.0 3.2 2.5 2.5
0.973
Av 0 . 8 6 f 0 . 3
T 21.7 21.7 21.7 21.7 21.7 21.7
7,000 10,000 10,000 13,500 14,000 70
= 261';
r+-
=
5.17
x
1.334 1.275 1.269 1.146 1.120 1.835ic0.015
10eg/sec 0.706 0.674 0.671 0.605 0.592 0.972
T 7,000 l U , 000 14,000 70
269"; r+- = 4.84X 1.296 1.208 1.099 1.866 f 0 . 0 1 6 =
g/sec 0.672 0.626 0.569 0.969
2 . 8 i0 . 5 1.2 1.8 2.1
1.24 1.13 1.07
Av 1 . 1 5 i 0 . 0 7 a
2 . 8 i0 . 3 2.2 3.4 3.3 2.8 2.5
1.00 0.86 0.88 0.93 0.96
Av 0 . 9 3 i c 0 . 0 5 21.7 21.7 21.7 21.7
r++x 109,
1 . 7 ic 0 . 5
Concentrations are in arbitrary units.
could be accounted for by diffusion. Nanis, Richards, and Bockris' state that at high Reynolds number the effective length is established within a revolution. This 1% change could therefore arise from the difference between the effective and actual lengths. Because any change in starting concentration caused by change of effective length has already automatically been accounted for by the 70-sec molten salt runs, only the length used in eq 1 contributes to the error. Thus at most a 2% correction to the diffusion coefficient would be necessary. Because of the small size of the correction and its uncertainty, no correction was made to the length of the capillaries. Note that this is in contrast to the findings of Nanis, Richards, and Bockris, who predict at the Reynolds numbers of 6 and 13 used in this experiment effective length corrections of about 6-7y0,. The differences may be attributed to the differences in geometry. I n this experiment the ends of the thinwalled tubes (0.004 mm) projected less than a millimeter above the supporting disk shaped holder, so the stream lines were not greatly distorted by stirring. Yet the mixing was adequate in the reservoir as was shown by additional dye experiments. The results are given in Table 11. I n general we can say that the over-all precision in the calculation of the diffusion constant is within *6%. This precision is excellent considering these long diffusion times with the salts a t rather high temperatures.
Discussion These measurements of the interdiffusion of Pr,P+ into Pr4nT+ ions in the fluoroborate melt should yield self-diffusion coefficients for Pr4N+ ion which are low by less than 6%. This assumes that the self-diffusion coefficients are proportional to the conductances which differ by 6% in Figure 1. This difference is within the limit of accuracy of the data. Should the diffusion coefficients be more closely related to the viscosity, the probable error increases to lO-l5%, but does not significantly affect any of our analysis. I n a previous paper, FC,' a friction formalism similar to that used by Laity*was shown to be useful. For 1-1 electrolytes the friction constants c are related to the equivalent electrical conductance and the self-diff usion coefficients by the equations A
(2)
= e2/(+-
Di-= kT/(c+-
+ l++)
(3) Table I11 contains the friction constants for PriNBF4 as well as those for KC1, and the former are shown in Figure 2 . The friction constants for the organic salt are four to five times as large as those for KCl, but the (7) L. Nanis, 5. R. Richards, and J. O'M. Bockris, Reu. Sci. Instr., 36, 673 (1965). (8) R. W.Laity, Ann. N . Y. Acad. Sci., 79, 997 (1960); J . Chem. Phys., 30, 682 (1959). Volume 79, N U W Z ~7E TJuly 1960
S.W. RUDICHAND JOHN E. LIND,JR.
2 102
1
5.5
a, '.
4.5
0
t
p 3.5
+
++
2.151
*+
LL LL
8 3.0 V
t.
+
E 2.0
L
POSITIVE LIKE ION
t
+
1.5
I
I
260
265
I
255
+
I
270
the effect does not depend directly on the coulomb potential. For with the organic salt this friction constant is quadrupled as is the one from the conductance, even though the coulomb energy is halved. If coulomb terms are important, they must appear in Boltzmann factors which are nearly the same for KC1 and the organic salt. This factor which might occur in a radial distribution function is exponentially related to the inverse of Tu+- in Table I11 where u+- is the anioncation collision diameter. As both salts are near their melting points, it is not surprising that their Boltzmann factors are about the same. Since no approximate intermolecular potential is known for Pr4NBF4,a hard-sphere model will be used and the results will be compared with KC1 data treated in the same fashion. For the hard-sphere model, ignoring the coulomb forces, the friction constants are of the form
TEMPERATURE, 'C Figure 2.
Friction constants for Pr4NBF4.
rati.0 of the like ion friction constant for the larger ion to the anion-cation friction constant is of the same size for both salts. KC1 was chosen for comparison because the size ratio of C1 diameter to K-C1 distance is about the same as Pr&+ diameter to Pr4N-BF4 distance. Thus {++ for PrrN+ must be compared with {-- for
c1-.
Table I11 : Transport Parameters Salt
Temp, "C 10y+-, g/sec (obsd) 109b++ (obsd) loof-, (obsd) A U--,
li
ff+-(U),
obsd
g + - ( ~ ) , calcd
irdp/6
TU+?I+-, OP
9tt
9-7
9
(calcd) (obsd)
PrrNBF4
252 5.61 2.8
...
6.26 2.76 20.0 7.13 0.516 3287 0.883 0.269 (0.14) 1.29 2.17
K C1
783 1.44 0.73 0.94 2.71 1.56 3.99 2.15 0.255 2862 0.259 0.047 0.112 0.418 1.17
900 1.18" O.5lb 0.74* 2.63 1.51 3.57 1.87 0.215 3085 0.185 0.029 0.076 0.290 0.80°
a I. S. Yaffe and E. R. Van Artsdalen, J . Phye. Chem., 6 0 , l l (1956). J. O'M. Bockris, S. R. Richards, and L. Nanis, ibid., 69, 1627 (1965). ' I . G. Murgulescu and S. Zuca, 2. Phys. Chem. (Leipzig), 218,379 (1961).
I n FC,' the conductance of KC1 was estimated accurately by the Brownian diffusion model, but the like ion friction constants were underestimated. It was surmised that the underestimate was caused by the neglect of coulomb interactions. If this be true, The Journal of Physical Chemistrz,
where p is the total particle density. The radial distribution function g is based on the total density so that it can be compared with distribution functions for single component systems. The experimental friction constants were used to calculate g+- for Table 111. The value of u+- for this calculation was that used to fit the hard-sphere equation of stateU2)OBy assuming the nearest neighbors in the salt are always of opposite sign, the salt can be treated as a one-component fluid and the radial distribution function was calculated for these salts from the scaled particle theory.'" These theoretical values are low compared to those calculated from the friction constants. The values of g(a) for a salt are expected to be higher than those for a nonelectrolyte of the same density because of the shortening of the nearest-neighbor distances by the coulomb field.'l However, the value of g(u) from the friction constant for Pr4NBF4is even so somewhat large. This larger value arises from the absorption of the effects of rotation and of the greater reflection or rattling in the cage of nearest neighbors in the fused organic salt, for it is at about twice the hard sphere density ( r a 3 p / 6 ) of KC1. The reflections are probably also enhanced by the chains on the cation. The rotational contribution can be estimated in the Enskog approximation by the rough sphere model of Pidduck,l2 and it can account for an increase of the friction constant of only 15%. Thus the density effects are dominant. As an estimate such effects when no coulomb field is present, g ( u ) can be caluclated for ccl4 from its diffu(9) F. H. Stillinger, J. Chem. Phys., 35, 1581 (1961). (10) H. Reiss, Advan. Chem. Phys., 9, 1 (1965). (11) G. (1968).
Morrison and J. E. Lind, Jr., J. Chem. Phys.,
49, 5310
(12) S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge University Press, London, 1961.
FRICTION CONSTANTS FOR
TETRA-7%-PROPYLAMMONIUM
TETRAFLUOROBORATE
sion coefficient'3 a t 25". The hard-sphere density of CCL, based on its cornpressibility,'O is 13% less than that of the organic salt but yet g(u) from diffusion is 12, a little over twice the theoretical value of 5 computed from the density. The rough sphere model would only account for a 18% decrease in g(u) calculated from the diffusion coefficient. Thus sizable density effects can arise without the coulomb field. If direct coulomb terms are important, they should affect the viscosity more than the friction constants. The viscosity in Table I11 is calculated from the friction constants by the equation
v
= '/lop =
v+-
[U+-Y+-
+
+ v++ + v-
1/2(uf++2f++
+
u-2s--)
1
2103
fused salt. Secondly, from the Brownian model in FC we see that the viscosity averages over a wider range of interparticle distances than the friction constant, and thus a larger effective u should be used to calculate the viscosity from the hard sphere model. The same calculation of the viscosity of Pr4NBF4 is more adequate. This better agreement might arise from smaller coulomb terms, but it is probably masked by the general increase of momentum transport because of the high packing density of the organic salt as well as its hindered rotation. Thus no clear separation of the effects can be made at present.
Conclusion (5)
where p is the total particle density. The contribution of the anion-anion interactions to the viscosity of Pr4NBF4 is relatively small and is scaled from the cationcation interactions on the basis of the KC1 data. Single ion diameters are scaled from u+- on the basis of the ratio of Pauling diameters for KC1 and the diameter of BF4- is taken as 2.76 A . 3 Equation 5 is valid in the Enskog approximation where the radial distribution function absorbs the unknown density and geometrical dependence. For the equation to be valid, these effects can only be space coordinate dependent. The viscosity of KC1 is drastically underestimated for several reasons. The 20% correction for the perturbation of the distribution function has been neglected, since it is uncertain how well it applies to high density fluids, especially a
The friction constants for Pr4NBF4 are four to five times larger than those for KC1. However, the relative size of the two friction constants for each salt is about the same. Thus the Nernst-Einstein approximation would be equally poor in both cases. The results also suggest that direct coulomb effects are small. The viscosity of Pr4NBF4is better approximated than that for KC1 from the friction constants by the hard-sphere theory. However, the effects of the much higher organic salt density on its viscosity cannot at present be separated from the effect of the coulomb field. Acknowledgment. This study was aided by grants from the Office of Saline Water, U. S. Department of the Interior. (13) R. E. Rathbun and A. L. Babb, J. Phys. Chem., 65, 1072 (1961).
Volume 79, Number 7 July 1069
