J . Phys. Chem. 1987, 91, 210-215
210
was fitted to a single or a double Lorentzian and corrected for the shimmed line-width values of the magnetic field (of the order hertz). To be certain in exciting and observing all deuterons the signal amplitude of the free induction decay was checked with the expected value for each sample. Conclusion
The increasing rate of the transverse relaxation of the deuterons in sodium polyacrylate is correlated with the increasing persistence length on diluting the solution. This is confirmed through varying the ionic strength at a fixed polymer concentration by adding NaCl to the solution. A simple rotation diffusion model was presented to describe the experimental data. Two interpretations of this model were given to obtain a relation between the dynamic parameters (correlation times and dynamic persistence lengths) and the static
parameters (intrinsic and electrostatic persistence length). It is found as a first approximation that at high concentrations a reorienting sphere with internal rotation adequately connects the relaxation rates with the persistence length. At low concentrations the persistence length is better approximated by a prolate ellipsoid model. At present it seems that this model is very well capable of reproducing the double-exponential relaxation rates. Other model^^^-^^ on polymer motion (main chain dynamics) will be treated in a forthcoming paper. Registry No. CD,PAA.xNa, 104876-23-5. (29) Valeur; Jarry, J. P.; Geny, F.; Monnerie, L. J . Polym. Sci., Polym. Phys. Ed. 1975, 13, 667. (30) Bendler, J. T.; Yaris, R. Macromolecules 1978, 11, 650. (31) Skolnick, J.; Yaris, R. Macromolecules 1982, 15, 1041.
Measurement of Binary and Ternary Mutual Diffusion Coefficients of Aqueous Sodium and Potassium Bicarbonate Solutions at 25 "C John G. Albright,* Roy Mathew, Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129
and Donald G. Miller University of California, Lawrence Livermore National Laboratory, Livermore. California 94550 (Received: April 17, 1986: In Final Form: August 22, 1986)
Binary mutual diffusion coefficients are reported for NaHC03-H20 and KHC03-H20 at 25 OC for concentration ranges of 0.015-0.9 M. Experimental results are compared with those diffusion coefficients calculated from the Onsager-Fuoss theory. Thermodynamic diffusion coefficients are calculated. Ternary diffusion coefficients for NaHCO, (0.25 M)-KHCO, (0.25 M)-H20 at 25 OC are reported. The results suggest that, up to 0.5 ionic strength for this system, ternary diffusion coefficients may be estimated within 1-2% accuracy from the experimental binary diffusion coefficients and the limiting ternary diffusion coefficients which are calculated from limiting ionic conductances. A few important partial molar properties of the ternary system are calculated by using Young's rule. Density measurements are reported for the KHCO,-H,O system up to 1.0 M.
Introduction
Sodium bicarbonate and potassium bicarbonate are important solutes in many aqueous systems found in nature, yet good experimental diffusion data for these systems have not been available. In 1941 Vinograd and McBain' reported data for the sodium bicarbonate system obtained by the diaphragm cell method. However, at that time, techniques for the method had not been refined, and the results obtained were subject to considerable error. Recently good quality data for the sodium bicarbonate system were reported by Leaist and Noulty2 which had been obtained by the restricted-diffusion conductometric methods3 However, their data are for low concentrations and only extend up to about 0.1 M. Thus, there is still a need for good data at higher concentrations. Experimental work reported here gives diffusion coefficients at 25 OC for sodium and potassium bicarbonate salts in aqueous binary solutions up to almost 0.9 M. The approximate saturation concentrations for NaHCO, and KHCO, at 25 OC are I . 18 and 3.15 M, re~pectively.~The diffusion Coefficients were measured by the free-diffusion method with either Rayleigh or Gouy types of interferometer^.^^^ Thermodynamic diffusion coefficients were (1) Vinograd, J. R.; McBain, J. W. J . Am. Chem. SOC.1941, 63, 2008. (2) Leaist, D. G.; Noulty, R. A. Can. J . Chem. 1985, 63, 2319. (3) Harned, H. S . ; French, D. M. Ann. N.Y. Acad. Sci. 1945, 46, 267. (4) Linke, W. F. Solubiliries of Inorganic and Metal Organic Compounds,
4th ed.: American Chemical Society: Washington, DC, 1956; Vol. 2.
0022-3654/87/2091-0210$01.50/0
calculated by using activity coefficients tabulated by Peiper and Pitzer' and by Roy et ale8 Also presented here is one set of ternary diffusion coefficients measured at 25 OC at a mean concentration of 0.25 M for both solutes in water. The results suggest that ternary diffusion coefficients may be estimated from binary diffusion coefficients at concentrations up to 0.5 ionic strength for the system NaHCO,-KHCO,-H,O. Density measurements were made by pycnometry up to 1.15 M concentrations for aqueous solutions of potassium bicarbonate at 25 "C. At the time, the data of Larson et aL9 were unavailable. Experimental Section
Chemicals. Mallinckrodt reagent grade NaHC0, and KHCO, were used without further purification for all experiments. Distilled water was used for preparation of all solutions. Values of 84.01 and 100.12 were used as the respective molecular weights of N a H C 0 3 and KHC03. Preparation of Solutions. At both TCU and Lawrence Livermore National Laboratory (LLNL), ah solutions were prepared (5) Gosting, L. J. Aduances in Prorein Chemistry, Academic: New York, 1956; Vol. 2, pp 429-554. ( 6 ) Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids, Butterworths: London, 1984. (7) Peiper, J. G.; Pitzer, K. S . J . Chem. Thermodyn. 1982, 24, 613. (8) Roy, R. N.; Gibbons, J. .I. Wood, ; M. D.; Williams, R. W.; Peiper, J. C.; Pitzer, K. S . J . Chem. Thermodyn. 1983, 25, 2017. (9) Larson, J. W.; Zeeb, K. A,; Hepler, L. G. Can. J . Chem. 1982, 60, 2141.
0 1987 American Chemical Society
Mutual Diffusion Coefficients of N a H C 0 3 and K H C 0 3 by weight to the nearest tenth milligram with precision 1 kg Mettler balances. Buoyancy corrections were made in all cases. Diffusiometers. Diffusion measurements at TCU were made with a Beckman Model-H electrophoresis-diffusion apparatus. Optically this is a double-pass instrument in which light is reflected back through the cell. The instrument was operated in the Rayleigh optical mode. Photographs of the Rayleigh fringe patterns were taken on Kodak Metallographic glass plates. A mercury vapor lamp and filters provided a 546-nm wavelength light source. A multiple source slit yielded Rayleigh fringe patterns that were of the order of 12 fringes wide in the horizontal direction of the cell, Le., y direction. Diffusion experiments were performed in Beckman double channel glass cells. Free-diffusion boundaries were formed in one channel, whereas the other channel contained homogeneous top solution. All cell fittings were made of Teflon. A double-slit mask fit directly and reproducibly on the front of the glass diffusion cell. A thin wire was attached horizontally across the cell mask at a position 2.5 cm below the vertical center of the cell. The shadow of the wire appeared as an unexposed line across the fringe patterns on the photographic plates. This line was used as a reference position for the alignment in the x direction of the fringe patterns when they were analyzed. The diffusion cells were immersed in the water bath of the instrument, which was maintained at 25 O C to within fO.01 deg. A calibrated Brooklyn thermometer with a 1-deg range was used to measure the bath temperature. The magnification factor for the optical system was measured by photographing a reflecting ruled scale placed on the mirror, which is located just behind the cell position and which is the effective center of the cell. By measuring the separation distances of the ruled lines on both the scale and the photograph with a Davis-Mann Measuring Machine, it was determined that the magnification factor was 1.0017. However, when this value was used, it appeared that the diffusion coefficients calculated for the bicarbonate-water systems were slightly higher than the values obtained a t LLNL on an instrument which has a much higher inherent accuracy. Since the measurements at TCU were made by using two different diffusion cells, one for the measurements of sodium bicarbonate and one for potassium bicarbonate, it was decided that a magnification factor would be determined for each cell by bringing the TCU measurements into agreement with the LLNL measurements. This was done for each solute by a least-squares method. This calibration technique is similar to that used at LLNL before its Model H was realigned.I0 The magnification factors used for the sodium bicarbonate experiments and potassium bicarbonate experiments were 1.0037 and 1.0030, respectively. This amounts to an adjustment of the TCU diffusion coefficients by 0.40% and 0.26%,respectively. Diffusion measurements at the LLNL were made with the “Gosting” diffusiometer operating in the Gouy optical mode.lla,b This instrument was originally built at the University of Wisconsin under the direction of Dr. Louis J. Gosting. The thermostat of the instrument was maintained at 25 f 0.01 O C , measured with a calibrated 1-deg range thermometer. The distance from the center of the diffusion cell to the photographic plate, Le., the b distance, was found to be 3.088 852 m. The diffusion cell was made of quartz, and the overall geometrical quality of the cell was good. The inside dimension of the diffusion cell, the a distance, was 2.5064 cm. Automatic Plate Scanner. At TCIJ the photographs of the Rayleigh fringe patterns were analyzed with the computerized plate scanner mentioned in ref 12. The unit is now interfaced to an Apple IIe computer through an ADALAB board. During an experiment, one base line, three fraciional parts of a fringe, and nine Rayleigh free-diffusion patterns are usually photo(10) Albright, J. G.; Miller, D. G. J . Pfiys. Cfiem. 1972, 76, 1853. (1 1) (a) Gosting, L. J.; Kim, H.; Lowenstein, Id. A,; Reinfelds, G.; Revzin, A . Reu. Sci. Instru. 1973, 44, 1602. (b) After thi: death of Prof. Gosting, the instrument went to Prof. Kegeles at the Universiiy of Connecticut. It is now at LLNL, thanks to the courtesey of Prof. Kegeles and the University of Connecticut. (12) Sherrill, B. C.; Albright, J. G. J . Solution Chem. 1979, 8, 217.
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 211 graphed. Measurement of the separation of Creeth fringe pairsI3 by the plate reader yields, after analysis, an apparent diffusion coefficient, D’, for each free-diffusion pattern. These D’ fit a straight line of D’vs. the reciprocal of elapsed time, I/t’, typically with a standard deviation of about 0.03%. Although this would imply a precision of better than 0.1% for the measured diffusion coefficients, Le., the value of D’extrapolated to l / t ‘ = 0, other technical problems and imperfections of the diffusion cells limit the accuracy. This may be 0.25% for well-behaved systems. Because of problems encountered with these bicarbonate systems, such as formation of small bubbles, particularly at higher concentrations, the accuracy is somewhat less. A more detailed description of the use of the plate reader for the measurement of diffusion coefficients by the Rayleigh interferometric method is available in the microfilm edition of the journal (see Supplementary Material Available paragraph at the end of this article) and may also be obtained by writing one of the authors, J.G.A. Plate Reader at LLNL. The Gouy interference patterns obtained in experiments at L L N L were read with a Gaertner tool makers’ microscope. This unit also came from the laboratory of Dr. Gosting and is fitted with a scanning device that allows the determination of fringe minima to within 1.0 pm.’491S Fringe positions are punched onto IBM cards for analysis by a CDC 7600 computer. Pycnometers. Two-stemmed Ostwald-type pycnometers were used to measure density of the KHCO, solutions at TCU. The volumes of the pycnometers were about 30 cm3. Because of problems due to formation of very small bubbles, possibly due to outgassing of COz, the accuracy of these measurements should be considered to be only fO.Ol%. Density measurements at LLNL were performed with single-stem pycnometers of about the same volume. Again the accuracy of the measurements should be considered to be f0.01%. Measurement of Binary Diffusion Coefficients. The general procedures for performing the measurements are given by Albright and Miller.I0J6 Diffusion coefficients were calculated by averaging data from Creeth pairs13which met the criteria 0.25 < zG) < 1.0. Here z o ) = erf inv ( ( J - 2j)/4, where j is the lower-numbered fringe of the symmetrical fringe pair and J is the total number of fringes. Measurement of Ternary Diffusion Coefficients. Four diffusion coefficients are necessary to describe one-dimensional isothermal diffusion in a three-component system. The flow equations may be written as -J1
= DII(dCl/dx) + D1Z(dCz/dx)
(1)
-J2
= D 2 l ( d C l / d ~+ ) DZZ(dC2/dx)
(2)
The four diffusion doefficients, Dik,relate the flows of the solutes Ji to the concentration gradients of the solutes. Here the diagonal coefficients, Dit, are referred to as the main-term diffusion coefficients, and the off-diagonal coefficients, Dik ( i # k ) , are cross-term diffusion coefficients. In this study, the four diffusion coefficients were measured for the NaHC03-KHC03-H20 system at 25 “ C at a concentration of 0.25 M of each solute. The subscripts 1 and 2 are used to denote the solutes NaHCO, and KHCO,, respectively. Ternary measurements were performed at LLNL by using the free-diffusion Gouy interferometric method. The experimental procedures have been described p r e v i ~ u s l y . ~ * ’ ~ ~ ~ * The theory for the analysis of Gouy data was developed by Fujita and Gosting.lg A refinement of the method18,z0uses the (13) Creeth, J. M. J . Am. Chem. SOC.1955, 77, 6428. (14) Albright, J. G. Ph.D. Thesis, University of Wisconsin, Madison WI, 1962. (15) Wendt, R. P. Ph.D. Thesis, University of Wisconsin, Madison, WI, 1960. (16) Albright, J. G.; Miller, D. G. J . Solution Cfiem. 1975, 4, 809. (17) Dunlop, P. J.; Gosting, L. J. J . Am. Chem. SOC.1955, 77, 5238. (18) Woolf, L. A.; Miller, D.G.; Gosting, L. G. J . Am. Chem. SOC.1962, 84, 317.
212
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987
TABLE I: Density Measurements of KHC03-H20 Solutions at 25
Albright et al. TABLE II: Data for the System NaHCOq-H20 at 25
OC
molality, mol/kg
molarity, mol/dm3
density, kg/dm3
0.10021 0.10074 0.200 52 0.21224 0.300 72 0.401 06 0.501 22 0.501 32 0.599 34 0.601 58 0.701 80 0.701 88 0.802 07 0.90000 1.002 67 1.002 95 1.10326 1.203 43
0.099 61 0.099 63 0.198 55 0.209 98 0.296 83 0.394 32 0.491 14 0.491 15 0.584 95 0.587 16 0.682 53 0.682 49 0.777 22 0.869 06 0.964 3 1 1.056 74 1.057 13 1.14873
1.003 97 1.003 55 1.010 03 1.01033 1.01675 1.022 66 I .029 03 1.028 86 1.034 53 1.034 79 1.040 85 1.040 67 1.046 8 1 1.052 60 1.058 26 1.063 88 1.064 00 1.069 53
values of the area under the Q graph vs.f((), denoted as Qo,and the reduced height area ratio, DA. We note here that the values of DA were obtained from each ~ /fringes ~ fringe pattern by a linear extrapolation of C, V S . ~ ( { )of 1-6, 8, and 10. Nine fringe patterns were photographed at evenly spaced values of l/t'where t'is the clock time from the start of the experiment. The values of DA from all the fringe patterns were then linearly extrapolated vs. l/tr, to l / t r = 0 to yield the correct value of DA for the experiment. Areas, Qo,of the Q graphs were obtained as follows. For each fringe, j , the Q, values from the nine photograpohs of the fringe patterns are averaged. This was done for fringes 0-6, 8, 10, ..., 78. Curves were tit through the values of Q,vs.f(z) by a running four-point interpolation formula (for unevenly spaced points), and the integral was calculated from the series of these curves. A three-point formula was used at each end. The integration starts at = O,f({) = 0 and ends at Q = O , f ( { ) = 1. Although only two experiments are required to measure the four diffusion coefficients, four experiments were performed in order to improve the data and give statistical information.
Results and Discussion Binary Systems. Density Measurements. Densities of aqueous solutions of KHCO, measured at TCU at 25 "C are shown in Table I. In order to calculate mean molar concentrations, C, for the diffusion experiments, the following equation was fit by the method of least squares to these data pz =
0.997045
+ 0.06516C + 0.002662C3/20.00444c
(0
< C < 1.0 M) (3)
where pz is the density of KHCO, in g cm-,. The standard error of the fit, 6,is 2.1 X The densities presented here agree within 0.02% of values calculated from the apparent molar volumes of the KHC03-H20 system at 25 OC obtained by Larson et aL9 Because density data for the system NaHC03-H20 at 25 "C were available from Perron21 and from Larson et al.,9 measurements were not made at TCU for this system. For the purpose of calculating molar concentrations, the following equation was fit to their data by the method of least squares
+ 0.061367C - 0.005155C3/2+ 0.001248c + 0.0001162C5/2 (0 < C
= 0.997045
for which 6 is 0.7
X
p,
< 0.9 M) (4)
is the density of NaHCO, in g cm-,.
(19) Fujita, H.; Gosting, L. J. J . Am. Chem. SOC.1956, 78, 1099. (20) Fujita, H.; Gosting, L. J. J . Phys. Chem. 1960, 64, 1256. (21) Perron, G.; Desnoyers, J. E.; Millero, F. J. Chem. J . Chem. 1975, 53, 1134. The actual data used for the calculation were obtained from the Depository of unpublished data, CISTI, National Research Council of Can-
ada, Ottawa.
AC,
C, mol/dm3
mol/dm3
J
0.0000" 0.0301 0.0446c 0.0498 0.0744 0.0978' 0.1048 0.1488 0.203 I 0.2460 0.3019 0.3485' 0.3970 0.4934' 0.4992 0.5888 0.5938 0.6900 0.7856 0.7995 0.8894
0.0602 0.0892 0.0694 0.0694 0.1194 0.0800 0.0790 0.0789 0.1841 0.0785 0.1836 0.0741 0.1773 0.0860 0.0966 0.0772 0.0779 0.0764 0.1000 0.1039
62.17 93.98 71.29 71.06 60.98 80.83 79.02 78.11 91.03 76.63 89.03 71.06 84.40 81.31 89.99 72.49 71.25 70.26 90.50 93.38
O C
1090,, m2/s
1094 m2/s
1.225 1.178 1.156 1.156 1.140 1.128 . 1.128 1.108 1.099 1.083 1.069 1.059 1.047 1.024 1.027 0.995 0.993 0.978 0.961 0.965 0.937
1.225 1.276 1.268 1.272 1.272 1.271 1.274 1.267 1.271 1.260 1.251 1.244 1.234 1.211 1.215 1.180 1.178 1.161 1.141 1.146 1.112
Limiting diffusion coefficient calculated by using Nernst-Hartley equation with the limiting equivalent conductances listed in ref 27. Experiments performed at L L N L on Gosting diffusiometer. e Experiment performed at LLNL on Beckman-Spinco diffusiometer.
'
TABLE III: Data for the System KHC03-H20 at 25 "C
he,
C, mol/dm3
1094,
1 0 9 ~ ~
mol/dm3
J
m2/s
0.0000" 0.0149 0.0299 0.0496* 0.0497 0.0721 0.1044 0.1980 0.2400* 0.3007 0.3927 0.475 1 0.4994' 0.5914 0.6803 0.7747 0.8683
0.0299 0.0598 0.0993 0.0793 0.0745 0.0594 0.0707 0.1397 0.078 1 0.0782 0.0582 0.1497 0.0742 0.0758 0.0750 0.0745
31.25 62.00 50.70 82.44 76.28 60.00 70.82 72.15 76.53 76.18 55.56 70.53 69.55 70.39 68.51 67.36
1.476 1.393 1.374 1.356 1.353 1.338 1.325 1.294 1.277 1.262 1.246 1.224 1.225 1.212 1.195 1.181 1.168
m2/s 1.476 1.480 1.489 1.496 1.493 1.498 1.507 1.518 1.514 1.517 1.525 1.520 1.527 1.534 1.534 1.538 1.544
Limiting diffusion coefficient calculated by using Nernst-Hartley equation with the limiting equivalent conductances listed in ref 27. Experiments performed at LLNL on the Gosting diffusiometer.
'
Density measurements were performed on the solutions that were prepared for the ternary diffusion experiments. It was found that a form of Young's rule22applied within 0.01%. Consequently, for a specified temperature, densities of ternary mixtures may be related to densities of binary mixtures by the equation P = Y l P l ( 0 + Y2P2(1)
(5)
Here y, and y, are the ionic strength fractions of N a H C 0 , and KHCO,, respectively; I = CI C2 is the ionic strength of the ternary solution; and C, is the molar concentration of i. The densities pl(Z) and p2(Z) are for the corresponding binary solutions at concentrations equal to I, and for these 1:1 mixtures y, = C,/(C,
+
+ CZ).
Binary Diffusion Coefficients. Binary diffusion coefficients for the system NaHC0,-H20 at 25 OC are given in Table 11, and values for KHC0,-H,O at 25 OC are given in Table 111. Since the volume change on mixing is small, these diffusion coefficients, D,,are for the volume-fixed frame of reference. The (22) It has been shown by one of us (D.G.M.) that eq 5 is a consequence of Young's rule for apparent molar volumes (to be published).
Mutual Diffusion Coefficients of N a H C 0 3 and K H C 0 3
t
The Journal of Physical Chemistry, Vol. 91, No. 1, 1987 213 The diffusion coefficients for the two binary bicarbonate systems were fit to polynomials in powers C1I2by the method of least squares. In order to have equations that would be valid at low concentrations, 20 evenly spaced points calculated from the Onsager-Fuoss t h e ~ r yfor ~ ~each , ~ solute ~ in the range 0.001-0.020 M were included in the data sets for the least-squares process. The resulting equations are
1. 2
D, X 109(NaHC03)= 1.2550 - 0.6299C'i2
+ 1.1178C-
1.3736C3I2
+ 0.55319C (6)
D, X 109(KHC03) =
1.4761 - 0.8246C1/2+ 1.6079C- 1.9143Ql2
+ 0.81580cZ (7)
0.9 0
0.1
0.2
0.3 0.4
0 . 5 0.6 0 . 7 0.8 0.9
MOLARITY
mol dm-3
Figure 1. Binary diffusion coefficients of NaHC0,-H20 system as a thermodynamic, A; (-) function of molarity: ( 0 )experimental; (0) theoretical from Onsager-Fuoss;26 (X) at low concentrations from Leaist and Noulty.2
Diffusion coefficients calculated from the above equations have a standard deviation from the measured diffusion coefficients for the tabulated in Tables I1 and I11 of less than 0.004 X sodium bicarbonate system and less than 0.003 X m2 s-I for the potassium bicarbonate system, or 0.2-0.3%. Thermodynamic diffusion, A, were calculated from the equation A=
1.45 L
-
1.35
IL
E.
log y =
1.25
0 1. 15
1.0 5 0
+ m(d In y/dm)
(8)
Values of the derivative of the logarithm of the activity coefficient with respect to molality were obtained by first fitting the activity coefficients for NaHC03-H207 and KHC03-H208 to an equation of the form
N
x
0,
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 M O L A R I T Y , PI d m - 3
Figure 2. Binary diffusion coefficients of KHC03-H20 system as a function of molarity: ( 0 )experimental; (0)thermodynamic, A; (-) theoretical from Onsager-Fuoss.26
tables include values of the mean concentrations, C, the concentration difference across the initial boundaries, AC, the total number of fringes, J , and the thermodynamic diffusion coefficient, A, for each experiment. Experiments are identified as having been performed at T C U or LLNL. These results are plotted in Figure 1 for sodium bicarbonate and in Figure 2 for potassium bicarbonate. Included in Figure 1 are data points for the N a H C 0 3 - H 2 0 system at 25 "C that were obtained at low concentrations by Leaist and Noulty.2 As may be seen from the figure, the data presented here are in good agreement with their results. Also included in Table I1 and shown in Figure 1 is a diffusion coefficient which had been measured at LLNL on a Beckman-Spinco instrument. Since about 1% of the bicarbonate ions disproportionate to carbon dioxide and carbonate ions, the measured diffusion coefficients relate net concentration gradients to net flows. Nevertheless, most of the anion transport is due to the bicarbonate ion itself. Discussion. Diaphragm-cell diffusion data for bicarbonate systems had been reported by Vinograd and McBain' in 1941. However, their results are as much as 10% higher than ours above 0.1 M. Although the magnitude of the error was probably due to problems with early experimental techniques, imprecise results at higher concentrations have been typical for the diaphragm-cell method because of the need to differentiate data.10*23-25 (23) Rard, J. A.; Miller, D. G. J. Solution Chem. 1979, 8, 701, 755. (24) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Robinson, R. A. J . Solution Chem. 1980, 9, 467.
A11'12 + AzI
1
-+ A3I3I2
(9)
+ B1I1l2
This nonstandard form is easily cast into a linear form for least-squaring and adequately fits the data. Here I is the molality in Figures 1 and ionic strength. The points represented by (0) 2 are for the values of A. The values of A , , A2, A3, and B , are -0.5092, -0.015 522, 0.009 144, and 0.905 89 for N a H C 0 3 and -0.5092, 0.01 1899, -0.066739, and 0.835 516 for KHC03, respectively. The curve with no points specified, as shown in Figures 1 and 2 are theoretical values of D, calculated from the Onsager-Fuoss theory.26 For these calculations, the activity coefficient gradient was calculated in the same manner as it was for A, except that ionic strength was based on molarity instead of molality. The corrections A I and A2 for the 1st and 2nd order terms of the electrophoretic effect were calculated on the assum tion that the minimum ion pair separation distance was 4 . Limiting equivalent conductances of the sodium, potassium, and bicarbonate ions were taken from Appendix 6.2 of Robinson and Stokes.27 The agreement between theory and measured values appears to be reasonably good up to 0.05 M. The theoretical values become too high for sodium bicarbonate and too low for potassium bicarbonate above this concentration. There is some discrepancy in the calculated values because disproportionation of the bicarbonate ion was ignored in the calculation, but this error should be under 0.5%. Ternary System. Diffusion Coefficients. Results of the ternary experiments are given in Table IV. The accuracy of the measured diffusion coefficients should be considered to be f0.02 X m2 s-' . Discussion. It is interesting to compare these results to the values of the Dik calculated for a 1:l ratio of the solutes at zero concentration assuming that the water does not ionize. These values may be calculated from the limiting equivalent conductance of the ions according to Nernst-Hartley equations given by Gosting5 or by Miller.28 For these calculations, tracer-diffusion
w
(25) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Albright, J. G. J. Phys. Chem. 1984, 88, 5739. (26) Onsager, L.; Fuoss, R. M. J . Phys. Chem. 1932, 36, 2689. (27) Robinson, R. A.; Stokes, R. H. ElecfrolyfeSolutions, 2nd ed., Butterworths: London, 1970; pp 294-299.
214
The Journal of Physical Chemistry, Vol. 91, No. 1. 1987
Albright et al.
TABLE IV: Data for the Ternary System N a H C 0 3 (0.25 M ) - K H C 0 3 (0.25 M)-HIO at 25 OC
N aHC03,
KHCO3,
expt
AC,, mol/dm'
AC2, mol/dm3
exptl
calcd
104Qo
exptl
calcd
1 2 3 4
0.0000
0.2000 0.1600 0.0400 0. IO00
94.214 94.260 95.040 94.912
94.167 94.402 95.105 94.753
17.46 16.49 2.58 12.41
1.207 1.172 1.061 1.110
1.209 1.169 1.060 1.112
0.0400 0.1600 0.1000
VI,
partial molar properties NaHC03 KHC03 diffusion coeff, m2/s measured limiting
1090,, m2/s
J
cm3/mol
Hi, g/mol
cm/mol
26.895 36.590
57.183 63.614
10.280 10.277
1o ~ R , ,
1090,~
1O9DI2
1o9D2,
109D22
1.065
-0.197 -0.182
-0.043 -0.05 1
1.434 1.690
1.300
Here 0,and D: are the diffusion coefficients for the binary system containing solute i. The D,, and D,: are the main-term diffusion coefficients at a given ratio of C1/(Cl CJ for the same solute 2.0 in the ternary mixture. The DIoand D,,' are values at infinite dilution and D, and D,, are values for systems with ionic strength 7 equal to C, + C,. c The cross-term diffusion coefficients in this system, while €1.0 relatively large, are not too different at I = 0.0 and I = 0.5 M. P This suggests that up to I = 0.5 M, the cross-term diffusion X 0coefficients may be estimated from the values at I = 0.0 at the m2 s-l. same ratio of Cl/(Cl + C2) to within f0.02 X Values of H I ,the partial derivative of density with respect to molarity, C,, were calculated from eq 3 and 4 and Young's rule (eq 5). These values, combined with the diffusion coefficients and initial concentrations of the free-diffusion experiments, were used to test for stability of the diffusion boundaries. Criteria given by Wendt30s31for gravitational static stability and by McDouga1132,33a,b for dynamic stability were used for the tests All exFigure 3. Calculated dependence of ternary diffusion coefficients of periments listed in Table IV were stable. However, an experiment NaHC03-KHC0,-H20 system at infinite dilution on the ratio C , / ( C , with ACl = 0.200 and AC, = 0.000 was predicted to be unstable + C*). at the two boundary edges, so was not used in analyzing the data. Young's rule was also applied to determine the partial molar coefficients may be calculated according to the Nernst equation; volumes. These are listed as in Table IV and are for the mean see, for example, eq 11.49 in ref 27. Equivalent equations are given by Leaist as limiting equations from Chen and O n ~ a g e r . ~ ~ concentrations of 0.25 M of each solute. Values of R,listed in Table IV correspond to An/ AC,, the change of refractive index Values of Dlk for infinite dilution are included in Table IV for with respect to molarity of each solute. These were obtained by comparison. The calculated dependence of the four diffusion a least-squares procedure using the cell length, a, and the values coefficients at infinite dilution on the ratio C1/(CI + C2) is shown of J , ACl, and AC2 of each experiment. in Figure 3. Let us now consider the main term D,, at 0.5 M total concenSummary tration compared to infinite dilution values. The measured cross This study shows that the diffusion coefficients of sodium and terms are negative and significant, D I 2being nearly 19% of Dll. potassium bicarbonates decrease as concentration increases from The main terms are poorly predicted by the infinite dilution 0.0 to 0.9 M. This contrasts with the behavior of corresponding approximation, which gives values that are in error by +22% and halides whose diffusion coefficients decrease and then increase +18% for D , , and D,,, respectively. The cross terms are fairly in the same concentration range.27 The results of bicarbonates well predicted however. In comparison with the corresponding are more consistent with available data for the behavior of corbinaries at 0.25 M, D,, is fairly close, but DZ2is 16% larger. responding nitrates.34 The cross-term diffusion coefficients of It may be noted that the difference between DIl at 0.0 and 0.5 the ternary systems are negative and significantly large. One may m2 s-l, is similar ionic strength in the 1:l mixture, 0.23 X note that the thermodynamic diffusion coefficients for both sodium to the difference of the binary diffusion coefficients of N a H C 0 3 and potassium bicarbonate first increase and then decrease as in water at 0.0 and 0.5 ionic strength, which also rounds to 0.23 concentration increases. Thus the thermodynamic term alone X m2 s-l. For the solute K H C 0 3 , values of D22 in a 1:l tends to "over correct" the diffusion coefficients at low concenmixture give a difference of 0.26 X lo+' m2 s-I at 0.0 and 0.5 ionic trations. This is typical of both strong and weak electrolytes. strengths, and the binary diffusion coefficients have a difference m2 at 0.0 and 0.5 ionic strengths. Again the of 0.25 X Acknowledgment. J.G.A. thanks the National Science Foundifferences are similar. This suggests that, for the NaHC03-Kdation Grant CHE-8112575 and the TCU Research Foundation HC03-H,0 system, the ternary main-term diffusion coefficients may be estimated from the binary diffusion coefficients and the limiting ternary D,, at all ratios of C,/(C, + C2) up to at least (30) Wendt, R. P. J . Phys. Chem. 1962, 66, 1740. 0.5 ionic strength. Thus (31) Reinfelds, G.;Gosting, L. J. J . Phys. Chem. 1964, 68, 2464. (32) McDougall, T. J. J . Fluid Mech. 1983, 226, 379. (33) (a) Vitagliano, P. L.; Della Volpe, C.; Vitagliano, V. J . Solution = D,,o + ID,(I) - Q01 (10) Chem. 1984, 13, 549. (b) Miller, D. G.; Vitagliano, V . J . Phys. Chem., 1986,
+
rv
0'
ml)
(28) Miller, D. G.J . Phys. Chem. 1967, 71, 616. (29) Chen, M.; Onsager, L. J . Phys. Chem. 1977, 82, 2017.
90, 1706. (34) Janz, G . J.; Lakshminarayanan, G. R.; Klotzkan, M. T.; Mayer, G. E. J . Phys. Chem. 1966, 70, 536.
J . Phys. Chem. 1987, 91, 21 5-2 18 Grant 5-23656 for their support. Portions of this work were done under the auspices of the Office of Basic Energy Sciences (Geosciences) of the U.S. Department of Energy by LLNL under contract No. W-7405-ENG-48. We thank Mr. Charlie Brannick for his assistance in making density measurements on the K H C 0 3 system and Dr. Joseph A. Rard of LLNL for measuring the
A Proton NMR Relaxatlon Study of (CH,),N+ Mn2+ in Dilute Aqueous Solutions
215
diffusion coefficient of NaHCO, at 0.0446 M. Registry No. NaHCO,, 144-55-8; KHCO,, 298-14-6.
Supplementary Material Available: Detailed description of the use of the automatic plate reader ( 5 pages). Ordering information is given on any current masthead page.
and CH,COO- Interacting with Hydrated
P. H. Fries, N. R. Jagannathan, F. G. Herring,* and G. N. Patey* Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada, V6T 1 Y6 (Received: May 16, 1986)
This paper describes a study of the relative motion of the ion pairs CH,COO-/Mn(D,O)~+ and (CH3)4N+/Mn(D20)62+ in dilute DzO solution at 25 O C . We report the interparticle spin-lattice relaxation rates at 400 MHz of the CH3COOand (CH3)4N+protons due to their dipolar interaction with the electronic spins of the paramagnetic Mn(D,O):+ ion. The experimental results are compared with theoretical calculations for model electrolytes.
I. Introduction In several recent articles we have described theoretical' and experimental24 investigations of the relative motion of pairs of attractive and repulsive ions in dilute aqueous solution. The purpose of this work was to obtain a theoretical description of the ion dynamics, and to test the theoretical results with nuclear magnetic resonance ( N M R ) experiments. The present paper is a further contribution to this end. We report and compare experimental and theoretical results for the ion pairs CH,COO-/ MnZ+and (CH3),N+/MnZ+in dilute DzO solution. The N M R experiments are analogous to those carried out in our earlier ~ o r k , and ~ - ~consist of measuring the interparticle dipole-dipole relaxation rate of nuclear spins I located on diamagnetic ions due to their interaction with electronic spins S associated with charged paramagnetic particles. The measured relaxation rates strongly depend upon the relative ion distribution and motion and provide a sensitive test of theoretical results. In most of our previous measurements the nitrosodisulfonate anion was used as the paramagnetic species. In ON(S03)22-(S = the present experiments the manganese(I1) cation Mn2+ (S = 5 / 2 ) serves this purpose. The manganese(I1) ion has long been recognized as an N M R relaxation agent which is particularly suitable for studies of the structural and dynamical properties of its microscopic environm e ~ ~ t . ~ -This ' is especially true for liquid systems at normal temperatures and pressures in which the characteristic fluctuation times of the electronic spins on Mn2+ are long compared with the correlation times for molecular motion. Hence these electronic spin fluctuations do not contribute to the quantum dynamics of the It is interesting to compare this situation with (1) Frics, P. H.; Patey, G. N. J . Chem. Phys. therein. __._. - _.. .
TABLE I: Experimentally Measured Relaxation Rates, l/Tl, of the CHKOO- Protons at 400 MHz"
i,
[CH3COONa],
[MnCI2], mol L-'
mol L-'
1/TI, s-l
0.02 0.02 0.05
0.001 0.005 0.001 0.005 0.001 0.005
0.023 0.035 0.053 0.065 0.103 0.115
19.38 79.37 15.34 73.10 12.58 49.50
mol L-'
0.05 0.10 0.10
"The observed relaxation rate without MnCI2 l/T,O = 0.129 s-l. TABLE II: Experimentally Measured Relaxation Rates, l/Tl,of the (CH&N+ Protons at 400 MHz"
1,
[(CH,hNClI, mol L-'
[MnCM, mol L-'
mol L-'
1/TI, s-l
0.005 0.005 0.005 0.005
0.005 0.010 0.020 0.040
0.020 0.035 0.065 0.125
2.30 4.50 9.17 19.01
"The observed relaxation rate without MnCI, ] / T I 0= 0.104 s-I. previous studies'@'4 of ion-ion interactions in aqueous solution which have used Ni2+as the paramagnetic s p i e s . In these earlier studies the theoretical analysis of the experimental measurements differs significantly from that required in the present work. The difference stems from the fact that, unlike the case for Mn2+,the electronic relaxation processes for Ni2+are fast compared with the correlation times for relative diffusional motion.I2 Under these conditions, the relaxation times for nuclei on the diamagnetic ion
1984,80,6253and references
(2)Fries,P. H.; Jagannathan, N. R.; Herring, F. G.; Patey, G.N. J . Chem. Phys. 1984,80,6267. (3)Fries, P. H.; Rendell, J.; Burnell, E. E.; Patey, G. N. J. Chem. Phys.
(9) Kowalewski, J.; Nordenskiold, L.; Benetis, N.; Westlund, P. 0.Prog. Nucl. Magn. Reson. Spectrosc., in press. (10) Hwang, L.P.; Krishman, C. V.; Friedman, H. L. Chem. Phys. Lett.
1985,83,307. (4) Fries, P. H.; Jagannathan, N. R.; Herring, F. G.; Patey, G. N. J. Phys. Chem. 1985. 89, 1413. (5) Bloembergen, N.; Morgan, L. 0. J . Chem. Phys. 1961,34, 842. (6)Dwek, R. A. Nuclear Magnetic Resonance in Biochemistry; Clarendon: Oxford, U.K.,1973. ( 7 ) Nicollai, N.; Tiezzi, E.; Valensin, G. Chem. Rev. 1982,82,359. (8) Rubinstein, M.; Baram, A.; Luz,2.Mol. Phys. 1971,20, 67.
1973 -. . -, 20.,
0022-3654/87/2091-0215$01.50/0
- - -391 - -.
(1 1) Weingartner, H.; Muller, C.; Hertz, H. G. J. Chem. SOC.,Faraday Trans. 1 1979,75, 2712. (12)Friedman, H. L.;Holz, M.; Hertz, H. G. J. Chem. Phys. 1979,70,
3369. (13) Hirata, F.; Friedman, H. L.; Holz, M.; Hertz, H. G. J . Chem. Phys. 1980,73, 6031. (14) Hertz, H. G.; Holz, M. J. Magn. Reson. 1985,63, 6 4 .
0 1987 American Chemical Society
