Relative Viscosities of Some Aqueous Rare Earth Chloride Solutions

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F. H. SPEDDING AND M. J. PIKAL

Relative Viscosities of Some Aqueous Rare Earth Chloride Solutions at 2 5 O I

by F. H. Spedding and M. J. Pikal Institute for Atomic Research and Department of Chemistry, Iowa State University, Ames, Iowa (Received December 17,1966)

~~

~

~~

~~

The relative viscosities of aqueous solutions of LaCL, NdC13, SmCL, TbC13, DyC13, HOC&, and ErCb were determined a t 25” over a concentration range of about 0.05 m to saturation. The B coefficients of the Jones-Dole equation were determined for these salts. The relative viscosities of several dilute PrCL solutions were also measured to allow the B coefficient for this salt to be determined. The concentration dependence of the relative viscosity of a rare earth chloride solution is discussed in terms of the Vand theory for large nonelectrolyte molecules. It is proposed that the major contribution to the viscosity of a rare earth chloride solution at moderate to high concentration arises from the interference of large hydrated rare earth ions with the stream lines in the solvent. The B coefficients and the relative viscosities at “iso-molalities” are studied as a function of the ionic radius of the rare earth ion, and irregularities are noted near the middle of the rare earth series. These irregularities are discussed in terms of a change in preferred water coordination number of the rare earth ion.

Introduction A number of recent investigations carried out in this laboratory on apparent molal apparent molal heat capacities,2band heats of dilution3of aqueous rare earth chloride solutions have shown that these thermodynamic properties are not simple functions of the rare earth ionic radius. Irregularities were noted near the middle of the rare earth series when these properties were examined as a function of rare earth ionic radius. The apparent molal volumes at infinite dilution2*show the usual decrease with decreasing ionic radius from La to Nd and from T b to Yb. However, in the region from Nd to Tb, the apparent molal volumes at infinite dilution increase with decreasing rare earth ionic radius. The heats of dilution3 at moderate concentrations show phenomena similar to those observed for the apparent molal volumes at infinite dilution. At moderate concentrations, the apparent molal heat capacitieszb of Lac& and NdC13 are nearly the same, and those of DyC13, ErCb, and YbC13 are nearly the same. However, the apparent molal heat capacities of LaCL and NdC13 are significantly lower than those of DyCl3, ErC13, and YbC13. For each of these thermodynamic properties, the irregularities noted above were discussed in terms of ion-water interactions. This paper The Journal of Physicat Chemistry

is an extension of the above work to include viscosity measurements on aqueous rare earth chloride solutions. The relative viscosity, qr, is defined as the ratio, qr = q/qO,where 7 is the viscosity of the solution and qo is the viscosity of the solvent at the same temperature. The relative viscosity of dilute electrolytes has proved to be an effective method for studying ion-solvent intera c t i o n ~through ~ ~ ~ use of the Jones-Dole equation6J qr =

1

+ Ac”’ + BC

(1)

where c is the molar concentration. The quantity A is a coefficientwhose value depends upon the particular electrolyte, temperature, and solvent under consideration and may be theoretically calculated.* The quan(1) Work was performed in the Ames Laboratory of the U. 5 . Atomic Energy Commission; Contribution No. 1838. (2) (a) F. H. Spedding, M. J. Pikal, and B. 0. Ayers, J. Phys. Chem., 70, 2440 (1966); (b) F. H. Spedding and K. C. Jones, ibid., 70, 2450 (1966). (3) F. H. Spedding, D. A. Csejka, and C. W. De Kock, ibid., 70, 2423 (1966). (4) M. Kaminsky, Discussions Faraday Soc., 24, 171 (1957). (5) R. W. Gurney, “Ionic Processes in Solution,” McGraw-Hill Book Go., Inc., New York, N. Y., 1953. (6) G. Jones and M. Dole, J. Am. Chem. SOC.,51, 2950 (1929). (7) G . Jones and 5. Talley, ibid., 55, 624 (1933).

RELA.TIVE VISCOSITIESOF SOMEAQUEOUS RAREEARTH CHLORIDE SOLUTIONS AT 25”

tity B, which is normally called the B coefficient, is an empirical parameter whose value depends upon the electrolyte, temperature, and solvent being considered. The theoretical interpretation of the B coefficient has been discussed in detail by Kaminsky4 and by Gurney.6 Briefly, the B coefficient is an additive property of the ions, and it is normally considered to be a function only of ion-solvent interactions. Furthermore, the ionic B coefficient is normally interpreted as a measure of the order the ion introduces in the solvent surrounding the ion, a large B coefficient indicating a higher degree of order. The rapid increase in viscosity of a nonelectrolyte solution with increasing concentration of large solute particles was explained by Vandg as due to interference of the particles with stream lines in the liquid. Treating the liquid as a viscous continuum containing a suspension of rigid obstructions at the surface of which the liquid is at rest, Vand obtained a result which may be written as In vir =

k18c

+ r2(k2- k1)ir2c2+ . . . 1 - Qirc

- &’C)

cerning ion-water interactions at high concentrations, as well as in dilute solutions. This paper will examine the B coefficients and viscosities of some aqueous rare earth chloride solutions as a function of rare earth ionic radius. Because the rare earth ions may be expected to be highly hydrated in aqueous sol~tion,~~-’’ the concepts of Stokes and cow o r k e r ~ ~ may ~ - ’ ~be expected to apply to aqueous rare earth chloride solutions. Thus, the applicability of eq 3 to solutions of the rare earth chlorides will be also investigated.

Experimental Section Preparation of Solutions. Preparation of rare earth chloride solutions has been described elsewhere.2* Viscometers. Four size 25 and two size 75 CannonUbbelohde filter stick viscometers, purchased from the Cannon Instrument Co., were used to determine the viscosities of the solutions studied in this research. These viscometers are capillary flow viscometers of the suspended-level Ubbelohde design, so it was not necessary to charge the viscometer with the same volume of liquid each time the viscometer was used. Furthermore, with a suspended-level viscometer, the viscometer constant is effectively independent of temperature, and no surface tension corrections are necessary. The tops of the viscometers were modified lsaib

(2)

where ir is the molar volume of the obstructions, c is the molar concentration, kl is the “shape factor” for single solute particles, kz is the “shape factor” for collision doublets, r2 is the collision time constant, and Q is a hydrodynamic interaction constant. For rigid, nonsolvated spheres without Brownian motion, the following values were derived theoretically by Vand: kl = 2.5; k2 = 3.175; r2 = 4; and Q = 0.60937. Vand has shown that this theory is in agreement with experiment. lo Vand’s theory may not be expected to apply rigorously to electrolytes. However, it is interesting and perhaps significant to note that a modification of Vand’s equation in the form

hl q r = A$/(l

2431

(3)

where A3 and Q’ are adjustable parameters, gives an excellent representation of the viscosities of many “strongly hydrated” electrolyte solutions in the region of moderate to high concentration.“,l2 According to Stokes and ~o-workers,~’-’~ this success of eq 3 suggests that the major contribution to the viscosity of a “highly hydrated” electrolyte at moderate to high concentrations arises from the “obstruction effect” of large hydrated ions. This implies that, except at extreme dilutions the effect of ion-ion interactions plays a minor role in determining the viscosity of a “highly hydrated” electrolyte, and therefore, viscosity data might be expected to yield valuable information con-

(8) H. Falkenhagen and E. Vernon, Physik. Z.,33, 140 (1932). See also H.S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1958. (9) V. Vand, J. Phys. Chem., 52, 277 (1948). (10) V. Vand, {bid., 52, 300,314 (1948). (11) R. H. Stokes, “Mobilities of Ions and Uncharged Molecules in Relation to Viscosity-A Classical Viewpoint,” in W. J. Hamer, Ed., “The Structure of Electrolytic Solutions,” John Wiley and Sons, Inc., New York, N. Y.,1959,p298. (12) R. Robinson and R. Stokes, “Electrolytic Solutions,” 2nd ed, Butterworth and Co. Ltd., London, 1959. (13) R. H. Stokes and R. Mills, “Viscosity of Electrolytes and Related Properties,” Pergamon Press, London, 1965. (14) The hydration theory of Bernal and Fowler16indicates that, due to the high charge to radius ratio of a rare earth ion, the fist coordination sphere of a rare earth ion should contain as many water molecules as is sterically possible. Experimental evidence seem8 to support this conclusion. A fisbsphere water coordination number of either 8 or 9 was found for aqueous Gd3+,’6 compared to a value of only 4 for aqueous K+.” One may also expect that the effect of secondary hydration (number of water molecules in the second coordination sphere of the ion) would increase as the charge to radius ratio of the ion increased. Thus, it seems likely that the total “hydration number” of a rare earth ion would include some contribution from secondary hydration, although it seems probable that the second sphere water molecules would be bound much less firmly than the water molecules in the first coordination sphere. (15) J. Bernal and R. Fowler, J. Chem. Phys., 1, 515 (1933). (16) L. Morgan, ibid., 38, 2788 (1963). (17) G. Brady and J. Krause, ibid., 27, 304 (1957). (18) (a) L. Ubbelohde, J . Inst. Petrol., 19, 376 (1933); (b) L. Ut+ belohde, ibid., 23, 427 (1937).

Volume 70,Number 8 August 1966

F. H. SPEDDING AND M. J. PIKAL

2432

slightly to allow the interior of the viscometer to be sealed off from the atmosphere, yet allowing the suspended-level feature to be retained. Also, viscometer holders were constructed so the viscometers could be placed in the same vertical position in the constant temperature bath each time they were used. The modifications to the viscometers mentioned above and the viscometer holders are described in detail elsewhere. l9 Calibration of Viscometers and Viscosity Cdculation. The equation relating efflux time, 2, and viscosity, 7, niay be written as20 7 =

pCt - pE/t2

(4)

where p is the density of the fluid and C and E are constants for a given viscometer. For efflux times greater than about 300 sec, the term in E may be neglected for the viscometers used in this research, so eq 4 reduces to q = pct

(5)

The values of the viscometer constant, C, were determined by the calibration procedure described below. The size 25 viscometers were calibrated using pure water at 20" as the calibration fluid. The efflux times were in excess of 600 sec, so the values of C were obtained directly from eq 5, using 1.002 cpZ1for the viscosity of water at 20". The values of C obtained were 1.4813 X 1.4901 X 1.5353 X and 1.5948 X l W 3 cp/sec. Using the size 25 viscometers and eq 5, the viscosities of water at 5, 25, and 45" were found to be 1.5179, 0.8903, and 0.5952 cp, respectively. These values are in satisfactory agreement with the results of Hardy and Cottington.22 The size 75 viscometers were calibrated using water at 5, 20, 25, and 45" as the calibration fluids. For this calibration, the viscosities of water at 5, 20, 25, and 45" were taken as 1.5179, 1.002, 0.8903, and 0.5952 cp, respectively. Except for the viscosity of water at 20", which was the ultimate standard used in the calibration, these viscosity values represent the values determined in this research. The values of C were obtained using eq 4. Specifically, the quantity, q/pt, was plotted as a function of l/t3 and the intercept at l/t3 = 0 was taken as C for that viscometer. The values of C obtained were 6.741 X and 6.166 X cp/sec. The values of E were about 20 cp/sec2 for both viscometers. The viscosities determined in this research were calculated from eq 5 . The density data of Saeger and SpeddingZ3 and Spedding, Brown, and Grayz4 were used in the calculations. The size 25 viscometers were The Journal of Physical Chemistry

used for the rare earth chloride solutions below about 1.5 m, and the size 75 viscometers were used for the more concentrated solutions. The relative viscosity was calculated using 0.8903 cp for the viscosity of water at 25". Two independent relative viscosity determinations, using different viscometers, were made for each solution, and the mean of the two results was taken as the relative viscosity of that solution. Errors. The temperature of the water bath was maintained at 25.00 0.01". It is estimated that the probable error in the relative viscosity measurement of a given solution is about *0.05%. However, the uncertainty in the concentration of the solution (probable error *0,05%) must also be considered when the viscosities of different solutions are being compared. For the rare earth chlorides studied in this research, a probable error in concentration of *0.05~ohas the same effect, when viscosity data on different solutions are being compared, as a probable error in viscosity of about *O.O% at zero concentration, *O.l% at 2 m, *0.2% at 3 m,and *0.4% at 4 m. It should be mentioned that the concentration error of each solution is due mainly to the error in concentration of the stock solution, and this error will be a systematic error for all solutions prepared from a giver1 stock solution. As a further check on the accuracy of the method used, the relative viscosities of several aqueous electrolytes were determined and compared with the corresponding literature values. The results of these comparisons are summarized in Table I. Since an error in the viscosity determination of *0.05% and a concentration error of less than *0.05% could account for the differences between the relative viscosities determined in this research and the corresponding literature values, the agreement is quite satisfactory.

*

Results The relative viscosities of aqueous solutions of LaC13, NdC13, SmC13, TbCl3, DyCl3, HoC13, and ErC4 were determined at 25" over a concentration range of about 0.05 m to saturation. The relative viscosities of three dilute PrC4 solutions were determined to allow the B coefficient for PrCL to be calculated. The experi(19) M. J. Pikal, Ph.D. Dissertation, Iowa State University of Science and Technology, Ames, Iowa, 1966. (20) M. R. Cannon, R. E. Manning, and J. D. Bell, Anal. Chem., 32, 355 (1960). (21) J. F. Swindells, J. R.Coe, Jr., and T. B. Godfrey, J . Res. Natl. Bur. Std., 48, 1 (1952). (22) R. C.Hardy and R. L. Cottington, ibid., 42, 573 (1949). (23) V. W.Saeger and F. H. Spedding, U. S. Atomic Energy Commission Report I8338 (Iowa State University of Science and Technology, Ames, Iowa, Institute for Atomic Remarch), 1960. (24) F. H. Spedding, M. Brown, and K. Gray, unpublished work.

RELATIVE VISCOSITIESOF SOMEAQUEOUS RAREEARTHCHLORIDE SOLUTIONS AT 25”

Table I: Relative Viscosities at 25’ Determined in This Research Compared to the Literature Values ?r

Density

(this research)

(lit.)

1.3443 1.341 1.289

1.757 4.247 15.78

1.758” 4. 254b 15.73c

Molality,

Salt

m

K2CrO.4

2.910 12.90 19.19

LiNOs LiCl

tlr

a G. Jones and J. Colvin, J . Am. Chem. Soc., 62, 338 (1940). A. N. Campbell, G. H. Debus, and E. M. Kartzmark, Can. J . Chem., 31, 617 (1953). S. Lengyel, J. T a m b , J. Giber, and J. Holderith, Magy. Kem. Folyoirat, 70, 66 (1964).

2433

tration, c k . The B coefficient, B, for a given rare earth chloride was taken to be the weighted mean of the B, values for each of the solutions studied having a concentration less than about 0.1 m, as indicated by eq 7. The weighting factor, wk,was taken to be the inverse square of the probable error in Bk, calculated assuming a probable error in ( 7 r ) k of *0.05%. The theoretical values of A were calculated using the conductivity data given by Spedding and AtkinsonZ8and the equations and tables given by Harned and Owen.25 The relative viscosity data reported in this paper are given at the experimental molalities, m. To calculate Bk from eq 6, the molality was converted to the molar concentration, c, using the equation

+

c = pm/(i 10-31n~~) (8) mental relative viscosities, qr, determined during the course of this investigation are given in Table 11. The where p is the density of the solution, and M z is the corresponding concentrations are expressed in terms molecular weight of the solute. The probable error in * B coefficientswas estimated to be about *0.005. of molality, m, and the corresponding d e n s i t i e ~ , ~d,~ > ~the are also listed. The quantity, A, represents the relative One of the objectives of this research was to compare difference, [ (qF) experimental - (?lr) calculated] X the viscosities of rare earth chloride solutions at “iso102/(qr) calculated, where, except for PrC13, the calcumolalities.” For this purpose, some form of empirical lated value refers to the relative viscosity calculated equation accurately representing the experimental from eq 16 and 17 with the appropriate parameters. relative viscosities as a function of molality was needed. For PrCI3, the calculated relative viscosities were obThe relative viscosities of the rare earth chloride solutained from the Jones-Dole equation for this salt. tions studied in this research changed by roughly a facThe B coefficients of the Jones-Dole equation are tor of 20 over the concentration range studied. Furusually obtained by evaluating both A and B from thermore, the relative viscosities were not a simple the experimental data. However, the theoretical exfunction of molality. Therefore, a simple and accurate pression for A has been well verified for a number of representation of the relative viscosities in terms of a electrolytes and temperature^.^^ In particular, the power series in either ml/= or m containing a reasonable relative viscosity data of Jones and StaufferZ6for Lac& number of parameters was not possible. and those of K a m i n ~ k yfor ~ ~ CeC13 show that the The assumptions made by Vand deriving eq 2 are Jones-Dole equation, eq 1, is obeyed for these eleccertainly not valid for electrolytes. However, the extrolytes up to about 0.1 M and that the experimental ponential form of the Vand equation predicts a rapid values of A are in excellent agreement with the theoincrease in viscosity with increasing concentration in retical values. Consequently, it was felt that more acconcentrated solutions, which was observed for the rare curate B coefficients could be obtained from the relative earth chloride solutions. I n fact, as will be discussed viscosity data obtained in this research if the values of later, the Vand equation in the form of eq 3 represented A were calculated from theory, and only the B coeffithe data within about ~ 1 % .However, there were cients were determined from the data. The B coeffisystematic deviations which were significantly larger cients obtained in this research were calculated using than the estimated experimental error. PittsZ9has exthe equations tended the viscosity theory of Falkenhagen and Vernon8 to include the effect of the a parameter, a, of the and

where Bk is the B coefficient calculated from the relative viscosity of a given solution, ( T , ) ~ , of molar concen-

(25) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1958. (26) G. Jones and R. Stauffer, J. Am. Chem. Soc., 62, 335 (1940). (27) M.Kaminsky, 2. Physik. Chem., 8, 173 (1956). (28) F. H. Spedding and G. Atkinson, “Properties of Rare Earth Salts in Electrolytic Solutions,” in W. J. Hamer, Ed., “The Structure of Electrolytic Solutions,” John Wiley and Sons, Inc., New York, N. Y., 1959, p 319. (29) E. Pitts, Proc. Roy. soc. (London), A217, 43 (1953).

Volume ‘70,Number 8 August 1966

E'. H. SPEDDINQ AND M. J. PIKAL

2434

Table I1 : Experimental Relative Viscosities at 25" m

d

m

d

0.01104 0,01606 0,04665 0.08038

0.9996 1.0007 1.0076 I . 0151

1.0085 1.0108 1.0324 1.0523

-0.04 -0.15 0.08 0.00

0,05293 0.06613 0.09959 0.25320

1.0096 1.0126 1.0204 1.0556

1.0361 1,0449 1,0636 1.1639

0.04 0.11 -0.08 0.13

0.09821 0.10213 0.20139 0.49440

1.0191 1.0199 1.0418 1.1047

1.0626 1.0658 1.1286 1,3316

-0.06 0.02 0.07 -0.14

0.49269 0.64067 1.0058 1.4553

1.1094 1.1420 1.2200 1.3118

1.3342 1.4541 1.8219 2.4548

-0.02 -0.13 0.01 0.11

0.64686 1.0076 1,4108 1.6927

1.1365 1.2094 1.2875 1.3395

1.4584 1.8153 2,3563 2,8530

0.04 0.02 0.07 -0.08

1.7024 1.9480 2.2566 2,5524

1.3603 1.4071 1.4641 1.5167

2.9180 3,503 4.466 5.713

-0.07 -0.04 -0.01 -0.04

1.9750 2.2517 2.5649 2.8324

1.3901 1.4382 1,4907 1.5340

3.498 4.305 5.520 6.900

0.03 -0.05 -0.02 -0.04

2,8974 3.2499 3.5901 3.9292

1.5758 1.6336 1.6870 1.7379

7.765 10.867 15,352 22.07

3,2896 3,6003 3,8959

1,6050 1.6512 I.6943

10.391 14.012 18.91

0.11 0.15 -0.05

?r

A%

SmCla

?r

A%

-0.03 0.03

0.10 -0.04

TbCla

*

0.03

0.04788 0,08136 0.10090 0.25812

1,0089 1.0171 1.0218 1.0597

1.0361 1.0592 1.0736 1.1871

0.00 -0.01 0.06 0.04

1,3655 1.6781 1.9176 2.4689

0.00 -0.04 -0.08 0.12

0,49015 0.73211 1,0005 1.3007

1.1144 1.1700 1.2301 1.2951

1.3764 1.6131 1.9327 2.3868

-0.05 -0.02 -0.05 0.04

2.8988 3.562 4.061 5.652

-0.09

1.6309 1.8759 2.1862 2.4998

1,3643

-0.01 -0.04

1.4140 1,4751 1,5347

3.039 3.657 4.671 6.052

0.07 -0.02 -0.07 -0.05

-0.04 0.07 0.33 -0.29

2.7954 3.1003 3.3803 3.5735

1.5910 1.6432 1.6911 1,7234

7.820 10.287 13.403 16.226

0.08 -0.02 -0.05 0.04

1.0412 1.0574 1.0713 1.1808

-0.08 0.06 -0.14

0.04955 0.07894 0.10195 0.25290

1.0097 1.0171 1.0229 1.0604

1.0376 1,0597 1,0759 1,1851

-0.01 0.12 0.11 -0.06

1.1175 1.1523 1.2352 1,3297

1.3866 1,5278 1.9624 2.6737

-0.02 -0.14 0.18 0.00

0.49457 0.74130 1.0021 1.2884

1,1192 1.1777 1.2379 1.3020

1,3881 1.6391 1.9680 2.4175

-0.08 -0.05 0.09 -0.06

1.3803 1.4369 1.4983 1.5506

3.208 3.980 5.122 6.462

0.09 -0.05 -0.14 -0.11

1.5907 1,8945 2.2007 2.4903

1.3675 1,4313 1.4934 1.5501

3.042 3,860 4.971 6.384

0.08 -0.07 -0.06 -0.03

0.04717 0.07660 0.09712 0.26869

1,0085 1.0154 1.0203 1.0605

1.0341 1.0525 1.0651 1.1795

0.51578 0.85276 1,0625 1.4371

1,1172 1.1921 1.2376 1.3161

1.6664 1,9453 2.1161 2.5233

1.3623 1.4176 1.4504 1.5260

2.8645 3.1788 3,5070 3.6401

1.5864 1.6398 1.6934 1.7144

0.05554 0.07650 0.09742 0.24914

1.0111 1.0163 1.0214 1.0586

0.49471 0.64312 1.0055 1.4371 1.6767 1.9529 2.2620 2.5342

The Journal of Physical Chemistry

7.606 10.165 13.995 15,883

0.09 0.03

-0.01

0.00

-0.01

RELATIVE VISCOSITIES OF SOMEAQUEOUS RAREEARTH CHLORIDE SOLUTIONS AT 25”

2435

Table I1 (continued) d

m

2.8530 3,1478 3.6310

1)r

1.6098 1.6626 1.7451

8.583 11.313 18.14

d

vr

A%

2.6713 2.7907 3.0919 3,3877

1.5847 1,6071 1.6623 1.7146

7.507 8.382 11.157 15.000

-0.01 0.07 0.06 -0.02

3.6942

1.7670

20.81

-0.04

0,05011 0.07986 0.10083

1.0087 1.0155 1.0203

A%

m

-0.14 0.14 0.18

ErC13

PrC13

0.05533 0,07409 0,10483 0.25090

1.0114 1.0161 1.0239 1.0607

1.0415 1.0555 1.0778 1,1857

-0.06 0.01

0.49531 0.77238 1.0096 1.4609

1.1209 1.1874 1.2427 1,3443

1.3927 1.6815 1.9889 2.7770

-0.05 -0.02 0.07

1,7153 1.9944 2.2695 2.5878

1.3994 1,4582 1.5143 1,5772

3.388

-0.02 -0.02

2.9182 3.2497 3.5379 3.7821

1.6401 1.7010 1,7520 1.7939

9.789 13.660 18.57 24.50

4,255 5.379 7,154

0.04 0.02

= 1

+ Ac’/’[l +

P(KU)]

-0.13 -0.01 0.07

0.00

-0.04 -0.01

0.08 0.08 -0.03 -0.05

Debye-Hiickel theory, on the viscosity of a dilute electrolyte and obtained a result which may be written in the form fr

1.0332 1.0526 1.0661

(9)

where the function P ( K u is ) defined by

The quantity A is the same coefficient which appears in the Jones-Dole equation. The quantity, K, is the reciprocal of the “radius” of the “ionic atmosphere,” which appears in the Debye-Hiickel theory of electrolytes, and is directly proportional to cl/’. Although the Pitts equation did not succeed in theoretically calculating the B coefficient of the Jones-Dole equation, which is normally attributed to ion-solvent interactions, the Pitts equation might be expected to give a good approximation for the “ion-ion” contributions to the viscosity in dilute solutions. Therefore, it seemed likely that a crude approximation to the relative viscosity of a rare earth chloride solution might be given by a combination of the Vand equation and the Pitts equation in the form

-

7, - Aml/’[l

+ P(4.831m1/’)] + exp((l

2*50 ) - 0.60937Om)

(11)

where 8 represents the molar volume of the hydrated solute in solution. Equation 11 states that the relative viscosity of an electrolyte solution is approximately given by the sum of the “ion-ion” effect calculated by Pitts and the “obstruction” effect calculated by Vand for the simplest case, where the molality, m, has replaced the molar concentration, c, and the higher order terms in the numerator of eq 2 have been omitted. The numerical factor, 4.831, appearing in the argument of the Pitts function, is a result of assuming an a parameter of 6 A for an aqueous 3-1 salt at 25”. If we define the “ion-ion’J contribution by

I = Am’/’[l

+ P(4.831m1/’)]

(12)

Equation 11may be rearranged to give (m[l/ln (sr - I)

+ 0.24375]1-’

2.5s

(13)

For convenience, the left-hand side of eq 13 will be defined by

Y

(m[l/ln

(fr

- I) + 0.24375]1-’

(14)

Volume 70,Number 8 August 1966

F. H. SPEDDING AND M. J. PIKAL

2436

Table I11 : Parameters for Viscosity Equations

’ Calculated from

Bb

BO

BI X 103

0.0285 0.0285 0.0286 0,0288 0.0293 0.0297 0.0295 0.0296

0.554 0.562 0.557 0.584 0.633 0.639 0.650 0.646

0.52145

-31.635

...

- 4.7569

...

...

0,52479 0.54687 0.59406 0.60165 0.60210 0.60731

-35.168 - 56.684 - 66.423 - 68.006 - 55,522 - 58.876

0,70453 5,4394 -0.16163 1,7828 - 4.6225 -2.4475

theory.

* Experimental

= bo

+ blm + b2m2+ b3m3

(15)

where the coefficients were determined by the method of least squares. The experimental values of Y were weighted using the inverse square of the probable error in Y as the weighting factor. The probable error in Y was computed by an application of the law of propagation of precision indexeslWassuming the probable error in both the molality and the relative viscosity was f0.05%. Using the definitions of Y and I given by eq 12 and 14, the relative viscosity may be written qr =

Am”’[l

+ P(4.831m1”)] +

)

exp(l- 0.24375m ym

(16)

From eq 15, Y may be conveniently written as

Y

=

Bo(l

+ Blm + B2m2 + Barn3)

(17)

I n all cases, eq 16 with Y given by eq 17 represents the experimental relative viscosity data determined in this research within the limits of experimental error over the entire concentration range studied. The B coefficients of the Jones-Dole equation, the values of A , and the parameters of eq 17 are given in Table I11 for each of the salts studied. As previously mentioned, for each salt the appropriate value of A appearing in the Jones-Dole equation and in eq 16 was calculated from theory. The limited amount of data The Journul of Phgeical Chemistry

B~

103

Be X 108

0.64282

... - 0,29122 - 0,74824 0,44780 0.0 1,0342 0.73771

B coefficient of the Jones-Dole equation.

If eq 11 were exact and d were independent of concentration, values of the defined quantity, Y , which may be calculated from the experimental data and the theoretical value of A , would be independent of molality. Actually, the values of Y calculated from the experimental data are about 0.5 and change by about 20% over the concentrat.ion range studied. However, it was possible to express accurately the concentration dependence of Y by empirical power series in molality of the form

Y

x

An

Salt

for PrCL was not analyzed in terms of eq 16 and 17, so only the theoretical values of A and the B coefficient are given for this salt.

Discussion The relative viscosity data given in Table I1 and the LaCI3 data illustrated in Figure 1 as a function of molality show that the relative viscosities of rare earth chloride solutions increase slowly with increasing concentration at low concentrations, but above about 2 m, they increase very rapidly as the concentration increases. Literature data for aqueous NaC113 are plotted in Figure 1 for purposes of comparison. It was noted earlier that the relative viscosities of many electrolytes containing strongly hydrated ions are well represented by a slight modification of the Vand equation, eq 2, of the form in eq 3, where A3 and Q’ are adjustable parameters. This success of eq 3 was interpreted by Stokes and c o - ~ o r k e r s ~as~ indicating -~~ the major contribution to the viscosity of “highly hydrated” electrolytes at moderate to high concentrations of the “obstruction” effect, owing to interference of large hydrated ions with the stream lines in the solvent. The relative viscosities of the rare earth chloride solutions investigated in this research may be represented by eq 3 within about f1% over the entire concentration range studied, provided A3 and &’ are suitably adjusted. The values of A 3 and &’ evaluated from the data are about 0.5 and 0.1, respectively, the exact value depending upon the particular rare earth chloride under consideration. If the “obstruction” effect is the dominant contribution to the viscosity of a rare earth chloride solution, comparison of the Vand equation, eq 2, with eq 3 indicates A3 should be approximately equal to 2.50 (assuming spherical hydrated ions), where d represents the molar volume of the hydrated ions, and the ratio &’/A3 should be approxi~

~~

(30) A. G . Worthing and J. Geffner, “Treatment of Experimental Data,” John Wiley and Sons, Inc., New York, N. Y., 1943.

RELATIVE VISCOSITIES OF SOMEAQUEOUS RAREEARTHCHLORIDE SOLUTIONS AT 25”

2437

Although the viscosities of the rare earth chloride solutions at a given concentration are somewhat similar, significant differences do exist. These differences are shown graphically by Figures 2-Fj. For the purpose of illustrating these differences on small-scale graphs, the ratio of the viscosity of a rare earth chloride solution to the viscosity of a LaC13 solution of the same molality, ~RClr/qLsclnwas calculated, using eq 16 and 17 with the appropriate parameters, and values of this ratio are plotted in Figures 2, 4,and 5 instead of the relative viscosity. From Figure 2, it should be noted that the viscosities of the rare earth chlorides increase as the atomic number of the rare earth ion increases at all concentrations below about 3.3 m. However, about this concentration, NdCl3 solutions have viscosities greater than SmC13solutions of the same concentration. This “crossover” will be briefly commented on later. In Figure 3, the B coefficients of the Jones-Dole M

Figure 1. Relative viscosity of aqueous Lac13 and of aqueous NaCl a1 25’ as a function of molality.

mately given by &/2.5 0.24. The quantity Q is the hydrodynamic interaction constant, which was theoretically calculated by Vand to be 0.60937 for the simplest case. The experimental ratios &’/A3 are about 0.2 and are therefore in good agreement with what one would expect on the basis of the Vand theory. The experimental values of @ are about 0.2 1. If it is assumed the chloride ions are unhydrated and have a radius of 1.8 A, and that the hydrated rare earth ion is spherical, a value of 0.2 1. for B implies the radius of the hydrated rare earth ion is about 4 A. This result is roughly in agreement with what one would expect if the rare earth ion were firmly coordinated with one row of water molecules, such that this “first coordination sphere” were held too firmly to participate in the viscous shearing process. The small but significant variation of A3 with rare earth ionic radius will be discussed later. The success of eq 3 in approximately representing the relative viscosity data of the rare earth chlorides with reasonable values for A Band Q’ suggests that, except at extreme dilutions, the major contribution to the viscosity of a rare earth chloride solution arises from the “obstruction” effect. It should be emphasized that the preceding statement does not imply that the “obstruction” effect is the only significant factor in determining the viscosity of a rare earth chloride solution, but states only that the largest effect at moderate to high concentrations is probably the “obstruction” effect.

‘4

1.28

1.20 -

Lac13

0961

0

1

4

I

.8

I

f

I

I

I

I

I

1.2 1.6 2.0 2.4 2.8 32 3.6 m

I

~ C some ~ ~ aqueous rare earth Figure 2. ( T ) R C I J ( ~ ) Lfor chloride solutions at 25’ as a function of molality.

-I

0.70

B

0.60

sm 0

Pr Nd

00

1.16

1.12

1.04

1.08

-r

1.00

0.96

(XI

Figure 3. Viscosity B coefficient a t 25’ &s a function of ionic radius for some aqueous rare earth chloride solutions.

Volume 70,Number 8 August 1966

F. H. SPEDDING AND M. J. PIKAL

2438

i?

La I I

’Nd

Sm

I

1

1

TbDvhiEr II I I

3.6 m

0

1.30

0

F

0

1.12

I

.

0

1.16

-

1.00

4

m

1

-

1.00

I&

Figure 4. ( v ) m 8 / ( ~ ) m ifor 8 some aqueous rare earth chlorides a t 25” as a function of ionic radius a t 0.4, 1.2, and 2.0 m.

equation are plotted as a function of ionic radius of the rare earth ion. The ionic radii are those of Pauling.,I Since the B coefficients have been shown to be an additive property of the individual ions, Figure 3 shows the effect of a changing ionic radius on the B coefficient of the rare earth ion. The B coefficients for La3+, Pr3+, and Nd3+ are equal within the experimental error of about *0.005 l./mole. However, the B coefficient of Sm3+ is slightly greater, and the B coefficients for the rare earth ions from Tb3+to Era+are about 0.1 l./mole greater than those of La3+,Pr3+,and Nd3+. The B coefficientsfrom Tb3+to Er3+also exhibit a slight increase with decreasing rare earth ionic radius. If the log% rithm in eq 3 is expanded for small c, one obtains the approximation qr E 1 A ~ c . Comparison of this equation with eq l, neglecting the small term in eq l proportional to cl/’, shows that the B coefficient of the Jones-Dole equation should be approximately equal to A,. I n practice, the experimental B coefficients differed only slightly from the corresponding experimental values of A,, so Figure 3 also represents the experimental values of A 3 as a function of rare earth ionic radius. The Vand equation, eq 2, indicates the numerical value of AB,or the B coefficient may be interpreted as the product of the “average” shape factor for the hydrated ions, ICl, and the total volume occupied by these hydrated ions, a. Thus, if the major contribution

+

The Journal of Phw-icd Chemistry

0

I .o 1.00

108

0

0 I

l

l

1

1

1

1.16 1.12 I.0801.04 1.00 0.96 -dA)

Figure 5. ( ~ ) R C I ~ / ( ~ for ) L ~some C I , aqueous rare earth chlorides at 25” as a function of ionic radius a t 3.6 m.

to the relative viscosity is due to the obstruction effect, we have B A3 &’ kfl. According to this interpretation, Figure 3 shows that the product of the shape factor and the volume of the hydrated rare earth ion is constant (within the experimental error of about d=0.005 l./mole) for the rare earth ions from La3+ to Nd3+, but this product increases by about 0.07 I./mole from Nd3+to Tb3+. For the rare earth ions from Tb3+ to Er3+, this product is again nearly constant, increasing by only about 0.015 l./mole. Therefore, either the shape factor or the volume (or both the shape factor and the volume) of the hydrated rare earth ions from Tb3+ to Er3+ is significantly greater than the corresponding quantity for the rare earths ions from La3+to Nd3+. An increase in the shape factor would imply the hydrated rare earth ions from Tb3+ to Er3+ are less spherical than the hydrated rare earth ions from La3+to Nd3+. The change in shape need not be drastic, since the numerical value of the shape factor is strongly dependent on the geometric shape of the obstruction. l a For spherical obstructions, the numerical value of the shape factor is 2.5, which is its minimum value. However, the numerical value of IC1 increases to a value of about 10 in the case of highly elongated obstruction^.'^ An increase in the “effective volume” of the hydrated (31) L. Pauling, “The Nature of the Chemical Bond,” 3rd ed, Corne11 University Press, Ithaca, N. y., 1960.

RELATIVE V~SCOSITIES OF SOMEAQUEOUS RAREEARTHCHLORIDE SOLUTIONS AT 25”

rare earth ion would indicat’e the “effective radius” of the hydrated ion had increased. If the shape factors for the hydrated rare earth ions remained constant froin Nda+to Tb3+,an increase in the “effective radius” of about 0.2 A would account for the observed behavior of the B coefficients. From the preceding discussion, it should be noted that the viscosity data suggest a small but significant structural change for the hydrated rare earth ion near the middle of the rare earth series. The ions from La3+ to S d 3 +appear to form one series in which the hydrated ions have a similar structure, whereas the ions from Tb3+ appear to form another series characterized by another basic structure for the hydrated rare earth ion. The hydrated samarium ion does not appear to be a member of either series, suggesting the change in structure is a continuous transition and possibly involves an equilibrium between two structural forms. According to the interpretation given the apparent molal volumes at infinite dilution of the rare earth chlorides,za the rare earth ion in water exists in an equilibrium between two possible water coordination numbers. Coordination number, in this discussion, refers only to those water molecules in the first hydration sphere. For the rare earth ions from La3+ to Nd3+, this equilibrium favors the higher coordination number. After Nd3+, :t displacement of this equilibrium toward the lower coordination number begins to take place that results in the lower coordination number becoming increasingly more favorable for the rare earth ions from Nd3+to around Tb3+. This shift toward the lower coordination number terminates around Tb3+, and the remaining rare earth ions have essentially the same coordination number. Experimental data for apparent molal heat capacities2 and for heats of dilution3 of a number of rare earth chlorides tend to support this interpretation. A shift to a lower coordination number is certainly a structural change in the rare earth ion, and it seems reasonable to interpret the irregularity in the B coefficients as being the result of a sudden but continuous change to a lower R3+-Hz0 coordination number. Unfortunately, the precise effect of a change in coordination number on the shape of the hydrated ion or on the “effective radius” of the hydrated ion cannot be unambiguously determined. However, there are at least two conceivable ways in which a change to a lower coordination number could increase the product kls. 1. If it is assumed the volume of the hydrated ion is not changed significantly by the change to a lower coordination number, the larger B coefficients for those rare earth ions from Tb3+ to Er3+ are a result of these hydrated ions having a less spherical shape (larger value

2439

of Icl) then the hydrated ions from La3+ to Nd3+. Although far from rigorous, this interpretation is plausible. 2. The increase in klt7 may also be interpreted in terms of a nearly constant value of kl for all hydrated rare earth ions and a larger volume for the hydrated ions with the lower coordination number. A larger value for the volume of the hydrated rare earth ion having fewer water molecules in its first hydration sphere would be plausible if and only if the change to a lower coordination number increased the extent of hydration in the second hydration sphere. I n fact, to be consistent with the experimental apparent molal volume datajZathe net effect of a decrease in first-sphere hydration and an increase in second-sphere hydration must be a decrease in the average density of the water medium surrounding the rare earth ion. At present, the two preceding interpretations seem to be equally plausible. Actually, a shift to a lower coordination may well increase both kl and g. In any case, it is plausible that a shift to a lower coordination number would increase the product of the shape factor and the volume of the hydrated rare earth ion. Thus, the observed variation of the B coefficients with rare earth ionic radius seems to be compatible with the proposed change in coordination number. It should be emphasized that neither the thermodynamic data, such as apparent molal volumes, nor the viscosity data presented in this paper allows the exact coordination number to be unambiguously determined. Therefore, the discussion given here is presented in terms of a change in coordination number. The magnitude of the change in B coefficients and apparent molal volumes2&on passing from Nd3+ t o Tb3+ does suggest the net change in coordination number should be small. As discussed elsewhere,*& coordination numbers of 9 for the rare earth ions from La3+to Nd3+ and of 8 for Tb3+through Yb3+seem to be a reasonable speculation, although certainly not the only possibility. The viscosity of a rare earth chloride solution cannot be interpreted in terms of ion size and ion-solvent interactions alone. However, it seems reasonable to expect that the effect of ion-ion interactions would be a smooth function of rare earth ionic radius for the rare earth chlorides. Thus, any irregularities in the viscosity as a function of ionic radius of the rare earth ion are probably the result of ion-water interactions. Figures 4 and 5 show values of the viscosity ratio, qRCls/qLsC1l, as a function of ionic radius of the rare earth ion at various concentrations. It should be noticed that the effect of ionic radius on the viscosity at 0.4 and 1.2 m is much the same as the effect on the B coVolume 70,#umber 8 August 1066

F. H. SPEDDING, M. J. PIKAL, AND B. 0. AYERS

2440

efficient, and the variations of the viscosity ratio with ionic radius at these concentrations may be interpreted using basically the same model and arguments presented when discussing the B coefficients. It may be noticed that the increase in the viscosity ratio with decreasing ionic radius becomes much more pronounced as the concentration increases. Although this phenomenon may seem surprising at first, it is a direct result ~ ~(kl. ~ of the positive sign of the quantity, ( k l i i ) O)LSCL,, and of the concentration dependence of the obstruction effect (ie.,eq 2 or eq 3).

However, at higher concentrations, the general shape of the viscosity ratio-ionic radius curve begins to change until at 3.6 m, shown in Figure 5 , it appears as through two distinct series exist, LaCh and NdC&forming one series and the rare earth chlorides from Sm through Er forming the other series. Figure 5 may also be interpreted as indicating the viscosity of NdCL is anomalously high, and the viscosities of the other rare earth chlorides form just one series. Viscosity data for concentrated solutions of CeC4 and PrCh are needed before it can be proved which interpretation is correct.

Apparent Molal Volumes of Some Aqueous Rare Earth Chloride and Nitrate Solutions at 2 5 O I

by F. H.Spedding, M. J. Pikal, and B. 0.Ayers Institute for Atomic Research and Departmmt of Chembtry, Iowa State University, Ames, Iowa (Received December 81, 1966)

The specific gravities and apparent molal volumes of aqueous solutions of 14 rare earth salts were determined over a concentration range of about 0.002-0.2 m. A magnetically controlled float apparatus was used to determine the specific gravities with an accuracy of about * 5 X lo-'. The apparent molal volumes showed significant deviations from the simple limiting law a t low concentrations. However, it was shown that, except for Nd(N03)3, these deviations are consistent with interionic attraction theory, provided the effect of the distance of closest approach parameter, a, is recognized. The partial molal volumes a t infinite dilution, Vz0, do not vary smoothly with ionic radius of the cation for either anion series. The Tz0values decrease with decreasing ionic radius from La to Nd and from T b to Yb, but from Nd to Tb the Vz0values increase with decreasing ionic radius. A change in preferred coordination number of the R3+-Hz0 species at a critical cation radius is postulated to account for this behavior.

Introduction Although the concentration dependence of apparent molal volumes in dilute solution have been investigated for many 1-1 eleCtrOlYteS,2'3accurate data On higher valence type electrolytes are very scarce. Therefore, it W a s decided to determine the apparent molal vohmes of Wme rare earth chlorides and nitrates Over the concentration range 0.002-0.2 m. It W a s expected that The Journal of Physical Chemistry

these data would provide useful information on ionsolvent interactions, hniting behavior of 3-1 salts, and (1) Work was performed in the Ames Laboratory of the u. 8.Atomic Energy Commission. This paper is based, in part, on the Ph.D. dissertation of B. 0. Ayers submitted to the Graduate Faculty of Iowa State University of Science and Technology, Ames, Iowa, 1954, and on the Ph.D. dissertation of M. J. Pikal submitted to the Graduate Faculty of Iowa State University of Science and Technology, Ames, Iowa, 1966; Contribution NO. 1836.