1'01, 7G
F. H. SPEDDING AND S. JAFFE
884
mean distance pf closest approach for RE2(S04)3 and about 3.6 A. for the mean distance of closest approach for (RE sO4),sO*. The ratio of the activity coefficients used is not very sensitive to the choice of the mean distance of closest approach. The values used here were just assumed as reasonable values. Calculations of the ratio from the simple limiting law gave nearly the same values at the concentrations used. Since Cis known, K may be calculated if X can be determined. X may be calculated from the conductance data by applying the Onsager equation for equivalent conductance in the form
complex and X03 is that for the sulfate ion. The are the interaction terms, in brackets, calculated from the Onsager equation as given above. The value of X can be obtained from the above relation by successive approximations using the experimental L and C and the calculated constants Xoi and b,. The value of X02 is not known and was, therefore, assumed to be about 40 conductance units, in agreement with the value assumed by Davies." Calculations show that the ionization constant is only affected very slightly by large changes in the value of X02 assumed. The ionization constants listed in Table I11 are a result of extrapolating the calculated constants to A, = A: - [o zm(1 - H]/Z),,~,Z,A: 21 ~ ~ I Z , I ] P / Z infinite dilution as a function of Because of in which the function (1 - H1/2),,r, is a result of the assumptions used in their calculation, these the solution of the time of relaxation effect and is values are probably in error as much as 1 0 . 2 5 fully described by Onsager and F ~ o s s . ' ~r) is the X lo-.'. Recently, Jenkins and Monk'? reported ional concentration which is equal to c C l Z 2 , = a value of 2.4 X Recently, Jenkins and for the Monk12 reported a value of 2.4 X (30C - 12X). I From the definition of the ionic specific con- ionization constant of LaSO4& while D a v i s " reported 2.8 x -1lthough these constants ductance were calculated by different methods, they are in h,C, 1, = good agreement with those reported here. 1000 and TABLE I11 bi's
+
El,
=
I
L
= E1000 A,c,
one may write the equation for the specific conductance in the form IOOOT, = (AO,
- hlr1/2)(2c- x ) 3
+ (vt - hqr%)x+ (A%
- h7r1/2)(3C-
XP
IONIZATION CONSTANTS OF SOMERARE EARTHSULFATE COMPLEXES Complex K X 10' Complex K X 104 2.4 GdS04' 2.2 LaSO, + 2.6 HoSO~ + 2 6 CeS04+ PrSOI 2 4 ErSOd 2 6 YdSOa 2 3 I'bSO, 2 6 SmSO, + 2 2 YSO.+ 3 4 +
+
+
in which XO1 is the rare earth ion equivalent conductance at infinite dilution, Xo2 is that for the (10) L Onsager and R. M. Fuoss, J . P h y s Ckem
[CONTRIRUTION S O . 277
, 36,
2689 (1932).
+
(11) C. W. Davies, E n d e a w u r 4 , 114 11943). (12) I. L. Jenkins and C. B. V o n k , THISJ O U R N A L , 72, 26913 (1950).
RESEARCH A N D DEPARTMENT O F CHEMISTRY, COLLEGE']
F R O M THE I U S T I T U T E FOR A T O M I C
IOWA STATE
Conductances, Transference Numbers and Activity Coefficients of Some Rare Earth Perchlorates and Nitrates at 25" BY F. H. SPEDDING AND S. JAFFE RECEIVED JULY 27, 1953 The equivalent conductances and cation transference numbers a t 25' of aqueous solutions of La( Clod),, Pr( C1O4)3,,Kd Sm(CIOa)d,Gd(CIOa)?,Ho(ClO4)?,Er( C104)a,Yb(C104)3: La(N0313, N(( N O J ) ~and , Gd("&h have been deterlnlned up t o 0.1 N. Also a method has been developed for approximating the mean distance of closest approach of the various ions in these solutions from conductance measurements. This method allows the calculation of activity coefficients from the Debye-Huckel IRW
Introduction This paper is the seventh in a series concerning the electrolytic behavior of aqueous solutions of rare earth compounds. The earlier paper^^-^ have presented data on the conductances, transference numbers and activity coefficients of several rare earth chlorides and bromides. This paper extends this investigation to the rare earth per(1) W o r k was performed in the Ames Laboratory of the Atomic Energy Commission. (2) F. H. Spedding, P. E. Porter and J. hl. Wright, THISJOURNAL, 74, 2055 (1952); 74, 2778 (19a2). (3) F. H. Spedding, P. E. Porter and J M. Wright, ibid., 74, 2781 (1952). ( 4 ) F. H. Spedding and I. S. Yaffe, ibid., 74, 2781 (1953),
chlorates and nitrates. As discussed in the first articles of this series,2 such information should be of considerable value in the study of the various factors which enter into the modern theories of electrolytic behavior. An extension of the electrophoretic correction in the Onsager equation for equivalent conductance has made the estimation of activity coefficients possible from the conductance data presented here.
Experimental Except as specifically discussed below, the experimental procedures and apparatus were the same as those previously All measurements were made at 25 i 0.02'. The rare earth oxides used in this research are from the
Feb. 5, 1954
885 RAREEARTH PERCHLORATES AND NITRATES:CONDUCTANCES AND TRANSFERENCES TABLE I
EQUIVALENr COSDUCTANCESOF SDME RAREEARTHPERCHLORATES AND NITRATES I N AQUEOUS SOLUTION AT 25"" Equivalent conductances, mhos. cm.-l Nurmality
La(ClOa),
Pr(C104d Nd(ClOd8 Sm(Cl0ds Gd(Cl0dr Ho(C1Ods
0.0000 137.0 136.8 .0002 133.4 133.3 .0004 131.9 131.8 ,0007 130.3 130.2 ,0010 129.1 129.0 ,0020 126.3 126.1 ,0040 122.3 122.1 .0070 118.3 118.5 .0100 115.8 115.8 .0200 110.2 110.2 .0400 104.3 104.4 .O'iOO 99.7 99.7 .lo00 96.8 96.7 a The data presented here are request.
137.2 133.4 131.8 130.1 128.9 125.8 121.7 118.0 115.5 110.0 104.1 99.5 96.5 smoothed
Er(ClO4r
135.8 134.5 133.8 131.9 130.6 130.1 130.4 129.0 128.6 128.8 127.5 127.0 127.7 126.4 125.8 124.8 123.6 122.9 120.9 119.7 119.1 117.1 116.1 115.5 114.5 113.4 113.0 109.2 108.1 107.6 103.5 102.2 101.8 98.7 97.6 97.3 95.8 94.9 94.4 values. Anyone desiring the
same source and have the same analysis as those used in the previous papers. 2-4 The rare earth salt solutions were prepared by dissolving the rare earth oxides in the proper acid. Doubly vacuum distilled perchloric acid and chemically pure nitric acid were used for this purpose, An excess of rare earth oxide was filtered off after the acid had completely reacted. However, the solutions still contained small amounts of rare earth hydrous oxides which were present in colloidal form. For this reason, the solutions were titrated to the equivalence points which were previously determined by pH titrations on aliquots of the solutions. Although these solutions hydrolyze slightly, the effect of the excess hydrogen ions was within the experimental error in the measurements presented here. Stock solutions of each rare earth salt were analyzed by precipitating rare earth oxalate, igniting this for 12 hours a t 900' and weighing as rare earth oxide. The rest of the solutions were prepared from the stock solutions by volume dilutions using calibrated glassware and conductance water mho-cm.-l. with a specific conductance of less than 1 X
Yb(CIO43
La(NO3)r
Nd(N0s)a
Gd(N0dr
133.5 133.4 140.6 141.3 138.7 130.0 130.0 136.5 137.4 134.9 128.5 128.5 134.9 135.7 133.3 126.9 126.9 133.1 133.9 131.6 125.7 125.6 131.7 132.6 130.3 122.8 122.7 128.3 129.0 126.9 119.0 118.9 123.9 124.2 122.5 115.3 115.3 119.5 119.5 117.9 112.9 112.8 116.3 116.0 114.5 107.5 107.4 109.4 108.8 107.5 101.7 101.6 102.1 100.7 99.9 97.2 97.1 96.0 94.2 93.5 94.3 94.2 92.0 90.0 89.4 exact data may obtain them from the authors on
normality may be used as an indication of the applicability of the Onsager limiting law since the theory is being followed when the slope of the curve is zero. The rare earth perchlorates are apparently beginning to obey the simple limiting law only a t around 0.0003 N while the nitrates seem to obey the theory out a t 0.001 N . I t will be shown that the agreement with the theory can be extended to almost 0.01 N solutions by the application of a graphical integration of the electrophoretic term of the Onsager equation. B. Transference Numbers.-The transference numbers of the rare earth perchlorates and nitrates were found to be linear functions of the square root of the normality. The least squares equations and the average deviations of the experimental points from these equations appear in Table 11. These data are represented in Figs. 1 and 2.
Results TABLE I1 A. Conductances.-The equivalent conNUMBER EQUATIONS FOR SOME RAREEARTH ductances of the rare earth salts are given in Table TRANSFERENCE P E R C H L O R A T E S A N D SITRATESAT25' I. The conductances of the rare earth perchlorates Least-squares eq.. Average generally show the same changes with atomic numSalt T+ deviation ber as is observed with the halides. The decrease La(C10da 0.5038-0.101 N'12 0.0002 in conductance with an increase in atomic number is Pr(ClO4),+ .5067- ,102 N1/' ,0004 attributed to the effective increase in ionic radii due Xd( c104)a .5067- .0931 N'Ia ,0002 to hydration, in spite of the decreasing ionic radii Sm (C104h .5072- .lo2 N'12 ,0002 as determined from crystallographic measurements. Gd(C0h .501 - .IO0 N I I t .0002 The equivalent conductances a t infinite dilution Ho(C101)~ .4909- .0924 NIIa .0003 were obtained by extrapolation of the Onsager Er(C1Oda .4947- .112 N I I a .0005 equation. The equivalent conductance a t infinite Yb(C104h .4930- .111 .0005 dilution of the perchlorate ion was taken to be La( XOd3 .4895- ,0597 N'I1 .oooz 67.32 as determined by Jones6 and that of the Nb( r\To3)3 .4922- .0698 N'Ia .0003 nitrate ion has the value of 71.44 from the work by Gd(N03)3 ,4857- .0746 N1/' .0005 MacInnes, Shedlovsky and Longsworth. The The solvent corrections on the transference numequivalent conductances a t infinite dilution of the bers were made in the usual way using the conrare earth ions calculated from the above were in reasonable agreement with those calculated from ductance data presented in Table I. However, since lithium chloride was used as indicator solution, other salts in earlier work. The values of A; were calculated from the the volume correction was modified slightly to experimental points by the method of S h e d l o v ~ k y . ~include the partial molal volume changes between The extrapolation to infinite dilution of the A; the lithium chloride and lithium perchlorate and corrections were made as values as a function of the square root of the nitrate solutions. These ( 5 ) J. H. Jones, THISJOURNAL, 67, 855 (1945). VRRE(Cl0,h = - 7jVCd 2 v w l r + f7clo,- - Vccr -
-
( 6 ) D.A. MacInnes, T.Shedlovsky and L. G. Longsworth, ibid., 64, 2758 (1932). (7) T. Shedlovsky, ibid., 64, 1405 (1932).
+
F. H. SPEDDWG AND S. JAFFB
886
vi
04950
-
04900
-
04050
-
0.4000
-
Vol. 76
W
f
f w ii
pa
04750 -
z
P
\
3 04700 -
a4650
-
04W
-
1
I
04550
01
02
a3
0
(NORMALITY)&,
Fig. 1.-Transference
I
ai
numbers of some rarc earth perchlorates at 25".
I
0.2 (NORMALITY)
I
0.3
a
numbers of some rare earth nitrates a t 25'.
Fig. 2.-Transference
and
numbers are consistent with the conductance data except for samarium which has transference numbers about equal to those of praseodymium while the conductances are lower than praseoin which dymium. This behavior also was observed in the case of the chlorides2 Since only three nitrates Vc1o4- - vell'I,if;lil* - c-l,,cl = " . l X have been measured so far, a discussion of their and behavior will be reserved until the series is more VNiso,-- Vc,- V L , s o 3 - PLii;l = 1O.xti nearly complete. The transference numbers of the rare earth perThe densities used in employing these corrections chlorates and nitrates do not approach the Onsager are listed in Table 111. limiting slopes in the range of concentrations TABLE 111 measured. However, the agreement between exDENSITIES OF AQUEOUSSOLUTIOSS OF RARE EARTH PER- periment and theory is greatly improved when the CHLORATES A N D SITRATES AT 25"" electrophoretic term in the Onsager equation is Salt Density Salt Density evaluated according to Dye and S ~ e d d i n g . ~ La(C104)3 Pr(ClO4)r Nd(ClO4)a Sm(C10,)a Cd(C1Oi)a Ho(C10~)a ErIC104)a
0 9970 ,9969
+ 0.334Cm +
,9971 f ,9968 ,9970 ,9970
+
+ +
.$I970
+
.358Cm .347Cm 364C,
,3726, .3G9Cm .382Cn
Yb(ClO4)i Ln(KOa)a Nd(N03)a Gd(N0a)s
I.iC104 LiNOl 1.1Cl
0.9970 + 0 . 3 8 3 C m .99iO .274Cm ,9971 ,279Cm ,9970 f ,289Cm ,9971 .0623Cm ,9971 .03Y8Cln ,9971 0241Cm
+
Calculation of Activity Coefficients A. Introduction.-The Debye-Hiickel limiting
+
law for activity coefficients may be used to calculate the mean activity coefficients of strong electrolytes in moderately dilute solutions. It has been shownst4that the activity coefficients of rare earth halides thus calculated agree remarkably well with those obtained experimentally from e.1n.f. niensureiiieiits. I t i i i a y be nssunietl, therefore, that the rare earth perchlorates and nitrates obey the same theory a t the same concentrations. However, in order to evaluate the activity coefficients from the equation
+ + +
The data for LiCIOl u-as obtained from u.ork by Jo11cs7 and the data for LiCl and LiX03 were calculated from data given in Harned and Owen's text.8 Other relations were obtained from measurements of densities over the range from 0.001 t o 0.1 iY with a 50-ml. pycnometer. C , is the molar concentration and m is the molal concentration. a
The cation transference numbers of the rare earth perchlorates are in relatively the same order LIS those of the halides.?,4 The transference (8) H. S. Harned and B. €3. Owen, "The Physical Chemistry of Electrolytic Solutions," Reinhold Publ. C o r p . , Inc., Kew York, h-,Y., 1950, p 556.
__
_.__
(9) J L Dye and F I< Speddmg, THISJ O U R N A L 7 6 , 888 ( 1 9 i i )
RAREEARTHPERCHLORATES AND NITRATES:
Feb. 5, 1954
the mean distance of closest approach for the solutes must be known. The distances of closest approach have not been evaluated previously for the rare earth perchlorates and nitrates. The mean distance of closest approach may be evaluated from the conductance data by a consideration of the dependence on 8. of the electrophoretic term in the Onsager equation for conductance. Dye and Speddingg have shown that the electrophoretic correction for 3-1 electrolytes may be expressed by the relations
CONDUCTANCES AND
152 -
TRANSFERENCES 887
153
151
-
I50
-
149
-
Ad
VALUES CALCULATED AT VARIOUS
ai
VALUES
Oi = 3.0! 01 . 4 . 0 ? 01 -4.4A 01 4.5A 01 =5.0ft Simple Onsoper Equation
0 0 A B 8
0
I I
-4 1480 W
3
-
147
3 0
2:
146-
0
145
=
FDkT 18007rve(/z+I
+
__
= 3.809
12-1)
and p = Kr. The values of the integrands may be plotted as functions of KY and the integrals may then be evaluated from the areas under the curves. The change in the conductance due to the electrophoretic effect is quite sensitive to the a value used, since this is the lower limit of the integration and the curve rises rapidly for small values of a. The procedure is to assume a series of a values for the perchlorates and nitrates and calculate the electrophoretic correction for each a a t various concentrations. These values are included in the Onsager equation for equivalent conductance and an extrapolation to infinite dilution of the calculated & values as a function of c ' / p is made. When the proper a value is used, the curve remains flat from infinite dilution to about 0.008 N . If the a value is too small, the curve rises from the
-
143 I42 141 144
140
-
1391
o
1
oa
I
a02
I a03
I a04
i 0.05
I
006
I I 007 a08
I 0.09
a
(NORMALITY)~.
Fig. 3.-Evaluation of the mean distance of closest approach from conductance data on neodymium nitrate.
flat portion a t very low concentrations, and if the a value is too large, the curve dips below the true A, value. Figure 3 is an example of the evaluation of a for Nd(N03)3by the method described above. It can be seen that a reasonable a value can be obtained when the best extrapolation curve is found . B. Results.-The changes in the equivalent conductances due to the electrophoretic effect were TABLE IV calculated for several c2 values a t several concenON CONDUCTANCE AT ELECTROPHORETIC CORRECTIONS trations by the graphical method described above. VARIOUSd VALUESFOR 3-1 ELECTROLYTES IN AQUEOUS Table IV lists a convenient selection of these SOLUTION AT 25" values. A series of curves were plotted from these At At At data to obtain interpolated values of the correc6 in A. 0.0100N 0.0035 N 0.0010 N tions a t intermediate B's and concentrations. The 28.34 15.60 7.15 3.0 8. values that were obtained for the rare earth 16.90 4.0 10.66 5.57 perchlorates and nitrates are listed in Table V. 14.20 9.04 5.07 5.0 The average d value for the rare earth chlorides 13.62 8.64 4.87 5.49 13.01 8.35 4.77 is about 5.7 A.6 This value supports the physical 5.92 12.30 8.00 4.65 6.5 picture that a monomolecular layer of water sur11.95 7.75 4.61 7.0 rounds the rare earth cation. This picture seems to hold for the perchlorates as yell, since the 11.50 7.25 7.5 4.50 perchlorate ion radius is about 1 A. larger than TABLE V that of the chloride and the expected 8. value is MEANDISTANCE OF CLOSEST APPROACH FROM EXTRAPOLA-about 6.7 A. This is in remarkable agreement TION OF CONDUCTANCE DATA with the values found by the conductance method. Salt i in A. Salt i in A. The value of 8. for the nitrates may also be made to La(C10dz 7.0 Er(C104)3 6.8 fit this picture if one considers that the effective Pr( Clod3 7.2 Yb( ClO4 )a 7.2 radius of the nitrate ion in this case is about 1 A. Xd( C1Oa)s 6.0 La(NO& 4.4 Since the rare earth ion is about 1 A. in radius Nd(N0sL 4.5 Sm(C10d3 6.8 and the water molecule is about 2.8 A. in diameter, Gd( NOa)3 4.4 Gd(C104)3 6.4 the value of 4.5 A. for the rare earth nitrates seems Ho( C104h 6.3 reasonable.
J. L. DYEAND F. H. SPEDDING
888
From these data, it is, of course, impossible to say how far the Debye-Hiickel limiting law involving perchlorates and nitrates will hold for activity coefficients. However, in light of the fact that a constant value of d results in agreement
[CONTRIBUTION No. 279
PROM THE
Vol. 76
of experiment and theory for the chlorides and bromides up to 0.1 N , it is reasonable to assume that the limiting law would hold equally well for the perchlorates and nitrates. AMES,IOWA
INSTITUTE FOR ATOMIC RESEARCHAND DEPARTMENT OF CHEMISTRY, IOWA STATE COLLEGE1]
The Application of Onsager’s Theory of Conductance to the Conductances and Transference Numbers of Unsymmetrical Electrolytes BY J. L. DYEAND F. H. SPEDDING RECEIVED JULY 27, 1953 Consideration of the transference number behavior of unsymmetrical electrolytes in aqueous solution led t o an examination of the mathematical development of Onsager’s theory of conductance. A mathematical treatment of Onsager’s theory of conductance is described, which employs graphical methods t o evaluate integrals which were only approximately evaluated by Onsager. The approximate methods, while satisfactory for 1-1 electrolytes, are unable t o explain the transference number behavior of unsymmetrical electrolytes. The new treatment of the theory was applied to the conductances and transference numbers of CaClz, XdCla and ErC13. The agreement between theory and experiment was greatly improved for both transference numbers and conductances of these unsymmetrical electrolytes.
Introduction expansion and subsequent neglect of high order In a theoretical study of the processes of diffusion terms in the usual treatment of the electrophoretic and electrical conduction in electrolytic solutions, effect, is invalid even a t quite low concentrations. Onsager2 developed equations which quantitatively In attempting to include the complete integral, describe the changes in conductance with con- however, the equation could not be put into an centration in sufficiently dilute solutions. Because analytical form, and certain integrals were evaluthe theory was developed for the individual ionic ated graphically. The results are compared with species in the solution, the results were also applic- experimental values of the transference numbers able to the transference numbers of the constituent anti conductances. ions. The theory predicted the transference numDevelopment of the Method ber to be a linear function of the square root of the concentration in dilute solutions. This has actuFrom the treatment of the electrophoretic effect ally been found to be the case for most electrolytes, upon conductance given by Onsager2 one may oband the slope of the resultant straight line agrees tain the electrophoretic correction to the conductquite well with the predicted slope for 1-1 elec- ance to be trolytes. I t has been pointed out, h ~ w e v e r , ~ - ~ that unsymmetrical electrolytes for which the transference numbers have been measured exhibit slopes which differ by as much as a factor of onein which n,i is the concentration of the i t h kind of fifth from the slope predicted by the Onsager equaion a t a distance r from a n y j ion and is given by tion. Because of this dependence upon charge type, the Boltzmann distribution to be approximately i t was deemed advisable to re-examine the mathematical development of the conductance equation. The change in the conductance of an electrolytic in which $! is the electrostatic potential a t a dissolution with concentration may be said to be due to two factors, which are called the “time of relaxa- tance r from the j t h ion and was found by Debye tion effect” and the “electrophoretic effect.” and Hiickel to be The electrophoretic effect normallv makes the larger contribution to the conductance. If the fundamental assutnptions of Onsager’s treatment T h e quantities in these equations are defined by of conductance are assumed to be correct, it can be 11 15 t h e viscosity of the solvent shown that [or icms of high charge ari exponential c , is the charge O I L the it11 ion 31id i~icludesthe slgil of the ( I ) Work \*as prrfurmed iu the I\rnrs Laboratury of the Atomic Energy Commission. (2) (a) L. Onsager, Physik %.. 28, 277 (10!27), (b) L. Onsager and R . M. Fuoss, J . Phys. Chem., 36, ?I389 (1932). (3) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” Second Edition, Reinhold Publ. Corp., h-ew York, N. Y . , 1950, p. 165. (4) L. G. Longsworth and D. A. blaclnnes, THISJOUKNAL. 60, 3070
charge on the ion k is the Boltzmanu constant D is the dielectric constant of the solvent a, is the mean distance of closest approach of the ions t o the j ion
(1038). (5)
F. H. Spedding, P. E:. Pixttlr a n d J
(1952).
51. %‘right, i h r d . . 74, 2778
the total number of kind$ of ions in the solution, :inti
s
ii
e
is t h e charge on the electron
