3317
J . Phys. Chem. 1985,89, 3317-3319
Lanthanide Thermodynamic Predlctlons. 7. Thermodynamics of 2+, 3+, and 4+ Aquo Ions and Standard Electrode Potentials at 298.15 K Steven G. Brats&* and J. J. Lagowski Department of Chemistry, The University of Texas, Austin, Texas 78712 (Received: December 26, 1984)
Simple equations are presented for the prediction of standard-state thermodynamicproperties of 2+, 3+, and 4+ lanthanide ions in aqueous solution at 298.15 K.
Introduction This paper continues our investigation of the thermodynamic properties of simple lanthanide compounds and ions (ref 1 and ref 1-5 therein). In this paper we present equations for the prediction of the standard enthalpies of formation ( A H f O ) , standard entropies ( S O ) , and standard Gibbs free energies of formation (AGfo) of lanthanide aquo ions M'+(aq) where z = 2+, 3+, and 4+. The prediction of AGfo(M'+,aq) is particularly valuable because it allows the calculation of standard electrode potentials ( E O ) connecting various oxidation states. The equations presented in this paper will be applied to the actinides in a future publication. All thermodynamic quantities tabulated in this paper refer to 298.15 K. As before,' we have attempted to maintain compatibility with the CODATA recommendationsZand we have taken all physical constants and energy conversion factors from Cohen and Tay10r.~ Thermodynamic properties of gas-phase ions and crystal ionic radii have been taken from ref 1. Thermodynamics of Single Ion Hydration Born-Haber thermochemical cycles have been constructed for the thermodynamic changes associated with the hydration of ions AYhydo(Mz+)where Y = H,S , or G.@* These cycles lead to AHhydo(M'+) = AHfo(M'+,aq) - AHfo(M'+,g) + [ U f o ( H + & ) + Mhydo(H+)lZ (1) &?hydo(MZ+) = So(M'+,aq) - So(Mz+,g)+ [So(H+,g) + hShydo(H+)]Z (2)
+
AGhydo(Mz+)= AGfo(Mz+,aq)- AGfo(M'+,g) [AGEo(H+,g) + AGhydo(H+)lZ (3)
In eq 1 and 3, the reference state for aquo ions is AYfo(H+,aq) = 0 while for gaseous ions it is AYfo(e-,g) = 0. In eq 2, the reference state for aquo ions is So(H+,aq) = 0 while for gaseous ions the absolute entropies may be obtained from statistical Values accepted in this paper for the hydration of the proton at 298.15 K are MhYdo(H+)= -1105.0 f 17 kJ mol-'
(4)
&?hydo(H+)= -130.9 f 1.4 J K-' mol-'
(5)
(1) Bratsch, S . G.; Lagowski, J. J. J. Phys. Chem., preceding paper in this issue. (2) CODATA Recommended Key Values for Thermodynamics. J . Chem. Thermodyn. 1978, 10,903. (3) Coben, E. R.; Taylor, B. N. J . Phys. Chem. Re$ Data 1973, 2, 663. (4) Rowinsky, D. R. Chem. Rev. 1965, 65, 467. (5) Mom. L. R. Chem. Rev. 1976, 76, 827. (6). (a) Johnson, D. A. "Some Thermodynamic Aspects of Inorganic Chamtry", 2nd 4.; Cambridge University Press: London, 1982; pp 120-122. (b) Ibid. p 8. (c) Ibid. pp 122-124. (7) Bratsch, S . G. Lagowski, J. J. J . Phys. Chem. 1984,88, 1086.
AGhydo(H+)= -1066.0 f 17 kJ mol-'
(6) The value for AGhflo(H+) is from Conway.8 The value for hShydo(H+) is derived from the absolute So(H+,aq) (-22.1 f 1.4 J K-' mol-'; ref 8) and the statistical-mechanical value for So(H+,g) (108.84 J K-' mol-'; ref 7). The value for AHhydo(H+) iS hGhydo(H+)+ ThShydo(H+). Combination of the selected AYh do(H+) values with the thermodynamic properties of H+ (8)Ty gives AHhydo(M'+) = AHfo(Mz+,aq)- AHfo(M'+,g) or AHfo(Mz+,aq)= AHf"(M'+,g)
+ 431.22
+ AHhydo(M'+) - 431.22
(7)
hShydo(M'+) = So(Mz+,aq)- So(Mz+,g)- 22.12 or So(Mz+,aq)= So(Mz+,g)+ AShydo(MZ+) + 22.12
= AGfo(Mz+,aq)- AGfo(M'+,g) AGhydo(MZ+)
(8)
+ 451.02
or AGfO(MZ+,aq) = AGfO(MZ+,g) + AG,,O(MZ+) - 451.02 (9) where Ho and Go are in kJ mol-' and So is in J K-' mol-'.
An Empirical Equation for S i l e Ion Hydration The Born-Bjerrum theory of ionic solvati~n~*'~ treats the solvent as a continuous dielectric and predicts that the standard enthalpy, entropy, and Gibbs free energy of solvation follow a functional dependence of zZ/r, where r is an ill-defined ionic radius. The Born-Bjerrum theory has been called unrealistic because it ignores the molecular nature of the solvent and specific ion-solvent interactions." Nevertheless, agreement with experiment can occasionally be achieved through empirical modifications (see, e.g., ref 4, 6c, 7, 8, and 12). A curious feature of the Born-Bjerrum theory is the prediction of a direct proportionality between the standard enthalpy and entropy of ionic solvation. Such a relationship actually exists (at least approximately), although it may be somewhat fortuitous because only a fraction of the standard entropy change is believed to be due to Coulombic (z2/r) i n t e r a ~ t i o n . ' ~ The J ~ relationship between A H h y d o and h S h for 2+ ions (Ca2+,Sr2+,and Baz+) and 3+ ions (Y3+ and Lag-Lu3+) is illustrated in Figures 1 and 2, respectively. (Data for 4+ ions are less reliable and not shown.) The assumption of a rough proportionality between M h y d o and &?hfioappears to be valid for afixed ionic charge over a limited ionic radius range. (8) Conway, B. E. J . Solution Chem. 1978, 7, 721. (9) Born, M. Z . Phys. 1920, 1, 45. (10) Bjerrum, N.; Larsson, E. Z . Phys. Chem. 1927, 127, 358. (11) Desnoyers, J. E.; Jolicoeur, C. In 'Comprehensive Treatise of Electrochemistry"; Conway, B. E., Bcckris, J. 0' M., Yeager, E., Eds.; Plenum Press: New York, 1983; Vol. 5, pp 1-109. (12) Latimer, W. M.; Pitzer, K. S.; Slansky, C. M. J . Chem. Phys. 1939, 7. 108. (13) Eley, D. D.; Evans, M. G. Trans. Faraday SOC.1938, 34, 1093. (14) Criss, C. M. J . Phys. Chem. 1974, 78, 1000.
0022-3654/85/2089-3317$01.50/0 0 1985 American Chemical Society
3318 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 -200
,
I
"A
r
-0 I
+ E
I
Y
Bratsch and Lagowski
TABLE I: Parameters for Eq 10
//
-220 -
7
I
Z
Y
Adz)d
rSz)e
2+ 4
H
-278.0 -44 -264.9 -613.4 -82.0 -589.0 -1079 -120 -1043
0.073 0.073 0.073 0.0784 0.0784 0.0784 0.080 0.080 0.080
S
G H S G
3+b
/
N
H
4+c
S
.240
G
I -260
"Experimental data for Ca, Sr, and Ba from ref 2 and 20. bExperimental data for Y and La-Lu from ref 5, 17, 20, and 24. 'Experimental data for Ce and Th from ref 5 and 20. dkJ-nm mol-'. enm.
/
-1600
-1500
-1400
-1300
TABLE 11: Predicted Standard Cibbs Free Energies of Formation of Lanthanide 2+, 3+, and 4+ Aquo Ions (kJ mol-', 298.15 K)" Figure 1. Correlation of AHhydo(M2+) and A&,o(M2+). passes through the origin.
The line drawn
AGfo(M2+,aq) b c -554 -559 -560
M
AGro(M3+,aa)b AGro(M4+.aa)b
-"; Ca Sr Ba
Y
La Ce Pr Nd Pm
-460
-202 -257 -377 -402 -414 -510 -535 -210 -319 -412 -398 -376 -453 -524 -169
Sm
AH'
(M3+1 I k J m o l - ' l
hYd
Figure 2. Correlation of mh*'(M3+) and L%~~,~(M'+). The line drawn passes through the origin.
On the basis of Figures 1 and 2, and in keeping with the ideas developed in ref 1, we propose AY,,,do(Mz+)= A y ( z ) / [ r M z + + rx(z)]
Eu Gd Tb DY Ho Er Tm Yb Lu Zr Hf Th
-322 -337 -343 -341 -343 -340 -249 -330 -332 -330 -302 -294 -301 -250 -222
-689 -693 -677 -677 -672 -667 -665 -567 -662 -659 -672
-508 -319 -197 -160 -134 +67 +99 -338 -198 -109 -100 -97 +41 +155 -46 1 -487 -704
-615
-676 -675 -627 -665
"Equations 9 and 10, using A d z ) and rx(z)from Table I and rMxt from ref 1. *Using lowest energy (ground-state electron configuration) AGfo(Mz+,g)from ref 1; no LFSE. CUsingAGfo(M2+,g,5d')from ref 1 and subtracting 120 kJ mol-' for LFSE (see text).
(10)
for the thermodynamics of hydration of lanthanide 2+, 3+, and 4+ ions. In eq 10, Y = H,S , or G , A d z ) is a constant for H, S, or G which depends on z, rM* is the crystal ionic radius of MTcl and rx(z)is a radius parameter to be added to the crystal ionic radius and which depends on z. Values of A d z ) and rx(z)for z = 2+, 3+, and 4+ are presented in Table I. These values are consistent with (and must be used in conjunction with) eq 4-9 and the crystal ionic radii in ref 1.
Predicted Standard Cibbs Free Energies of Formation of Lanthanide 2+, 3+, and 4+ Aquo Ions and Predicted Standard Electrode Potentials at 298.15 K Combination of eq 7-9 with the predictions of eq 10 gives AHfo(M*,aq), So(M*,aq), and ACfo(M*,aq). Predicted values of the latter quantity are listed in Table 11. A lanthanide 2+ ion with a 5d' electron configuration is subject to significant ligand field effects when placed in chemical environments. The standard enthalpies and Gibbs free energies of formation of chemical s p i e s containing such M2+ions are more negative than anticipated on the basis of eq 7 and 9, the difference being the ligand field stabilization energy, LFSE. The spectroscopic analysis of Johnsonls leads to LFSE values which decrease linearly across the lanthanideseries from about 110 to 80 kJ mol-'. (15) Johnson, D. A. J. Chem. SOC.,Dalton Trans. 1974, 1671.
L a Ce
P r N d Pm
Sm E u G d
Tb
Dy
Ho E r
Tm Yb
Lu
Figure 3. Predicted one-step standard electrode potentials (in volts), E0(M4+/M3+)and E0(M3+/M2+).
However, the standard electrode potentials Eo (M3+/M2+)compiled recently by Mikheev16indicate a higher LFSE of about 150 kJ mol-'. Taking both the uncertainty of the spectroscopic method and the difficulty of the electrochemical measurements into account, we adopt LFSE(M2+,aq,5d1)= 120 f 30 kJ mol-'. (16) Mikheev, N. B. Inorg.Chim. Acta 1984, 94, 241.
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3319
Lanthanide Thermodynamic Predictions
TABLE III: Predicted Standard Electrode Potentials (V, 298.15 K)' M Ca
Sr Ba Y La Ce Pr Nd Pm Sm Eu Gd Tb
DY Ho Er Tm Yb
Lu
M4+/M3+ M3t/M2+b M2+/Mb M"/M -2.87 -2.90 -2.90 -2.38 -2.39 -3.8' -1.67' -2.34 -3.5' -1.75' +1.7 -2.34 +3.7 -3.1 -1.95 -2.32 +4.9 -2.8 -2.08 -2.30 -2.6 -2.15 +5.3 -2.30 -1.6 -2.64 +5.5 -1.96 +6.6 -0.3 -2.77 4-7.9 -3.4' -1.71' -2.29 -2.28 -3.4' -1.72' +3.3 -2.32 +4.9 -2.7 -2.14 -2.33 +5.9 -2.9 -2.06 -2.34 -3.1 -1.95 +6.0 -2.33 +6.0 -2.3 -2.35 -1.1 -2.72 -2.17 +6.9 -4.6' -1.15' -2.30 +8.5
Zr Hf Th
M4+/M
I
-1.32 -0.83 -0.51 -0.41 -0.35 +0.17 +0.26 -0.88 -0.51 -0.28 -0.26 -0.25 +0.11 +0.40 -1.19 -1.26 -1.82
"AGfo(Mzt,aq) from Table 11. the more negative AGro(M2+,aq) from Table 11. 'M2+(aq) is predicted to have a 5d' electron config~ration.~~~~~'~~'~
- 1.54
-3.oj
I
, , ,
, , , , ,
,
, ,
J K-'mol-', corresponding to a perturbation in ACqydoof less than 3 kJ mol-'. Therefore, we have chosen to ignore this complication in our correlations. Predicted E0(M4+/M3+)and E0(M3+ M2+) in Table I11 generally agree with previous estimate^^-'^*' within f0.2 V. The level of agreement is gratifying because most of the values for M2+(aq),M3+(aq),and M4+(aq) in this paper have been derived independently of one another. The only significant disagreement with previous Eo estimates concerns those 2+ aquo ions which probably have 5d' electron configurations (Table 11). Our predictions for AGfo(M3+,aq)(Table 11) and Eo(M3+/M) (Table 111) are in general agreement with the experimental data.5v15.20 The standard Gibbs free energies of formation of Zr02(c) and HfOZ(c) given by NBSa may be combined with the solubility data of Baes and Mesmer21a to give values of AGfo(Zr4+,aq)and AGfo(Hf4+,aq)which are about 100 kJ mol-' more negative than our predictions. However, the Baes-Mesmer solubility data actually represent the combination of several measurements, some of which pertain to uncertain processes at high ionic strengths, and imply that the basicities of Zr02 and Hf02 are much greater than that of CeO,. This is difficult to rationalize when one observes that, among the oxides of the isoelectronic 3+ ions, the basicities of Y203and Lu2O3 are well established as being less than that of La203.5*20,21b We emphasize that our goal in this paper has only been to derive useful thermodynamic relationships and not to propose a theory of ionsolvent interaction. In particular, it is unnecessary that the values selected for proton hydration (eq 4-6) be highly accurate because the values of A d z ) and rx(z)listed in Table I have been chosen to give the best fit to experimental aquo ion data. As predicted by the Born-Bjerrum theory of ionic solvation?J0 the values of A d z ) and AG(z)in Table I vary approximately with z2, the dependence on z2 being somewhat closer for the latter quantity. These results suggest that AHhydo(M'+) and AGhydo(M'+) are predominately determined by Coulombic ( z 2 / r ) interaction for the ions considered. However, on the basis of our As(z) values it appears that hShydo(Mz+)varies with z rather than z2. A first-power dependence has been noted for other ions by Powell and Latimer.22-23
I
, , , , {
Ba La C e P r N d Pm Sm E u G d Tb D y Ho E r Tm Yb
L U Hf
Figure 4. Predicted overall standard electrode potentials (in volts), Eo(Mzt/M).
Predicted Eo values connecting various oxidation states are listed in Table 111. Trends in Eo are illustrated in Figures 3 and 4. Discussion
Spedding" has suggested that the entropy data for M3+(aq) indicate a gradual change in primary hydration number (perhaps from 9 to 8; see ref 18) between Nd and Tb. Figure 2 indicates that the effect of this change on h S h y d o is probably less than 10 (17) Spedding, F. H.; Rard, J. A.; Habenschuss, A. J . Phys. Chem. 1977, 81, 1069.
(18) Fidelis, I. K.; Mioduski, T. J. In "Structure and Bonding";Clarke, M. J., Ed.;Springer-Verlag: New York, 1981; Vol. 47, pp 27-51.
Acknowledgment. We are grateful for the generous financial support provided by the Robert A. Welch Foundation (Grant No. F081). Registry No. Ca, 7440-70-2; Sr, 7440-24-6; Ba, 7440-39-3; Y, 7440-65-5; La, 7439-91-0; Ce, 7440-45-1; Pr, 7440-10-0; Nd, 7440-00-8; Pm, 7440-12-2; Sm, 7440-19-9; Eu, 7440-53-1; Gd, 7440-54-2; Tb, 7440-27-9; Dy,7429-91-6; Ho, 7440-60-0; Er, 7440-52-0 Tm, 7440-30-4; Yb, 7440-64-4; Lu, 7439-94-3; Zr, 7440-67-7; Hf, 7440-58-6; Th, 7440-29-1. (19) Nugent, L. J.; Baybarz, R. D.; Burnett, J. L.; Ryan, J. L. J . Phys. Chem. 1973, 77, 1528. (20) NBS Technical Notes 270-3 through - 270-8; National Bureau of Standards: Washington, DC, 1968-1981. (21) (a) Baes, C. F.; Mesmer, R. E. "The Hydrolysis of Cations";WileyInterscience: New York, 1976; pp 152-158. (b) Ibid. pp 129-138. (22) Powell, R. E.; Latimer, W. M. J . Chem. Phys. 1951, 19, 1139. (23) Latimer, W. M. J . Chem. Phys. 1955, 23, 90. (24) Fuger, J.; Mom, L. R.; Brown, D. J. Chem. Soc.,Dalton Trans. 1980, 1076.
