Thermodynamics of complexed aqueous uranyl species. 1. Volume

J . Phys. Chem. 1990, 94, 7852-7865

7852

Thermodynamtcs of Complexed Aqueous Uranyl Species. 1. Volume and Heat Capacity Changes Associated with the Formation of Uranyl Sulfate from 10 to 55 OC and Calculation of the Ion-Pair Equilibrium Constant to 175 'Ct Chinh Nguyen-Trungt Centre de Recherches sur la Ciologie de I'Vranium, BP 23, 54501 Vandoeuure- IPS- Nancy Cedex, France

and Jamey K. Hovey*,§ Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2C2 (Received: January 16, 1990: In Final Form: July 9, 19901

Apparent molar heat capacities and volumes of aqueous solutions containing U02S04in dilute H2S04 have been obtained from IO to 5 5 "C. These results have been analyzed considering the ion-pair-formation reaction of uranyl with sulfate ion, and considering the contributions to the measured heat capacities from chemical relaxation. Young's rule was used to extract apparent molar volumes and heat capacities for the solute species U02S04from 10 to 5 5 OC. The temperature dependence of the standard-state heat capacity and volume functions for U02S04(aq)are represented by the following equations: P/(cm3 mol-') = -93.87 + 0.7784T - 0.001 0 9 3 p and Cpo/(J K-'mol-') = -2773.1 + 13.636T- 0.015 322P, which are accurate from 10 to 5 5 "C. The volumetric and heat capacity changes associated with the complexation of uranyl ion with sulfate are compared with other results for divalent and trivalent monatomic cations. The results for uranyl sulfate and other metal sulfates have also been examined in reference to the Eigen-Tamm three-step ion-pair-formation model. The highly negative standard heat capacities of this neutral solute along with the structural data for uranyl sulfate provide information regarding the enhanced solute-water dipole interactions not present in systems containing other neutral solutes.

Introduction Thermodynamic data for aqueous uranium(V1) species are required for a wide variety of applications involving classical thermodynamic calculations and for theoretical analysis of solute-solvent interactions of neutral solutes. For example, thermodynamic data for aqueous uranium(V1) species are fundamental for the evaluation and prediction of various natural and man-made processes. Calculations that employ such data are required to study or explain solubilities of various uranium minerals and the transport and precipitation of dissolved U(V1) species. Uranium(V1) can be present in aqueous solution in various forms depending on the number and nature of anions present, the concentration of hydrogen ion, the concentrations of dissolved uranium, and the relative oxidizing power of the solution. Most uranium in groundwater, or in solutions containing even small amounts of common anions such as hydroxide, carbonate, or sulfate, is in the form of uranyl cation [UOz2+(aq)]complexed with one or more of the anions present [UOz2+-Xz-(aq)]. Langmuir' has summarized numerous experimental studies that have indicated that dissolved uranyl cations form mononuclear and a variety of polynuclear complexes with many anionic ligands. Of particular importance in this context are the ion pairs formed between uranyl ion and carbonate or sulfate ions since these species are common in groundwater and relatively high sulfate ion concentrations are found in the alteration waters of sulfate-rich ore deposits. Of specific interest is the speciation of coexisting sulfate and dissolved uranyl ions in oxidized lixiviation solutions obtained from uranium ore deposits with sulfuric acid. Complexes formed between sulfate and uranyl ion are especially important in acidic solutions and may predominate the behavior of certain aqueous solutions containing dissolved uranium from near room temperature to the point of phase separation of uranyl sulfate-water solutions. Much of the interest in electrically neutral uranyl-ligand ion pairs (also other aqueous actinide ion pairs) stems from the increased mobility of these species compared to the mobilities of 'This paper is dedicated to Professor Kenneth S. Pitzer on the occasion of his 75th birthday. 'Visiting scientist at the Alberta Research Council, Oil Sands and Hydrocarbon Recovery Department, Edmonton, Alberta, Canada. f Present address: Department of Chemistry, Pitzer Group, University of California, Berkeley, CA 94720

0022-3654/90/2094-7852$02.50/0

charged uncomplexed species. Reliable thermodynamic quantities for the formation of these species are thus required for use in models that attempt to calculate the speciation and transportation of uranium(V1) aqueous species. Many applications require thermodynamic data for uranium(VI) species (mostly equilibrium constants) at or near room temperature, but even more applications require similar data for elevated temperatures or pressures. Complexation reactions of two charged species to form a neutral solute (as is the case here) are highly nonisocoulombic and thus have large standard-state heat capacity and volume changes. Equilibrium constants for reactions with such large heat capacity changes are thus strongly temperature dependent, and not accurately calculated at temperatures more than a few degrees away from the temperature of measurement using only the standard-state enthalpy change. Several recent studies have focused on measurements of the enthalpy of complexation of uranyl cation with various complexing ligands including ~ u l f a t e . ~ ~ ~ Because of the desire to predict equilibrium constants for reactions involving uranyl sulfate at higher temperatures, apparent molar heat capacities and volumes of aqueous solutions containing U02S04and H2S04were obtained from 10 to 5 5 OC. Small amounts of acid, which are required in solutions containing uranyl ion to prevent hydrolysis$ were used; this added acid complicates the subsequent data treatment. The two equilibria required to describe the present solutions are represented by the following equations: U022+(aq)+ S042-(aq) = U02S04(aq)

(1)

HS04-(aq) = H+(aq) + S042-(aq)

(2)

Calculations using equilibrium data selected by Smith and Martells indicate that almost all of the aqueous uranyl and sulfate ions are present as the neutral ion pair U02S04(aq)at the conditions of ( 1 ) Langmuir, D. Geochim. Cosmochim. Acfa 1978, 42, 547-569. (2) Ullman, W. J.; Schreiner, F. Radiochim. Acfa 1986, 40, 179-183. (3) Grenthe. 1.; Spahiu, K.; Olofsson, G . Inorg. Chim. Acfa 1984, 95, 79-84. (4) Baes, Jr. C. F.; Mesmer, R. E. The Hydrolysis of Cafiom; Wiley: New

York, 1976. ( 5 ) Smith, R. M.; Martell, A. E. CrificalSfabilifyComtanrs. Volume 4: Inorganic Complexes; Plenum Press: New York, 1976.

0 1990 American Chemical Society

Thermodynamics of Complexed Uranyl Species present interest, The presence of sulfuric acid in all of these solutions ensures that a larger fraction of the uranyl ions are present as the ion-paired species. There are, however, several complications associated with the following treatment because of the presence of small amounts of completely dissociated uranyl and sulfate ions. Also, the use of acid to prevent hydrolysis reactions complicates the treatment further because of the bisulfate-sulfate equilibrium. The temperature shift required for determination of the heat capacity alters the equilibrium concentrations of the various solute species present in solution. The extent of such changes depends on the magnitudes of associated enthalpy changes for the various equilibria present. Because these solutions contain uranyl sulfate in sulfuric acid, and because there are two coupled equilibria present in these solutions, all of the contributions due to chemical relaxation from all equilibria must be subtracted from the total measured properties in order to evaluate the desired properties of the neutral uranyl sulfate ion pair. Experimental Section Water used for all experiments was first passed through an activated charcoal filter and then through a Milli-R04/Milli-Q reagent grade mixed bed ion exchange and activated carbon system to yield purified water with a resistivity of 16 MQ cm or greater. Solutions of NaCl (Fisher certified, ACS) were prepared by mass after drying the salt at 1 I O OC until constant weight. Uranyl sulfate was obtained from Alfa (+96%) and was recrystallized several times from dilute sulfuric acid. Ion chromatographic analyses on the purified material showed no detectable impurities. Concentrated sulfuric acid was obtained from Fisher (certified, ACS). The stock solution of uranyl sulfate was prepared by adding the purified salt to a 0.05 rn sulfuric acid solution. Total uranium in the stock solution was determined gravimetrically by using 8-hydroxyquinoline as described by VogeL6 The total anion concentration in the stock solution (m2 m3,where m2 and m3 are the stoichiometric molalities of uranyl sulfate and sulfuric acid, respectively) was determined by passing weighed aliquots of the solution through an ion-exchange resin [Rexyn 101(H)] and titrating the eluent with a standard solution of sodium hydroxide. Since the solutions investigated here contain mostly the uranyl sulfate ion pair, widely varying amounts of stock solution were passed through the ion-exchange columns. The total anionic concentration in the solution was found to be independent of the amount of stock solution introduced into the ion-exchange column, indicating that complete exchange occurred. The amount of excess sulfuric acid in the stock solution was obtained by difference. The stock solution was diluted by mass with a standard solution of sulfuric acid with molality equal to approximately 0.05. The densities of all solutions were measured relative to water at each of the experimental temperatures f O . O 1 OC with a Sodev 03D vibrating tube densimeter,’ which was calibrated with pure water and a standard solution of NaCI. Densities of the standard NaCl solutions were obtained from the data listed by Millero.8 Heat capacities were measured relative to water with a Sodev CP-C Picker type flow micr~calorimeter.~Temperatures were maintained (constant to f O . O O 1 “C) with separate Sodev CT-L circulating baths and monitored with calibrated thermistors. Daily evaluation of the calorimetric heat loss correction factorlo by way of measurements on standard solutions of NaCl led t o y = 1.006 f 0.001 at 25 ‘C. Similar measurements at other temperatures indicate this term to be independent of temperature in the range from I O to 55 OC.

+

(6) Vogel, A. Textbook of Quantitative Inorganic Analysis, 4th ed. (revised by Bassett, J.: Denney, R. C.; Jeffrey, G. H.; Mendham, J,), Longman: London, 1978. (7) Picker, P.; Tremblay, E.: Jolicoeur, C. J. Solution Chem. 1974, 3, 377-384. ( 8 ) Millero, F. J . J . Phys. Chem. 1970, 74, 356-362. (9) Picker, P.; Leduc. P. A.: Philip, P. R.; Desnoyers, J. E. J . Chem. Thermodyn. 1971, 3, 631-642. (IO) Desnoyers, J. E.; de Visser, C.; Perron, G.; Picker, P. J . Solution Chem. 1976, 5. 605-616.

The Journal of Physical Chemistry, Vol. 94, No. 20, I990 1853 Results Experimental Apparent Molar Properties. As mentioned in the Introduction, a t low pH and high concentrations of uranyl and sulfate ions, most U(V1) is present as ion-paired or complexed species [U02S04(aq)]. At higher concentrations of sulfate ion, the disulfate complex [U02(S0,)22-] may also be present.’ A variety of monomeric and polymeric uranium-hydroxy species, including ( U 0 2 ) 2 ( O H ) yand (U02),(OH),+ at pH values above 2,’343113’2 may also be present. To reduce many of these complications, the measurements described here were performed on solutions containing sufficient sulfuric acid to permit neglect of hydrolyzed species, but also not high enough to lead to formation of higher sulfate complexes. “Experimental” apparent molar properties ( 4 F P t ’ ) of solutions containing two electrolytes are defined generally by (3)

where YSolnis the extensive property (volume or heat capacity) of a solution that contains 1 kg of solvent (water), n10and Y I o are the number of moles and molar volume or heat capacity of pure solvent (water), and rn2,i and m3,iare the stoichiometric (initial) molalities of uranvl sulfate and sulfuric acid. SDecific equations for calculations of these defined quantities are IOoo + m2,~M2 + m 3 , ~ M 3 9pxptl

EI

d m2.1

@Cpexptl

=

cp(I OOO

- -1000 dlO

+ m3,~

+ m2,,M2+ m3,,M3)- I O O O C ~ , ~ ~ m2.1

+ m3,1

Here Mi represents the solute molar mass of species i, cp and d are the specific heat capacity and density of the solution, and cp,lo and d10are the specific heat capacity and density of pure water. Values for the density and specific heat capacity of pure water at each of the experimental temperatures were obtained from Ke1113x14and had the values dlo = 0.999 700, 0.997 047, 0.992 219, 0.985695 g and cP,’O= 4.1919, 4.1793, 4.1783, 4.1821 J K-’ mol-’ at 10, 25, 40, and 55 OC, respectively. The experimentally determined relative densities ( d - dlo)and heat capacity ratios [cpd/(cp~lodlo) - I] along with calculated experimental apparent molar properties for all of the aqueous solutions containing U 0 2 S 0 4and H2S04 are listed in Table I. Calculations with appropriate equilibrium constants for the equilibria represented by eqs 1 and 2 for each of the experimental temperatures must precede further calculations with the experimental apparent molar properties for evaluations of the nature and number of species present in solution. For analyses of results from the heat capacity measurements, contributions due to “chemical relaxation” must also be considered. Details of literature selections, speciation calculations, and chemical relaxation calculations are described in the following sections. Selections of Literature Data. Several authors have reported equilibrium constants for 25 OC for the reaction represented by eq I . Most of the available data have been summarized and in some cases critically evaluated in several reviews and compilat i o n ~ . ~ , ” -Values ’~ reported for the equilibrium constant at zero (11) Baes, Jr. C. F.; Meyer, N. J. Inorg. Chem. 1962, I , 780-789. (12) Sylva, R. N.; Davidson, M. R. J . Chem. SOC.,Dalton Trans. 1979, 3, 465-47 1, (13) Kell, G. S. J . Chem. Eng. Data 1967, 12, 66-69. (14) Kell, G.S. Thermodynamic and transport properties of fluid water. In Watet-A Comprehensiue Treatise; Franks, F., Ed.;Plenum: New York, 1972; Vol. I, pp 363-412. ( I 5 ) Sillen, L. G.; Martell, A. E. Stability Constants of Metal-Ion Complexes; Chemical Society: Burlington House, London, 1964. (16) Sillin, L. G.;Martell, A. E. Stability Constants of Metal-ion Complexes. Supplement No I ; Chemical Society: Burlington House, London, 1971.

7854

Nguyen-Trung and Hovey

The Journal of Physical Chemistry, Vol. 94, No. 20, 1990

TABLE I: Compositions of Solutions and Experimental Apparent Molar Volumes and Heat Capacities for UOfiO, and H2S04 Mixtures 102(d - d'O), mvexptl, IO2((cpd/ QCCXPtl, m2, m3, mol kg-' mol kg-I g cm-' cm3 mol-' cp,1040) - 1) J K-'Pmol-' IO 0.8725 0.5008 0.3766 0.2500 0.1800 0.1313 0.09635 0.07272 0.06591

0.6606 0.3998 0.3 127 0.2238 0. I747 0.1406 0.1161 0.09947 0.09469

30.0568 18.0559 13.8380 9.4169 6.9385 5. I824 3.91 1 1 3.0526 2.8018

0.8725 0.5008 0.3766 0.2500 0.1800 0.1313 0.09635 0.07272 0.06591

0.6606 0.3998 0.3127 0.2238 0.1747 0.1406 0.1161 0.09947 0.09469

29.6598 17.7875 13.6234 9.3322 6.8234 5.0937 3.8405 2.9881 2.7401

0.8725 0.5008 0.3766 0.2500 0.1800 0.1313 0.09635 0.07272 0.06591

0.6606 0.3998 0.3127 0.2238 0.1747 0.1406 0.1 161 0.09947 0.09469

29.2651 17.5362 13.4234 9.1113 6.71 14 5.0107 3.7742 2 9285 2.6897

0.8725 0.5008 0.3766 0.2500 0.1800 0.1313 0.09635 0.07272 0.06591

0.6606 0.3998 0.3 127 0.2238 0.1747 0.1406 0.1161 0.09947 0.09469

29.1624 17.4608 13.3651 9.0690 6.66 16 4.9708 3.743 I 2.9019 2.6549

25

O C

4 1.906 39.454 38.381 37.172 35.912 35.066 34.132 32.938 32.652

-4.768 -3.255 -2.640 -1.917 -1.486 -1.153 -0.9046 -0.7292 -0.6 77 2

35.94 7.37 -5.07 -17.97 -28.52 -33.73 -38.02 -41.72 -42.07

43.725 41.724 40.828 38.451 38.607 37.822 36.990 36.244 36.065

-3.895 -2.531 -1.996 -1.412 -1.055 -0.7970 -0.6059 -0.4744 -0.4374

68.16 51.19 44.88 32.54 34.03 33.01 33.17 34.38 35.02

45.308 43.568 42.8 16 42.150 40.941 40.101 39.388 39.021 38.542

-3.470 -2.159 -1.676 -1.148 -0.8466 -0.6293 -0.4709 -0.3591 -0.3291

86.07 75.86 72.23 70.83 67.88 67.84 69.25 73.45 73.06

45.1 I I 43.454 42.693 42.073 41.370 40.620 39.944 39.7 13 39.872

-3.177 -1.958 -1.484 -1.026 -0.747 -0.552 -0.4 1 1 -0.3 1 1 -0.282

92.75 84.04 82.68 80.44 80.53 81.04 82.47 87.15 89.97

O C

40 O C

55

ionic strength and 25 OC range from 580 to 8500 for the reaction represented by eq 1 .19-z5 It appears that the "best" values for the equilibrium constant are around 1400 under these conditions. We chose to refit what we considered to be the best values: extended Debye-Huckel equations of the form

were fit to the data reported by Wallace22 and Ahrland and Kullberg.26 Values for A, and B, were taken from Robinson and (17) Hogfeldt, E. Stability Constants of Metal-ion Complexes. Part A: Inorganic Ligands: IUPAC Chem. Data Series No. 21; Pergamon: Oxford, U.K., 1982. ( I 8) Fuger, J.; Khodakovsky, I. L.; Medvedev, V. A,; Navratil, J . D.The Chemical Thermodynamics of Actinide Elements and Compounds Part 12. The Actinide Aqueous Inorganic Complexes. International Atomic Energy Agency: Vienna, in press. (19) Davies, E. W.; Monk, C. B. Trans. Faraday Soc. 1957,53,442-449. (20) Allen, K. A. J . Am. Chem. Soc. 1958, 80, 4133-4137. (21) Brown, R. D.; Bunger, W. B.; Marshall, W. L.; Secoy, C. H. J. Am. Chem. Sor. 1954, 76, 1532-1535. (22) Wallace, R. M. J . Phys. Chem. 1967, 71, 1271-1276. (23) Majchrzak, K. Nukleonika (NUKLAS) 1973, 18, 105-1 19. (24) Nikolaeva, N. M. Izu. Sb. Otd. Akad. Nauk SSSR, Ser. Khim. Nauk. (IZSKAB) 1970, 6, 62-66. (25) Nikolaeva, N. M. Int. Corros. Conf. Ser. 1976, NACE-4 (High Temp. High. Pressure Electrochem. Aqueous Solutions, Conf.), 146-1 52. (26) Ahrland. S . ; Kullberg, L. Acra Chem. Srand. 1971, 25, 3677-3691. See also: Ahrland, S . Acta Chem. Scand. 1951, 5 , 1151-1 167.

O C

TABLE 11: Equilibrium Constant Expressions for Uranyl Sulfate Complexation (Eq 1) from 10 to 55 O C ' 1, "C A,, L1/2 B,, A-' mol-2 L1/2 log K

10 25 40 55

0.4989 0.51 15 0.5262 0.5432

0.3264 0.3291 0.3323 0.3358

3.009 3.185 3.365 3.548

"ate that all of these coefficients pertain to solutions of known molarity. The concentration-dependent parameter B in eq 6 was found to be 0.143 a t 25 O C and was taken to be temperature independent.

Stokes2' and are listed in Table 11. The value A = 7.0 8, that was a result of fits by Wallace22was used in the present calculations. Values for log K and B obtained from these fits are listed at 25 O C (along with derived values for other temperatures from calculations described later) in Table 11. It should be noted that the results of both Ahrland and Kullbergz6and Wallacezz were obtained for solutions with compositions expressed as molarity, so all parameters in eq 6 pertain to this temperature-dependent concentration scale. Several authors have reported enthalpies of reaction associated with the formation of neutral uranyl sulfate. Several of these values have been derived by way of (a In K l a T ) , and other preferred values by direct calorimetric measurements. The best results are those of Bailey and Larson28who have obtained AHo (27) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; butterworths: London, 1959. (28) Bailey, A. R.; Larson, J . W. J . Phys. Chem. 1971, 75, 2368-2372.

The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7855

Thermodynamics of Complexed Uranyl Species

= 20& kJ mol-I and Ullman et aL2 who have obtained AHo = 19.6 kJ mol-'. Ahrland and Kullberg26have derived AH = 18.23 kJ mol-I at an ionic strength of 1 M. Although the measurements of Bailey and Larson% are probably the best, they were given equal weight and averaged with the results from Ullman et aL2who used equilibrium data more in accord with the present selections. This average standard-state enthalpy of reaction was combined with results from Ahrland and Kullberg26to obtain AH/(kJ mol-') = 20.21 - 1.9791 (7) for 25 O C and ionic strength less than approximately 1 M. Because of the presence of excess sulfuric acid in all of these solutions, accurate thermodynamic properties are required for sulfuric acid that are to be used in calculations of speciation and relaxation in solutions containing uranyl sulfate and sulfuric acid. P i t ~ e r ' selegant ~~ treatment of the properties of sulfuric acid provides the basis for these calculations. Because of the concentrations we used and the temperature range of these experiments, we can neglect many reactions such as the formation of neutral sulfuric acid and polynuclear uranyl sulfate complexes. Thus the two reactions required to describe the solutions of present interest written in terms of the fractional extents of reaction ( a and 0)are given as U022+(aq) + S042-(aq) = U02S04(aq) (1

-

(I

ah2

-

a)m2+Bmi

In K 2 = 199.0185 - 6658'95 - 31.8123 In T T Speciation of all of the solutions can now be calculated by solving the various ionic equilibria equations. These equilibrium concentrations of species can then be used to calculate apparent molar volumes. However, further analyses that consider the effect that a temperature shift has on the speciation of these solutions are required before calculations of apparent molar heat capacities can be completed. These speciation calculations are thus presented after an outline of the calculations required to correct for chemical relaxation in the following section. Relaxation Calculations. Measurements of the heat capacities of aqueous solutions that contain uranyl sulfate and sulfuric acid are complicated by contributions to the total measured properties due to chemical relaxation. Detailed accounts of chemical relaxation in relation to heat capacities of aqueous solutions have been presented by Woolley and Hepler,)' Barbero et al.,j2 Mains et and Hovey et al.34 For systems that contain a variety of species associated by various coupled equilibria such as reported here, the total heat capacity effect caused by the change in speciation due to the heat capacity measurement itself is given byj4

(1)

"2

HS04-(aq) = H+(aq) + S042-(aq) (2) ( 1 - B)m3 ( I + B)m3 ( I - 4 m 2 + Bm3 where (Y represents the amount of complexed uranyl sulfate and /3 the amount of dissociated bisulfate ion. The molalities of ionic species appropriate for the Pitzer equations that follow can be expressed in terms of these fractional extents of reaction a and p. The pertinent equations from Pitzer et al.29are listed below. The activity coefficient ratio for the bisulfatesulfate equilibrium (~H+YSO,Z-/YHSO,-), which we denote by y2, has the form29

where AHl and AH2 correspond to the first and second equilibria [uranyl sulfate complexation (eq 1 ) and the dissociation of bisulfate ion (eq 2)], and all AH values correspond to the enthalpy change at the total ionic strength of the solution. The equations that were derived from differentiation and rearrangement of the equations cited above yield the desired quantities (aa/dT)p", and (ap/aT),,, as given by eqs 24 and 25: a

In y2 = 4 p - 4 p m 3 B H l+ 2 [ ( 1 - a ) m 2 + ( 1 + 2 P ) m 3 ] P f i+ 2(1 + /3)mj[2(1- a ) m 2 + ( I + 3P)m3lCH2 + 4mj2(1 P2)B"I, ( 8 ) in which certain quantities are defined by

BHI =

+ @#{[-I

BHI, =

p g ] [I - ( I

+ 21'1~) e~p(-21'/~)] 21

QI

= ( 1 - a)[(1-a ) m 2

K2 =

(18)

+ om3]

( 1 + @ ) [ ( I - a)m2 + "lY2 1

(19)

-P

(10)

+ ( I + 2PI2 + 21) e ~ p ( - 2 / ~ / ~ ) ] ( 1 1)

212 Here pgl, pgj, and CH2 = a 2 / 2 j I 2parameters are independent of ionic strength but are functions of temperature as given by the following: 46.040 pfl] = 0.05584 + T 336.514 = -0.65758 + T 98.607 Pflj = -0.32806 T 63.124 Cfi2 = 0.25333 - T The equilibrium constant presented by Pitzer et al.,29modified slightly to include the temperature dependence of the enthalpy,30 is given below

@I,

f J =

+

(29) Pitzer, K. S.;Roy, R. N.; Silvester, L. F. J . Am. Chem. SOC.1977, 99, 4930-4936. (30) Hovey. J. K.; Hepler, L. G. J . Chem. Soc., Furaduy Trans. 1990.86, 2831-2839.

2m2@j - 4m2m,CH2(1 + p)

-

(

16m2mj2(1 - p2)

a:yl')Tp]

( 3 1 ) Woolley, E. M.; Hepler, L. G. Can. J . Chem. 1977, 55, 158-163. (32) Barbero, J. A.; Hepler, L. G.; McCurdy, K . G.; Tremaine, P. R. Can. J . Chem. 1983, 61, 2509-2519. (33) Mains. G. J.; Larson. J. W.; Hepler, L. G. J . Phys. Chem. 1984, 88,

1257-1261. (34) Hovey. J. K . ; Hepler, L. G.;Tremaine, P. R. Can.J . Chem. 1988, 66,

881-897.


1856 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990

Thus the desired heat capacity contribution that is due only to the molecular species is related to OTpand ” O c p f k by

mcp,,= m@px~tl

e = m3Q,(1- a) + 2m3(l - a)f2(

z)

(30)

P .T

(39) Speciation Calculations. The speciations of all of these solutions at 25 “C were calculated by using Newton’s root-finding algorithm along with various expressions listed above and appropriate mass and charge balance equations. The calculation scheme involved two steps; each step was iterated I O times, and the overall sequence was also iterated 10 times. Initially a and /3 values were estimated (ignoring activity coefficients) from the data for aqueous solutions containing the pure solutes. The molal ionic strength was estimated from these values and converted to molar ionic strength for use in several of the equations cited above. First the parameters for sulfuric acid (y2)were calculated, which were then used in calculations of total speciation. The results were found to converge quite quickly and the large number of iterations employed here led to differences in the calculated a or @ values of less than between the 9th and 10th iterations. The results of these speciation calculations were used along with the analysis given in the next section to obtain the standard-state partial molar heat capacity of U02S04(aq)at 25 O C . As mentioned earlier, the reaction of uranyl ion with sulfate to form the neutral uranyl sulfate complex has a large ACpo [here a value of 168 J K-l mol-’ was calculated from data presented in the next section and from refs IO, 30, and 3 5 - 3 8 ] . Refinement of both the equilibrium constant expression and the enthalpy change for uranyl sulfate complexation using the heat capacity calculated later for 25 “C leads to the equations log Kl = -52.0830

Q1was used instead of K ,and y , for calculations involving uranyl sulfate because of the availability of these data. The quantities y2 and Q I depend on both ionic strength and temperature; hence some care is required in obtaining derivatives that we show by consideration of a generic quantity g that is a function of I , T , and p : g = g(I,T,p)

(33)

Since the pressure is constant the last term in this equation is equal to zcro. Dividing through by d T and imposing constant pressure leads to

The ionic strength in these solutions that contain uranyl sulfate and sulfuric acid is given by I = 4(1 - a ) m 2 + ( 1 + 2/3)m3 (36) Thus replacing (al/aT), in eq 3 5 with the appropriate expression derived from eq 36 leads to

Equations of type represented by eq 37 were used to evaluate appropriate derivatives of Q , and for various other derivatives required for evaluation of relaxation contributions of sulfuric acid.M The total contribution from chemical relaxation to the “exptl” apparent molar heat capacities defined by eq 3 has the form

-

“e1 P.l:

+ 1562‘26 + 20.218 log T T ~

(40)

AHl’ = 20.21 + 0 . 1 6 8 1 ( T - 298.15) kJ mol-’ (41) These expressions were used for subsequent evaluation of the amounts of species present and for calculations of chemical relaxation. The results of these speciation calculations are given along with the contributions from chemical relaxation in Table 111.

Apparent Molar Properties of U02S04. The tabulated “experimental” apparent molar properties listed in Table I contain contributions from all of the species present and also contributions from chemical relaxation (for heat capacities). The heat capacity measurements have been “corrected” to “species properties” for the solutions by subtracting these contributions as given by eq 39. The resultant apparent molar properties of the solutions that correspond to only the species present (“p,, for heat capacities and @ P x P t l = @V,for volumes) can now be separated into terms representing each of the species present in the solution with Young’s n mi @Y,= C-+Yy, + 6 (42) r=2 C mi i=2

Here all myy values , apply at the total ionic strength of the solution, mi are equilibrium molalities for all of the aqueous species present ( m 2 : U022++ (m3: H+ HS04-), (m4: U02S04),and ( m 5 : 2 H + + S042-), and 6 is an excess mixing term. The magnitude of 6 is expected to be small here due to the large amount of complexed uranyl sulfate that should not have interactions as strong as many ionic species would with other oppositely charged species in solution. We have set 6 equal to zero for all subsequent calculations because of the expectation that it is small and also because we do not have sufficient information to do otherwise.

+

(35) Rogers, P. S. Z.; Pitzer, K. S. J. Phys. Chem. 1981,85, 2886-2895. (36) Pitzer, K. S.; Peiper, J. C.; Busey, R. H. J. Phys. Chem. Ref. Data 1984, 13, 1-102. (37) Allred, G . C.; Woolley, E. M . J . Chem. Thermodyn. 1981, 13,

147- 1 54. (38) Tremaine, P. R.; Sway, K.; Barbero, J. A. J . Solution Chem. 1986, 1 5 , 1-22. ( 3 9 ) Young, T. F.: Smith, M. B. J. Phys. Chem. 1954. 58, 716-724.



TABLE 111: Compositions, Speciation, Ionic Strengths, Chemical Relaxation Contributions, and Apparent Molar Heat Capacities for Solutions Containing U02S04and H#04

IO "C 0.872 49 0.500 8 I 0.376 59 0.249 97 0.17998 0.131 33 0.096 35 0.072 72 0.065 91

0.660 64 0.399 84 0.3 12 68 0.223 84 0.17473 0.14059 0.11605 0.099 47 0.094 69

0.903 85 0.890 73 0.883 49 0.87290 0.86445 0.856 51 0.848 99 0.842 48 0.840 3 1

0.249 23 0.244 34 0.248 07 0.259 94 0.273 98 0.290 21 0.307 77 0.32423 0.329 96

1.3255 0.8141 0.6433 0.4673 0.3681 0.2976 0.2457 0.2098 0.1993

73.40 73.16 74.08 76.49 79.28 82.60 86.43 90.32 91.75

-37.46 -65.79 -79.15 -94.46 -107.80 -1 16.33 -1 24.45 -1 32.04 -133.82

0.872 49 0.500 8 1 0.376 59 0.249 97 0.179 98 0.131 33 0.096 35 0.072 72 0.065 91

0.660 64 0.399 84 0.31 268 0.223 84 0.174 73 0.14059 0.11605 0.099 47 0.094 69

0.912 66 0.900 39 0.893 98 0.884 88 0.877 87 0.871 37 0.865 28 0.86005 0.858 30

25 OC 0.149 IO 0.14637 0.150 26 0.161 70 0.175 16 0.19078 0.207 8 1 0.223 92 0.229 55

1.1625 0.7 164 0.5664 0.4113 0.3239 0.26 18 0.2162 0.1847 0.1755

73.12 73.76 75.37 79.08 83.12 87.82 93.15 98.52 100.49

-4.96 -22.57 -30.49 -46.54 -49.09 -54.81 -59.98 -64.14 -65.47

0.872 49 0.500 8 1 0.376 59 0.249 97 0.179 98 0.131 33 0.096 35 0.072 72 0.065 91

0.660 64 0.399 84 0.3 I2 68 0.223 84 0.174 73 0. I40 59 0.11605 0.099 47 0.094 69

0.91639 0.904 73 0.898 91 0.890 95 0.884 93 0.879 50 0.874 47 0.870 19 0.868 76

40 OC 0.068 08 0.065 42 0.068 54 0.078 31 0.090 16 0.10423 0.11985 0.13485 0.140 I4

1.0424 0.6430 0.5078 0.3679 0.2891 0.2332 0.1922 0.1641 0.1 558

61.28 63.43 65.78 70.54 75.46 8 1.04 87.28 93.50 95.78

24.79 12.43 6.45 0.29 -7.58 -13.20 -18.03 -20.05 -22.72

0.872 49 0.500 8 1 0.376 59 0.249 97 0.179 98 0.131 33 0.096 35 0.072 72 0.065 91

0.660 64 0.399 84 0.3 12 68 0.223 84 0.174 73 0. I40 59 0.11605 0.099 47 0.094 69

0.91628 0.904 86 0.899 35 0.892 03 0.886 67 0.881 91 0.877 58 0.873 92 0.872 70

55 O C 0.007 30 0.003 37 0.005 28 0.01 2 89 0.022 84 0.035 09 0.049 05 0.062 72 0.067 59

0.9625 0.5931 0.4676 0.3376 0.2643 0.21 25 0.1746 0.1486 0.141 1

45.51 48.53 51.10 56.00 60.91 66.41 72.52 78.59 80.82

47.24 35.51 31.58 24.44 19.62 14.63 9.95 8.56 9.15

The relationship of the equilibrium molalities to the stoichiometric molalities of uranyl sulfate and sulfuric acid are given by m2 = ( 1 - a)m2,,,m3 = ( 1 - P)m3,i,m4 = am2,+and mS = ( 1 - a)mZ,i

+ Bm3j.

The apparent molar properties of neutral solutes in water in the presence of various ionic species are satisfactorily represented by expressions of the type 'Yq =

Ydo+ B y I

+ CyP

(43) where T 4 O is the standard-state partial molar property (equal to the apparent molar property at infinite dilution) and BY and CY are concentration dependent parameters. Expressions for the apparent molar volumes and heat capacities for all of the other solutes present in these solutions are also required. Apparent molar volumes and heat capacities of the species (2H+ + S042-) were estimated from two different schemes, the first at 25 OC, and the second for the three other temperatures. 25 OC: @Y(2H+ Sod2-) = @Y(2K++ S042-) - 24Y(K+ + CI-) + 24Y(H+ + C1-) (44)

+

+

10, 40, 55 "C: @Y(2H+

=

#Y(2Na+ + S042-) - 2@Y(Na++ CI-)

+ 24Y(H+ + CI-)

(45) At 25 O C , volumetric and heat capacity expressions of the general type corresponding to fitting non-extended Debye-Huckel equations to experimental data for (2K+ + S042-) were taken from Hovey et a1.,40heat capacity expressions were taken directly from

+

+

Desnoyers et for (K+ Cl-) and (H+ Cl-); these same types i f expressions were fit to the apparent molar volume data from Fortier et al.,41and finally volumetric data for (H+ + CI-) were 1:aken from Allred and W~olley.~'Volumetric properties for (Na+ f C1-) and (2Na" + S042-) were calculated from the parameters I isted by Connaughton et a1.,42 for (H+ CI-)the values from 4llred and Woolley3' were used at 10 and 40 O C , and the exmssion from Herrington et a!l3 was used at 55 O C . Heat capacity :xpressions applicable at 10,40, and 55 O C for (Na+ + CI-)were lerived from Pitzer et al.P6 for (H++ CI-)from Tremaine et md for (2Na+ S042-) from Rogers and P i t ~ e r . ~ ~ Apparent molar volume and heat capacity expressions for :ompletely dissociated uranyl sulfate (U022++ S042-) were de:ived using the following scheme: V(UO?+ + so:-) = 4Y(U0,2+ + 2C10,) 24Y(H+ + C104-) +Y(2H+ + S042-) (46)

+

+

+

Expressions for uranyl perchlorate and perchloric acid for each if the required temperatures were taken from Hovey et aLMand (40) Hovey, J. K.; Hepler, L. G.;Tremaine, P. R. Thermochim. Acta 1988, 145-253. (41) Fortier, J. L.; Philip, P. R.; Desnoyers, J. E. J . Solution Chem. 1974, 3, 523-538. (42) Connaughton, L. M.; Hershey, J. P.; Millero, F. J. J. Solution Chem. 1986, 15, 989-1002. (43) Herrington, T. M.; Pethybridge, A. D.; Roffey, M. G . J . Chem. Eng. Data 1985, 30, 264-267. (44) Hovey, J. K.; Nguyen-Trung, C.; Tremaine, P. R. Geochim. CosI nochim. Acta 1989, 53, 1503-1509.


7858


TABLE I V Standard-State Partial Molar Volumes, Heat Capacities, and Ionic Strength Parameters for U02S04(aq)' As Determined by Fits Using Eq 43

r,

f,

cm3 kg

IO 25 30 55

39.47 f 0.39 40.92 f 0.13 43.63 f 0.29 43.02 f 0.50

O C

J K-I mol-'

c;

IO

-140.1 -70.4 -1.7 50.9

-_ 75 30 55

CY,

BY,

PO,

cm3 mol-'

O C

cm3 kg2 mol-3

13.57 f 1.4 12.36 f 0.5 11.15 f 1.3 9.93 f 2.4 5c

3

J kg K-' mol-2

f 1.2

f 1.0 f 2.1 f 1.6

-4.20 -3.68 -3.17 -2.65

f 0.9 f 0.4 f 1.1 f 2.1

CC3

J kg2 K-' mol-'

182.2 f 4 150.9 f 4 106.4 f 9 88.39 f 7

-42.67 -38.30 -27.88 -29.57

f3 f3 f7 f6

Reported uncertainties are those derived from the fits with unconstrained values of the 5, and C y parameters: Bdl0-55 "C) = 36.44 0.08078T. CdlO-55 "C) = -13.98 + 0.03455T. &(IO-55 "C) = 772.61 - 2.085T. C,(10-55 OC) = -125.07 0.2910T. T/K = r / T 273.1:

+

+

r

-100

11

I -1LO

0

05

10

15

!/mol kg

Figure 2. Apparent molar heat capacities of U02S04(aq)as a function of ionic strength. The plotted curves are the result of fits using constrained values of the Bc and Cc parameters.

d

1

1 3

10

05

15

! / m o l kg

Figure I . Apparent molar volumes of U02S04(aq)as a function of ionic strength. The plotted curves are the result of fits using constrained values of the B y and CY parameters.

Hovey and H e ~ l e r respectively. ,~~ Appropriate expressions for 4Y(2H+ SO:-) were derived from the scheme described above. Expressions for mY(H+ + HS04-) were taken from Hovey and He~ler.~~ Fitting expressions of the type given by eqs 42 and 43 to the data given in Tables 1 and 111 resulted in definition of the standard-state properties y4' and concentration-dependent parameters B y and Cy. It was found that the temperature dependences of B, and C, parameters were accurately represented and those of BY and CY less accurately by linear equations in temperature given by

+

BY (and CY) = PI + P ~ T

(47) without significant loss in the overall standard deviations of the data fits. In this equation, p, are fitting coefficients and Tis the temperature in kelvin. Each data point was weighted as the inverse square of its estimated standard deviation as given in Table 1V. 'The results of fitting the data by the same procedure above using (45) Hovey. J . K.; Hepler. L. G . Can. J . Chem. 1989. 67, 1489-1495.

smoothed B and C parameters from eq 47 are reported in Table IV. The f values assigned to each of the parameters correspond to standard deviations from fits with unconstrained B and C parameters. Plots of @Y4against ionic strength are shown in Figures 1 and 2 . Discussion

Standard-State Properties to 55 "C. The temperature dependences of the presently derived standard-state volumes and heat capacities are well represented by expressions of the type

P = a + b T + cp

(48)

Equations of type 48 were fitted to the data given in Table 111 to obtain the following expressions accurate from 10 to 5 5 "C: P ( U 0 2 S 0 4 ( a q ) ) = -93.868

+ 0.778391" - 0 . 0 0 1 0 9 2 5 ~ (49)

~po(UOzSO,(aq))= -2773.10

+ 13.6367 - 0.015322p

(50)

These standard-state volumes and heat capacities are plotted along with the fitted equations in Figures 3 and 4.



t

-20

I

1

/

I

10

25

55

40

t /'C Figure 4. Standard-state heat capacities for (UO,SO,(aq)) from 10 to 55 O C .

Equilibrium calculations involving uranyl sulfate from 10 to

55 OC should employ these results rather than the less accurate equations that permit extrapolation to much higher temperatures that are described later. Literature Comparisons. Several studies have reported densities of aqueous solutions containing uranyl sulfate, although only the measurements by Manzurola and A ~ e l b l a are t ~ ~of sufficient precision for the calculation of apparent molar properties. Their study, however, considered neither the complexation reaction of uranyl with sulfate nor the effects of hydrolysis on aqueous uranyl ion. These results may be consistent with the present results once these effects are taken into account. Because the present work was designed to minimize all of these complications, the presently derived results are preferred. There do not appear to be any results that can lead directly to the standard-state heat capacity of uranyl ~ulfate.~'However, Yang and P i t ~ e have r ~ ~ presented a treatment of the thermodynamic properties of completely dissociated uranyl sulfate within the.framework of the Pitzer ion-interaction model. Although these calculations do not consider the ion-pair-formation reaction, the binary ion interaction terms may account for this, although the absence of the Pc2)term hampers such comparisons. The results of Yang and P i t ~ e can r ~ ~be compared with heat capacity results for uranyl sulfate when these results are transformed to a completely dissociated standard-state basis. Extrapolation of the second derivatives of equations out of their range of applicability from Yang and Pitzer4* lead to heat capacity differences (+Cp+Cpo)in fair agreement with those calculated from the present results for uranyl sulfate at 55 OC. This leads us to suggest that the calculations of Yang and P i t ~ e are r ~ ~consistent with the present calculations and are reliable at higher temperatures. High-Temperature Behavior. Heat capacities and volumes of hydrophobic neutral solutes such as argon approach + m as the critical point of water is appr~ached:~whereas the heat capacities and volumes of hydrophilic solutes such as all ionic solutes and ~

~~

~~~

~

~~

~

(46) Manzurola, E.; Apelblat, A. J . Chem. Thermodyn. 1985, 17, 575-578. (47) Smith-Magowan, D.; Goldberg, R. N. A bibliography of sources of

experimental data leading to thermal properties of binary aqueous electrolyte solutions; NBS S P 537; National Bureau of Standards: Washington, DC, 1979. (48) Yang, J.-z.; Pitzer, K. S. J . Solution Chem. 1989, 18, 189-198. (49) Biggerstaff, D. R.; White, D. E.; Wood, R. H . J . Phys. Chem. 1985, 89, 4378-4381,

several neutral solutes head toward -m as the critical point of water is approached. Recent results for the partial molar volumes of boric acid to high temperatures support this generalization.S0 A qualitative approach to the prediction of the high-temperature dependence of partial molar volumes and heat capacities for neutral solutes can possibly be based on the magnitude of the standard-state heat capacities near room temperature after appropriate considerations of other factors such as the atomic constituents of the species. Aqueous argon has a standard-state heat capacity of over 200 J K-I mol-' at 298 K whereas boric acid and uranyl sulfate have substantially lower values. The electron-rich oxygen ligands in these latter solutes provide potential H-bonding sites for discrete solute-water interactions that give rise to a disruption of the regular hydrogen-bonded water structure that is responsible for its high heat capacity. As discussed later, the standard-state heat capacities of some other neutral solutes, HIO,(aq) and MgS04(aq), are also comparably small. An elegant study of the heats of solution of several neutral solutes has been done by Olofsson et from which can be calculated standard-state heat capacities for He(aq), Ne(aq), Ar(aq), Kr(aq), and Xe(aq) of approximately 156, 166, 221, 241, and 271 J K-' mol-], respectively. Values for normal alkanes from C1 to C4 show an increase in the standard-state heat capacities from 278 to 477 J K-I mol-I. It is apparent from much of the theoretical work into the properties of fluids and multicomponent fluids [see for example the discussion by WheelerS2using the decorated lattice-gas model] that the divergent nature of thermodynamic properties of a solute nearing the critical point of the solvent can be described qualitatively in relation to the divergent compressibility (volumes) or in terms of the divergent partial molar enthalpy (heat capacities). Solutes that attract solvent molecules strongly will have standard-state volumes and heat capacities that tend to -m as the critical temperature of the solvent is approached, while those that exhibit only weak attractions or repulsions will diverge to + m as is the case for argon. The heat capacities of solutes as or more hydrophobic than argon will diverge positively and solutes as or more hydrophilic than boric acid will diverge negatively at increasing temperatures approaching the critical point of water. U02S04(aq) is almost certainly in this latter class of solutes. Further theoretical work that would describe the shift between positively and negatively divergent molecules is awaited. The Helgeson-Kirkham-Flowers equations of state for the standard-state volumes and heat capacities of aqueous solutes have the forms:53

Here u, 5, c, and 8 terms are fitting coefficients dependent on the specific solute. The AY,,, terms come directly from the relationship of the change in the free energy given by the Born equation to the changes in partial molar volumes and heat capacities. These equations can be modified by including terms representative of standard-state conversions, although these terms become important at much higher temperatures than considered here. For neutral solutes there is no contribution to standard-state properties from the Born equation because of the absence of a formal ionic charge. The HKF model has been modified by Tanger and Helgesod4 and also by ShockS5to include explicitly the divergent nature of (50) Grant-Taylor, D. F.; Read, A. J. Terra Cognita 1988, 8, 183. (51) Olofsson. G.; Oshodj, A. A,; Qvarnstrom, E.; Wadso, 1. J . Chem. Thermodyn. 1984, 16, 1041-1052. (52) Wheeler, J. C. Ber. Bunsen-Ges. 1972, 76, 308-318. (53) Helgeson, H. C.; Kirkham, D. H.; Flowers, G. C. Am. J . Sci. 1981, 281, 1249-1516. (54) Tanger, IV J. C.; Helgeson, H. C. Am. J . Sci. 1989, 288, 19-98. (55) Shock, E. L. Standard Molal Properties of Ionic Species and Inor-

ganic Acids, Dissolved Gases and Organic Molecules in Hydrothermal Systems. Ph.D. Thesis, University of California, Berkeley, 1987.

1860


TABLE V: Thermodynamic Properties" for Selected Aqueous Species at 25 O C and 1 bar species AGf0298. kJ mol'' So298,J K-' mol-' U0,2+(aq) -952.7 -97.1 UO;SO,(~~) -1715.4 51.8 Na+(aq) -261.905 59.0 -131.228 56.5 ClUaq) -8.52 182.0 C10,-( aq) S042-(aq) -744.53 20.1 0 0 H+(aq)

i.1

"Sources of these values are discussed in the text

Y

the standard-state volumes and heat capacities of neutral solutes when approaching the critical point of water. has introduced "effective" Born coefficients for aqueous neutral solutes that produce negatively divergent standard-state volumes or heat capacities (boric acid, uranyl sulfate) or positively divergent values (noble gases, hydrocarbons). It is, however, apparent from the available heat capacity res u l t ~that ~ ~the , ~partial ~ molar heat capacities of hydrophobic solutes begin to head rapidly toward + m at t = 300 "C. The completely empirical component of the H K F equations (minus the Born terms) were fit to the present results with the knowledge that calculations using the resultant expressions would be limited to temperatures well below 300 "C. The results of fitting equations of type 51 and 52 to the results in Table IV (with no Born terms) for uranyl sulfate leads to

C,,"(U02S04(aq)) 4028.332T = 5016.70 T - 62.0 J K-I mol-'

.,A

(54)

Several authors have reported the temperature dependence of the uranyl sulfate complexation reaction over reasonably wide temperature None of these results are in good agreement with any other sets of results, presumably due to experimental difficulties and/or data treatment methods. It is interesting to compare the equilibrium constants that can be calculated from the extrapolated equations cited above with those from the direct results. The complexation of uranyl ion by sulfate can be represented in terms of the following equation (all electrolytes totally dissociated except uranyl sulfate):

L.0

1 / 0

L o

Ntkolaeva (1976) 0 Nikolaeva (1971) 0 Lietzke and Stoughton (1960) - present calculations

4 50 100

OO

150

t /"C

Figure 5. Comparison of equilibrium constants for the association of

uranyl sulfate from direct experimental results, and predictions from extrapolations of the present results.

Both free energies and entropies for CI04-(aq), Na+(aq), Cl-(aq), and S042-(aq) were taken from the NBS tables.59 Similar values for UO?(aq) were taken from Fuger and Oetting.@ The selected values for the equilibrium constant and enthalpy of reaction for uranyl sulfate complexation that were cited earlier were used along with data for uranyl and sulfate ions to generate ilGf0B8= -1715.4 f 3 kJ mol-' and SO298 = 51.8 f 4 J K-' mol-' for U02S04(aq). All of the selected values for pertinent ionic and neutral species are listed in Table V . The Gibbs free energy of formation of a species can be calculated over all temperatures and pressures, referenced to the elements in their standard states at 298.15 K and 1 bar, according to = SGf"298 - S0298(T- 298.15)

+~

7 , 8 ~ d p To -

( U 0 2 * ++ 2CIO;) + (2Na+ + S042-)+ 2(Hf + CI-) = I_J02S0,+ 2(H+ + Clod-) + 2(Na+ + CI-) ( 5 5 ) Expressions that result from fitting H K F equations to data for ( U 0 2 2 + 2C104-) and (H+ + C104-) were reported by Hovey et aLMand Hovey and He~ler:~respectively. Gibbs free energies for (2Na+ + S042-)were calculated from the heat capacity results listed by Rogers and P i t ~ e r combined '~ with room temperature results from Desnoyers et a1.,I0 and volumetric data from Connaughton et al.42 The former data are applicable in the range 5-200 OC and were shown to be consistent to 280 'C by Pitzer and M ~ r d z e k .The ~ ~ latter results were fitted by HKF expressions for use above their range of measurement (0-100 "C). Equilibrium data for (Na' + Cl-) to 300 'C were taken from Pitzer et al.,36and Gibbs free energies for (H+ + CI-) were calculated from HKF fits to the heat capacities of HCl(aq) from 10 to 140 "C from Tremaine et a1.,j8 and fits to the volumetric data from Herrington et al.43and Allred and W ~ o l l e y . ~ ' Values of the Gibbs free energies and standard-state entropies for the participating species in reaction 55 were selected as follows.

where is equal to the standard-state partial molar entropy of the species at 298.15 K and 1 bar. The free energies for species not cited above were calculated from expressions of type 56. The variation in the Gibbs free energy for the reaction represented by eq 1 at temperature and pressure T and p was calculated from the Gibbs free energies for all of the solute species participating in the reaction at these conditions as outlined above. The standard-state changes in the logarithm of the equilibrium constant of uranyl sulfate complexation were calculated from 0 to 200 OC from the equations and data cited above. The results of these calculations are shown in graphical form along with the direct experimental data in Figure 5. There is qualitative agreement with all sets of data, yet these calculations are not in good agreement with any other set. From the present results and other results cited in Figure 5, it is clear that more experimental work is required to define unambiguously the equilibrium properties of uranyl sulfate solutions over moderate temperature ranges.

( 5 6 ) Biggerstaff, D. R. The Thermodynamic Properties of Aqueous Solutions of Argon, Ethylene, and Xenon up to 720 K and 34 MPa. Ph.D. Thesis, University of Delaware, 1986. (57) Lietzke, M . H.: Stoughton, R. W. J . Am. Chem. Soc. 1960, 64. 816-820. ( 5 8 ) Pitzer, K . S : Murdzek, 5 . S. J . Solufion Chem. 1982, / I , 409-413.

(59) Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I.; Bailey, S.M.; Churney, K. L.; Nuttall, R.L. J . Phys. Chem. Re/. Data 1982, 1I . Suppl. 2. (60) Fuger, J.; Oetting, F. L. The Chemical Thermodynamics of Acrinide Elements and Compounds-II. The Actinide Aqueous Ions. International Atomic Energy Agency: Vienna, 1976; pp 16-60.

+

Thermodynamics of Complexed Uranyl Species However, we have confidence that our calculated equilibrium constants are as good or better than any other direct determination at high temperatures. InnerlOuter-Sphere Complexes. There have been numerous spectroscopic and thermodynamic investigations of the formation of metal sulfate ion pairs or complexes.28*61-64 The thermodynamic properties of these types of ion-pair-formation reactions fall into two main categories that are characterized by the type of ion pair formed. The first type of ion pair is characterized by a pair of ions in direct contact without having any water molecules residing from one of the ions primary solvation shell (inner sphere). The second type of ion pair is a bound pair of ions separated by one or more solvent molecules normally consistent with solvent molecules from one of the ions primary solvation shell (outer sphere). Characteristic of inner-sphere complexes is the displacement of inner-sphere waters of hydration from the free ions upon formation of the complex, while outer-sphere complexes retain more of the hydration water molecules. Clearly volumetric changes associated with the formation of inner-sphere ion pairs would be larger in magnitude than for similar reactions forming outer-sphere pairs. Many metal ions form predominantly outer-sphere complexes with sulfate (monatomic bivalent metal ions62*64) while other more highly charged cationic species form predominantly inner-sphere complexes [ i r ~ n ( I I I ) ~Clearly ~l. the strength of the interaction between the two ions is dependent on normal factors such as the size, charge, and orientational possibilities of the species involved. Spectroscopic and ultrasonic evidence strongly supports mixtures of inner- and outer-sphere complexes such as for the indium(II1) sulfate ion pair thought to be a 50% inner- and outer-sphere mixture.62 These observations regarding relative distributions of inner- and outer-sphere species can account for many differences in the thermodynamic properties of these ions or deviations from existing theories. The present volumetric and heat capacity results for uranyl sulfate complexation are shown along with data for some other metal-sulfate complexation reactions in Table VI. The standard-state volume changes have been derived primarily from the work of Millero and co-workers. The heat capacity changes for metalsulfate complexation are known for only two species: these are the present results for uranyl sulfate and those from Woolley and Hepler3’ for magnesium sulfate. Differences in the changes in Gibbs free energies, enthalpies, and entropies seem to be consistent with the available data regarding classification as inner- or outer-sphere complexes, although by far the most sensitive properties for the elucidation of the type of ion-paired species formed are the standard-state volumes and entropies.

(61) Ahrland, S. Coord. Chem. Rev. 1972, 8, 21-29. (62) Larsson, R. Acta. Chem. Scand. 1964, 18, 1923-1936. (63) Larson, J. W. J . Phys. Chem. 1970, 74, 3392-3396. (64) Ashurst, K . G.;Hanccck, R. D. J . Chem. Soc., Dalton Trans. 1977, 1701-1 707. (65) Nair, V. S. K.; Nancollas, G.H. J . Chem. SOC.1959, 3934-3939. (66) Powell, H. K. J. J . Chem. SOC.,Dalton Trans. 1973, 1947-1951. (67) Spedding, F. H.; Jaffe, S . J . Am. Chem. SOC.1954, 76, 882-884. (68) Fay, D. P.; Purdie, N. J . Phys. Chem. 1969, 73, 3462-3467. (69) Hale, C. F.; Spedding, F. J . Phys. Chem. 1972, 76, 1887-1894. (70) Izatt, R. M.;Eatough, D.; Christensen, J. J.; Bartholomew, C. H. J . Chem. SOC.( A ) 1969, 47-53. (71) Lo Surdo, A.; Millero, F. J. J . Solution Chem. 1980, 9, 163-181. (72) Millero, F. J.; Gombar, F.; Oster, J. J . Solution Chem. 1977, 6 , 269-279. (73) Chen, C.-T.; Millero. F. J. J . Solufion Chem. 1977, 6 , 589-607. (74) Millero, F. J.; Masterton, W. L. J . Phys. Chem. 1974, 78, 1287-1294. (75) Hale, C. F.;Spedding, F. J . Phys. Chem. 1972, 76, 2925-2929. (76) Fisher, F. H.; Davis, D. F. J . Phys. Chem. 1967, 71, 819-822. (77) Inada, E.; Shimizu, K.; Osugi, J. Nippon Kagaku Zasshi 1971, 92, 1096-1 101. (78) Millero. F. J. Partial molal volumes of electrolytes in aqueous solutions. In Warer and Aqueous Solutions: Structure, Thermodynamics and Transport Processes, Horne, R. A., Ed.; Wiley: New York, 1972. (79) Spedding, F. H.; Shiers, L. E.; Brown, M. A,; Derer, J. L.; Swanson, D.L.; Habenschuss, A. J . Chem. Eng. Data 1975, 20, 81-88. (80) ,Hovey, J. K. Thermodynamics of Aqueous Solutions. Ph.D. Thesis, University of Alberta, 1988.

The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7861 TABLE VI: Changes in Thermodynamic Properties for the Formation of Selected Metal Sulfate Complexes by the Reaction MZt(aq) S04*-(aq) = MS041-*(aq) at 25 O c a

+

AGO, AHo, Ki“‘, species kJ mol-l kJ mol-’ J K-I mo1-I 62.0 MgZt -13.18 5.31 6.28 65.3 Ca2+ -13.18 8.61 72.0 Mn2’ -12.8, 66.5 -13.48 6.36 CO” 65.6 NiZt -13.2, 6.36 69.3 CU” -13.47 7.20 Zn2+ -13.26 5.69 63.6 Cdzt -13.18 9.00 74.4 20.2 128.7 UO? -18.17 -19.8, 14.4 114.7 Yjt La” -20.66 14.1 116.6 -20.47 15.2 119.6 Ce3’ 15.5 120.3 PrJt -20.6, Nd3+ -20.77 15.8 122.7 Sm3’ -2O& 16.8 126.4 -20.95 16.3 125.0 Eu3+ -20.88 16.1 124.0 Gd3’ 15.9 122.9 Tb3’ -20.75 Dy3+ -20.61 15.2 120.1 Ho” -20.46 15.1 119.3

AP, AC;, cm3 mol-’ J K-I mol-’ 5.8 127 25 (10.2) 7.4 10.9 11.4

11.3 10.0 3.4 20.6

168

22.8

25.6

‘All thermodynamic properties for the UO?’(aq) reaction were derived as discussed in the text. For MgZt,Ca2+, Ni2+, Cu2+,Zn2’, and Cd2+ the Gibbs free energies were derived from the choices of L a r ~ o n , ~ and enthalpy changes from his experimental results except for Ni2+ where the selected Gibbs free energy is based on the average of values from Larson63 and Nair and N a n c o l l a ~ .The ~ ~ Gibbs free energies for Mn2+ and CoZt were derived from selected averages of results from Nair and N a n ~ o l l a sand ~ ~ the corresponding enthalpy changes were derived from The thermodynamic properties for the rare earth elements were all derived from Gibbs free energies from Nair and N a n ~ o l l a except s ~ ~ those for Tb3+ and Dy” which were estimated by Fay and Purdie68 based on the results from Nair and Nancollas6’ and the value for Eu” which was taken from Hale and S ~ e d d i n g . ~ ~ All enthalpy changes for the rare earth complexation reactions were derived by calculating averages of results from Fay and Purdie68 and Izatt et aI.’O All entropy changes were calculated from the selected Gibbs free energies and enthalpies. The standard-state volume changes were derived from the results of M i l l e r ~ ~except I - ~ ~ that for Eu3+ which was taken from Hale and S ~ e d d i n g ’and ~ the value for LaJt derived from Fisher and Davis.76 The value listed in parentheses for calcium complexation was taken from the results of Inada et All conventional ionic volumes for the cationic species were taken from miller^^^ except Eu3’(aq) which was derived by using the perchlorate ~ a l t . ~ . ~ ~ The change in heat capacity for MgZt was derived from Woolley and Hepler.3’,80

Several theories have been proposed for calculation of the equilibrium constant for ion-pair reactions. The first of these was proposed by Bjerrw”l An improved electrostatic theory and related equation was proposed by F u o ~ s .This ~ ~ latter equation will be discussed in more detail in relation to the present and additional data cited in Table VI. Fuoss Equation. Fuoss’s theory for ion-pair formation considers cations to be charged spheres with fixed radius a and anions to be point charges. Ion-pair formation is characterized by anions on the surface or inside the volume element of the cation.27 The equation derived by Fuoss gives an expression for the equilibrium constant (on a molarity basis) as in K A . F ~= -~

4000rNAa3 exp( Iz+z-le2 - 3 4rtoeakT

)-

-2.52256 X- 103(a/A)3 exp(b) (57)

where NA, zt, z-, e, to, e , and k are Avogadro’s number, charge of cation and anion, charge of an electron, permittivity of vacuum, static dielectric constant of solvent, and Boltzmann’s constant, (81) Bjerrum, N . Kgl. Dansk. Vidensk. Selsk. Mar.-Fys.Medd. 1926, 7 , 1-48. (82) Fuoss, J. Am. Chem. SOC.1958, 80, 5059-5061.

7862


respectively. As correctly pointed o ~ t , this ~ ~equation 9 ~ ~ should be converted to the molality scale for differentiation to avoid temperature-dependent concentration units.8s Therefore, In K , = In KA,Fuoss+ In p (58) where In KA is the equilibrium constant defined in terms of molality and p is the density of the solvent. Larsod3 has shown that predictions using this equation (with a = 4.0 A) for bivalent monatomic metal-ion complexation reactions are extremely good for changes in free energy, enthalpy, and entropy. HemmesS4and MilleroS3 have shown similar good agreement for some of the associated volume changes for complexation of bivalent cations. The equation fails (in some instances) to mimic the trends with increasing or decreasing radii in the transition-metal ions but usually provides reasonable quantitative agreement. The general procedure for testing the Fuoss equation has been to derive the a parameter from forcing agreement with known equilibrium constants and then calculating the various other thermodynamic properties. If, as Eigen and Tammx6have suggested, the formation of metal sulfate ion pairs proceeds via three distinct steps, this application of the Fuoss equation is not valid. Instead, as many authors have shown, the Fuoss equation can be used to predict one sequence in a multistep reaction scheme whereby more meaningful results can potentially be derived for the remaining steps. It is of interest here to compare the directly measured volumes and heat capacities for uranyl sulfate complexation with those that can be calculated solely from the Fuoss equation. The volume change of a reaction is related to the equilibrium constant by AVO = - R 7 (a dIn pK )

The Fuoss equation can be differentiated with respect to pressure to give

[

Uranyl ion is well-known to be nearly hears9and X-ray diffraction of solid p-U02S0490 shows that coordinated uranyl has an CkU-0 angle of 176'. In aqueous solution the uranyl ion is effectively more spherical in nature due to the coordinated water molecules in the primary solvation shell that number probably 5 based on the X-ray diffraction and 'H N M R studies of aqueous uranyl perchlorate solutions9' and preferred geometries of uranyl sulfateg0 and uranyl sulfate h y d r a t e ~in~ the ~ , ~solid ~ state. Some studies also strongly support four-coordinated uranyl ion in aqueous solution based largely on IH NMR results.94 Based on known structural data, it is not surprising that the Fuoss equation does not agree (for volumes) with the experimental value for this or other inner-sphere complexes. Spectroscopic data for aqueous solutions of uranyl sulfate are scarce and thus for structural information we can rely partly on data for the solid hydrates, partly on the present volumetric results, and consideration of the entropy of complexation, along with a spectrophotometricg5 and some ultrasonic velocity result^.^^,^^ Since it is apparent that we cannot explain the observed results with the Fuoss equation for volumes first we will look at heat capacities and then return to a more realistic interpretation of the experimental results. For heat capacity changes associated with ion-pair-formation reactions, the Fuoss equation (eqs 57 and 5 8 ) can be differentiated in accord with

y)

AH' = R P ( a In K

P

(59) 7

A P = R T b ( a;;97]


to generate

=

0.909826 - 0.191 kJ mol-' at 25 "C ( 6 5 )

1,15326 - 1.12 cm3 mol- at 25 'C ( 6 0 )

where 16.0366 - 8.37 J K-' mol-' at 25 "C ( 6 6 ) 03 K

(61)

where

(62)

and all appropriate fundamental constants were taken from Taylor et al.87 At room temperature, (8 In t/ap), is positive and the compressibility term is small,'3*88thus leading to an expected A P for uranyl sulfate complexation of 8.34 cm3 mol-' based on the value of a from Bailey and LarsonZ8calculated to match the experimental Gibbs free energy change. This is in extremely poor agreement with the present AVO = 20.6 cm3 mol-'. What we feel is a more correct application of the Fuoss equation will be presented later in relation to the Eigen-Tamm three-step model for ion-pair formation. MilleroS3has noted that the Fuoss equation underestimates the AVO associated with the formation of inner-sphere complexes as exemplified by results for the formation of l a n t h a n ~ m ' ~and .~~ europium7ssulfate complexes and further supported by the present results for uranyl sulfate. Because the uranyl ion is substantially nonspherical, the approximation of a spherical cation for the Fuoss equation is limiting. (83) Millero, F. J . Thermodynamic models for the state of metal ions in seawater. In The Sea: Ideas and Observations on Progress in rhe Study of the Seas; Wiley: New York, 1977; Chapter 17, pp 653-693. (84) Hemmes, P. J . Phys. Chem. 1972, 76, 895-900. (85) Hepler, L. G. Thermochim. Acta 1981, 50, 69-72. (86) Eigen, M.; Tamm, Z . Elekrrochem. 1962, 66, 93-107. (87) Taylor, B. N.; Parker, W. H.; Langenberg, D. N. Rec. Mod. Phys. 1969, 41, 375-496. (88) Archer, D. A.: Wang. P. J . Phvs. Chem. Re5 Data 1990, 19. 371-41 1 .

Prue83s98derived the term in brackets eq 66 (and an expression for the change in entropy) but did not include the third term representative of the molarity based standard state. Values of (a In t/dT), and (a2 In t / d P ) , were taken from Archer and WangEXand are positive and negative, respectively, thus leading to a large positive term in the brackets of eq 66 for most ionpair-formation reactions. The third and fourth terms were calculated by using data from Kell13 and are both positive and small. Using a = 3.5 X m for uranyl sulfate leads to AC,,*' = 142 (89) Pyykko, P.; Laakkonen, L. J.; Tatsumi, K. Inorg. Chem. 1989, 28, I 80 1- 1 805. (90) Brandenburg, N . P.; Loopstra, B. 0. Acta Crystallogr. 1978, B34, 3734-3736. (91) Aberg, M.; Ferri, D.; Glaser, J.; Grenthe, I. Inorg. Chem. 1983, 22, 3986-3989. (92) Putten, N . van der; Loopstra, B. 0. Crysr. Sfrucr. Commun. 1974, 3, 377-380. (93) Brandenburg, N . P.; Loopstra, B. 0. Cryst. Strucf. Commun.1973, 2, 243-246. (94) Fratiello, A.; Kubo, V . ; Lee, R. E.; Schuster, R. E. J . Phys. Chem. 1970. 74., .3726-3730. ~.. .~ (95) Jedinakova, V.; Kovarova, A. Sci. pap. Prague Inst. Chem. Technol. Inorg. Chem. Technol. 1977, 8 2 2 , 59-83. (96) Emst, S.; Jezowska-Trzebiatowska, B. 2. Phys. Chem., Leiprig 1975, 256. 330-336. (97) Ernst, S . ; Jezowska-Trzebiatowska, B. J . Phvs. Chem. 1975. 79, 2113-2116. (98) Prue. J . E J . Chem. Educ. 1969, 46, 12-16. ~



J K-' mol-' that is in excellent agreement with the experimental value 168 J K-'mol-'. Reasons for the models' success in terms of heat capacities and failure for volumes will be discussed later. A similar calculation has led to the value AC,," = 123 J K-I mol-' for magnesium sulfate complexation in amazingly good agreement with the experimental value 127 J K-'mol-' from Woolley and H e ~ l e r . ~The ' problem with these calculations, however, is that they rely upon a value of a not realistic for the complexed species and also a value that does not provide agreement for all thermodynamic properties. Instead, the Fuoss equation should realistically give the correct changes when considering the formation of a predominantly outer-sphere ion pair. Three-Step Eigen-Tamm Model. The three-step model originally proposed by Eigen and Tamms6 has been successfully applied to certain ion-pair-formation reactions including some involving sulfate The model has been criticized because of some seemingly unrealistic changes in the volumes for some reaction steps as derived by fitting the model to various ultrasonic absorption spectra for metal sulfates.84 Hemmes has remarked that the two-step model proposed by Jackopin and YeagerIo' provides more reasonable volumetric changes associated with each step in the ion-pair-formation process; however, our analyses on a variety of systems suggest that this model is unsatisfactory when applied to a variety of ion-pairing reactions. In light of this we have analyzed our present results for uranyl sulfate and various other results in terms of the three-step model of Eigen and Tamm.86 The Eigen-Tamm model can be summarized by the following reaction scheme, here in a notation for steps I and I1 favored by us: M'+(aq)

k

+ S042-(aq)2 M(H20),S04z-2 k21

& M(H20)S04'-2 + (x - 1)H20(1) k M(H20)S04z-2& MS04'-2 + H20(1)

M(H20)$042-2

32

k43

(68) (69) (70)

The total thermodynamic equilibrium constant for the association reaction is related to the stepwise kinetic rate constants and equilibrium constants by KA

=

KIKllKlll

+

KIKll

+ KI

(71)

where Kl = k 1 2 / k 2 1KII , = k23/k32rand Klll = k34/k43. The various appropriate thermodynamic properties can be derived by using eq 71 and the thermodynamic relationships described by eqs 59, 63, and 64 and -RT In K = AH" - T A P . The appropriate equations derived from this model for other thermodynamic quantities are thus

(73)

(99) Atkinson, G . ; Kor, S. K. J . Phys. Chem. 1965, 69, 128-133. (100) Eigen, M.: Tamm, Z . Elekrrochem. 1962, 66, 107-121. (101) Jackopin. L. G . ; Yeager, E. J . Phys. Chem. 1970, 74, 3766-3772.

Because of the number of steps in this sequence, it is possible to fit observed thermodynamic quantities or ultrasonic absorption data without regard to the reality of each step so long as the total property is represented. These concerns were reflected early due partially to the derived volumetric changes for MnS04(aq) formation by Tamm.lo2 To obtain quantitative agreement the AV for step 11 was derived84 to be -13.3 cm3 mol-' contrary to any prediction or physical reality. However, as mentioned earlier, application of the two-step model gives in most cases reasonable fits to ultrasonic absorption data, although in many other cases the two-step model is not flexible enough for quantitative predictions. Here we have applied a slightly modified Eigen-Tamm three-step model to the thermodynamic properties associated with ion-pair formation, although we have a much less well-defined step I1 in the overall sequence meant to explain huge differences in certain thermodynamic properties for the formation of predominantly inner-sphere compared to outer-sphere complexes. For a discussion of the total thermodynamic properties associated with ion-pair-formation reactions, it is useful to analyze each of the three steps in the Eigen-Tamm model assuming that this is indeed an adequate description of reality. Since step I involves the outer-sphere or solvent separated ion pair that should be an artifact primarily of electrostatic interactions, we can estimate the thermodynamic properties of this step using the Fuoss equation and its various derived forms for other thermodynamic quantities. Steps I1 and 111 in the three-step model are applied here to correspond with primary solvation shell water loss (probably from sulfate ion) and accompanying secondary solvation shell loss or additional primary water loss in step 11, and the loss of one water molecule from the primary solvation shell (of the cation) in step 111. As such we have chosen to use approximate changes in the thermodynamic properties for solid hydrate dehydration for step I11 to represent expected similarities for ions of similar charge and size. We have thus used step I1 as the *unknown" quantity to explain either minute or substantial water loss in this step and thus formation of predominantly an inneror outer-sphere ion pair. We have listed the thermodynamic properties associated with each step in the ion-pair-formation reaction in Table VII. The equilibrium constants for step I were calculated by using the Fuoss equation and intermolecular distances from Magini et aI.lo3using the following scheme. We have assumed that step I forms a very loosely bound outer-sphere complex where the distance of closest approach ( a ) of divalent monatomic cations is given by rMaH2 + 2:H2O + rsO,z--,where rH = 1.40 A rs0,2- = 2.30 A, and rM4H2 is given by Magini et al.l* The total distance of separation was based on approximate agreement with ultrasonic evidence for KI values. For trivalent ions this distance of separation was decreased as is expected due to enhanced electrostatic interactions. Here the distance of closest approach ( a ) is given by rM+H2 + rH20+ rs0p + 0.65 8, which represents a decreased distance of approximately one half of a water molecule radius. Again this parameter was chosen to mimic trends in ultrasonic absorption data for KI for the rare earth sulfates. The KIIlvalues were all derived from experimental studies as listed in Table VII, and KII values were calculated from the known thermodynamic constants KA. The entropy change for step 111 was chosen to be constant for all dehydration reactions based on results for selected metal sulfate hydratess9 and the values suggested by LatimerIo4 and Marcus1osfor aqueous dehydration. Entropies for step I1 were thus calculated as were the enthalpy changes for all steps from these constraints. Sulfate is thought to be bound as a unidentate species in inner-sphere complexes,64although the observed thermodynamic properties for the ion-pairing reactions indicate very strong in(102) Tamm, K. Reports of the 6th International Congress on Acoustics; Paper GP-3-3, Tokyo, 1968. (103) Magini, M.; Licheri, G.; Paschina, G.; Piccaluga, G.; Pinna, G. X-ray Diffraction of Ions in Aqueous Solutions: Hydration and Complex Formation; CRC Press: Boca Raton, FL, 1988. (104) Latimer, W. M. Oxidation Potentials, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1952.


I864 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990

TABLE VII: Stepwise Changes in Thermodynamic Properties for the Formation of Selected Metal Sulfate Complexes for the Eigen-Tamm Three-Step Model at 25 OC' AH,', AHII', AHIIIO, MI', aSIIo, AVIO, AV1lo, ACp,la, ACp,llO. Mz+ a. A K I KII Kill kJ mol-' kJ mol-' kJ mol-' J K-l mol-l J K-I mol-' cm3 mol-' cm' mol-) J K-'mol-) J K-' mol-' 12.7 44.0 20.1 3.46 2.8 1 55.3 60.2 3.42 0.76 Mg2+ 7.21 50.0 2.659 0.16 13.8 43.2 26.4 3.28 8.87 3.28 2.67 Ca2+ 7.51 48.2 2.941 0.10 12.7 43.8 35.3 3.40 5.24 3.38 5.49 Mn2+ 7.30 49.4 2.248 0.16 13.8 44.0 26.3 3.46 9.3 1 3.43 2.49 Co2+ 7.20 50.1 3.271 0.10 13.3 44.2 25.4 3.50 10.2 3.45 2.37 Ni2+ 7.15 5 0 . 4 2 . 7 4 7 0 . 1 2 5 13.8 43.9 29.9 3.45 9.91 3.41 3.58 Cu2+ 7.23 49.9 3.271 0.10 13.8 43.5 23.6 3.35 8.53 3.33 1.81 Zn2+ 7.39 48.9 3.010 0.10 3.33 6.19 13.8 43.5 38.1 3.35 0.0 Cd2+ 7.39 4 8 . 9 2 . 8 8 0 0 . 1 0 3.43 43.2 64.6 3.27 16.1 52.7 61.4 14.5 UO?+ 7.52 48.2 4.000 6.675 3.27 4.86 68.3 32.0 6.03 17.0 5.45 5.76 La3+ 6.93 411.3 1.923 3.75 5.46 7.24 4.52 68.4 35.2 6.91 415.2 1.559 4.3 Ce'+ 5.42 68.6 38.1 5.48 7.08 6.89 419.0 2.237 3.0 Pr'+ 5.50 7.60 4.80 68.8 38.0 6.86 424.4 1.905 3.85 Nd'+ 5.53 9.03 4.08 69.0 40.5 Sm'+ 6.82 432.4 1.548 5 . 1 5 4.40 69.3 37.9 6.16 19.8 6.80 437.5 1.703 4.52 5.56 8.29 Eu'+ 5.57 8.17 4.20 69.3 37.9 Gd3+ 6.78 441.8 1.576 6.78 5.59 8.01 4.20 69.5 36.9 Tb3+ 6.76 446.4 1.469 4.9 69.6 35.1 5.60 6.99 4.77 6.75 449.3 1.651 3.9 Dy" 5.61 7.75 3.46 69.7 32.5 Ho'+ 6.73 452.2 0.987 6.6

'All thermodynamic properties for the reaction steps were derived as discussed in the text with the following selections for required data. The equilibrium constants for step I l l were derived from averages or selections from a variety of ultrasonic absorption studies for the divalent metal sulfates99.'00*'0bl''and the rare earth sulfates.l'*1'2-1'6 The value ASlllo = 27.3 J K-' mol-! based on mineral dehydration reactions59of what would be a primary solvation shell water molecule was used for all systems consistent with values proposed by LatimerIM and Marcus.'os The enthalpy changes were all calculated from AHo = AGO + T a S O . The value AVIIIo= 2.1 cm3 mol-) was taken from Miller0 and the value ACB:llIo = 42 J K-l mol-' also derived from mineral dehydration reactions was used. Because UOZ2+(aq)was the only polyatomic cation considered, a ifferent scheme was emplo ed here. The equilibrium constant for step I was calculated as described for other metals only that the U-OH2 distance was based on the results of i b e r g et aI?l Correlations of In KIIand In Kill with z2/rcryl,were reasonably linear especially considering the uncertainties in these values. We chose to consider UO?+(aq) as atomic uranium 6.7 for U022+.a crystal radius equal to that of four coordinate U(V1) as we expect that for step I I more so than step I l l , sulfate would react mostly on an electrostatic basis and since the bulk of the charge density lies directly on the U atom. These correlations yield KII values between 1.8 and 6.7 for UO?'. We chose the value KII = 4.00 and have calculated all other properties as previously described.

teractions in some cases and arguably bidentate complexing would be expected for uranyl cation. The values for the entropy and volume changes associated with step I1 in the reaction sequence should, if the implementation of the scheme is adequate, give details of the distribution of inner- to outer-sphere complexes as do the equilibrium constants for the steps. The magnitude of the volume and entropy changes for this step could also be used to derive the number of water molecules lost as done by other aut h o r ~ . ~ ~As* expected, ~ ~ * ' ~ the ~ smaller the ion and more highly charged it is leads to increased changes in these properties for step 11. We have plotted the dependence of AV,,O on z 2 / r in Figure 6. The regular trend toward more positive AVIIovalues and thus inner-sphere ion pairs is clear with increasing values of z 2 / r . Here we have plotted the uranyl ion using the radius for four-coordinate atomic U as discussed in the footnote of Table VII. As can be seen the correlation is quite good considering the limited data available except for one result for calcium sulfate complexation. The more positive calcium ion-pair volumetric change is suspect considering the vastly different values derived from the two cited experimental investigations. One study clearly indicates preferred inner-sphere f ~ r m a t i o n while '~ the other a larger ratio of outer-

(105) Marcus, Y . J . Solution Chem. 1987, 16. 735-744. (106) Fittipaldi, F.; Petrucci, S. J . Phys. Chem. 1967, 71, 3414-3417. (107) Atkinson, G.; Petrucci, S. J . Phys. Chem. 1966, 70, 3122-3128. (108) Purdie, N.; Farrow, M. M. Coord. Chem. Reu. 1973, 11, 189-226. (109) Pearson, R . G. J . Chem. Educ. 1961, 38, 164-173. ( I IO) Bechtler, A.; Breitschwerdt, K . G.; Tamm, K. J . Chem. Phys. 1970. 52, 2975-2982. ( I I I ) Fritsch. K.;Montrose, C. J.; Hunter, J. L.; Dill, J. F. J . Chem. Phys. 1970, 52, 2242-2252. ( I 12) Purdie. N.; Vincent, C. A. Trans. Faraday SOC. 1967, 63, 2745-2757. ( I 13) Fay, D. P.; Litchinsky, D.; Purdie, N. J . Phys. Chem. 1969, 7 3 , 544-552. ( I 14) Farrow, M. M.; Purdie, N.: Eyring, E. M. J . Phys. Chem. 1975, 79, 1995-1999. ( 1 1 5 ) Fay, D. P.: Purdie, N . J . Phys. Chem. 1970, 74, 1160-1166. (116) Qadeer. A . 2. Phys. Chem. Munich 1974, 91, 301-316.

3oc

i

Oca Ed' 0 0UO>

0

; 4 OCd

5

7 8 9 /r Figure 6. The derived change in volume calculated for the second step in the metal sulfate complexation reaction plotted versus z 2 / r . These AV11 values were derived according to the Eigen-Tamm model as discussed in the text. The sources for the two calcium values are discussed in the text and Table V I I . z=

to inner-sphere pairs." The entropies and enthalpies associated with step I1 have relatively large associated uncertainties due t o the relative uncertainties in the observed mA values used for these calculations. The heat capacity changes for magnesium and uranyl complexation with sulfate both appear very regular even though the volumetric changes are so different. Here the heat capacity change is very insensitive to inner- or outer-sphere ion-pair formation due partly t o contributions not arising from the heat capacities of the various forms of the ion pair but rather to a relaxational style heat

J. Phys. Chem. 1990, 94, 7865-7867 capacity as given by the second and third lines of eq 74. The higher heat capacity change of uranyl complexation, however, indicates more inner-sphere complexation than for magnesium, although more heat capacity data need to be gathered to make this more of a quantitative tool. The standard-state partial molar heat capacities of metal sulfate ion pairs (such as MgSO, and U02S04)are more negative than expected if based on an analysis of the number of possible molecular motions of the species similar to values for certain polyatomic anionic species (such as phosphate and sulfate ions). The heat capacities of several oxygen-containing neutral solutes (see have substantially higher heat cafor example C 0 2 and pacities more in accord with molecular expectations. The observed effects in the present metal-sulfate complex arise from a combination of the loosely bound oxygen atoms forming the complex and also from the charge distribution on the species that both lead to enhanced solutewater interactions. These solutes even though they formally have no charge decrease the heat capacity of water by the removal of much of the H-bonded structure as do ionic solutes. Solutes such as these that display these effects are expected to have negatively divergent standard-state volumes and heat capacities approaching the critical temperature of water as discussed previously. The trend to more positive values of the

7865

standard-state heat capacities are due both to decreased ion-water interactions but also the various forms of the metal sulfate ion pairs and thus the various equilibria relating these forms. More specific trends in the heat capacities for these types of ion-pairformation reactions are of much interest but require further experimental work. Acknowledgment. We are grateful to Professor Loren Hepler for the University of Alberta for advice and assistance with all aspects of this work. We are also grateful to Dr. Peter Tremaine of the Alberta Research Council for allowing us to use the laboratory facilities of the Alberta Research Council and for helpful discussions. Ms. Karen Jensen is kindly acknowledged for her preparation of all figures. J.K.H. is grateful to the Natural Sciences and Engineering Research Council of Canada for postgraduate and postdoctoral fellowships, and to Professor Kenneth Pitzer and the Lawrence Berkeley Laboratory for facilities and time to produce the final copy of this paper. C.N.T. is grateful to NATO for the provision of a visiting fellowship at the Alberta Research Council and to CREGU for authorization of this work. In closing we are pleased to acknowledge helpful comments given by Professor Pitzer especially now in commemoration of his 75th birthday.

Core Polarization in Aluminum P. A. Christiansen Department of Chemistry, Clarkson University, Potsdam. New York 13676 (Received: January 16, 1990; In Final Form: June 7 , 1990)

Core polarization in atomic AI was analyzed by using core-valence correlation potentials with relativistic effective potentials in quantum Monte Carlo simulations. The core-valence correlation potential included dipole, quadrupole, and higher order corrections within the constraints of the 10-electroncore (spdf) basis set. With core polarization included, relativistic effective potential quantum Monte Carlo (REP-QMC) simulations gave the first three AI ionization potentials to within 0.03 eV of the experimental values. Although the core polarization correction to the first ionization potential is negligible (due to cancellation), corrections to the second and third potentials are quite significant, 0.1 and 0.4 eV, respectively. We encountered no difficulty in carrying out Monte Carlo simulations, including branching, on the three-electron (Fermi) ground state.

Introduction Although metallic aluminum has a long history of use in solid rocket propellants, there are also possible advantages to the use of light metals in conjunction with liquid hydrogen-oxygen rocket fuels. However, to be practical, the metals must be stabilized in atomic or at most very small cluster form. It has been speculated' that the large dipole polarizabilities of the alkali and alkaline-earth metals might make it possible to form marginally stable (at liquid hydrogen temperatures) van der Waals complexes involving single metal atoms, or perhaps diatomics, surrounded by H2 molecules. Relatively little is known about van der Waals complexes in general and almost nothing about these in particular. And unfortunately the accuracy of alkali and alkaline-earth metal electronic structure predictions is greatly complicated by the large core polarizabilities, which can cause substantial core-valence correlation errors.2 Furthermore such complications may not be limited to the leftmost columns of the periodic table. In recent relativistic effective potential quantum Monte Carlo (REP-QMC) simulations we noted a 0.06-eV error in the AI ionization potentiaL3 The error was attributed to either an inadequate trial wave ( I ) Konowalow, D. D., private communication. (2) Muller, W.; Flesch, J.; Meyer. W. J. Cbem. Pbys. 1984, 80, 3297. (3) Christiansen, P.A. The Challenge of d and f Electrons, Theory and Compurcltion; Salahub, D. R., Zerner, M. C., Eds.; ACS Symposium Series 394; American Chemical Society: Washington, DC, 1989; Chapter 22, pp 309-321.

0022-3654/90/2094-7865$02.50/0

function used in the simulation or, considering the large corevalence correlation corrections in Mg, neglect of core polarization. Regardless of whether core polarization shifts the AI ground state or not, previous calculations have shown a 0.3-0.4-eV discrepancy between the AI2+one-electron REP energy and the experimental third ionization energy. This seems almost certainly due to the neglect of core-valence correlation. If accurate predictions of van der Waals complex stabilities are to be made, whether for the alkali or the alkaline-earth metals or perhaps even aluminum, the problem of core-valence correlation will have to be resolved. The most economical approach to the problem would probably involve algorithms such as those developed by Meyer and co-workers2 and others4 as well as by ourselves.s Our particular method has the advantage that it can readily include corrections of order higher than just dipole, and because of the REP, relativity (including even spin-orbit coupling) can be properly treated. Also, in contrast to most other approaches, ours requires at most only a minor empirical adjustment. In this paper we evaluate the core polarization corrections for the AI ground state as well as for the first two positive ions. We use the method proposed earlier5 in conjunction with REP-QMC. We are able to compute ionization energies and compare our values (both with and without core polarization) with experiment. One (4) Silberbach, H.; Schwerdfeger, P.; Stoll, H.: Preuss, H. J. Pbys. B 1986, 19, 501. ( 5 ) Christiansen, P. A. Cbem. Phys. Left. 1986, 127, 50.

0 1990 American Chemical Society

Thermodynamics of complexed aqueous uranyl species. 1. Volume

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