J. CHEM. SOC. DALTON TRANS. 1990 1 Thermodynamic Properties of Co3+(aq) David A. Johnson* Department of Chemistry, The Open University, Milton Keynes MK7 6AA Peter G. Nelson School of Chemistry, University of Hull, Hull HU6 7RX Improved values of the thermodynamic properties of Co3+ (aq) at 298.1 5 K have been calculated: AfHg(Co3+,aq) = 79 -341 k 25 J K-I mol-l. The value of S",(Co3+,aq) is considerably more negative than that of Sz( Fe3+,aq), a difference which can be related mainly to the small ionic radius associated with the d6 low-spin state. Other calculations suggest that the stabilizations with respect to the high-spin state are AH; = 46 & 20 kJ mol-I and AG; = 38 k 25 kJ mol-I. 7 k J moI-l;AfG~(Co3+,aq) = 131 k 4 k J mol-I; S;(Co3+,aq) = The thermodynamic properties of the blue aqueous ion Co3+ (as) are of particular interest.One reason is that it is the only dipositive or tripositive aqueous ion of the first transition series which occurs in the low-spin state.',, Another is that its stabilization with respect to the high-spin state has a crucial bearing on competing explanations of the anomalous rates of its electron-transfer r e a ~ t i o n s . ~ , ~ Unfortunately c o 3 +(as) slowly oxidizes water, and this has made its thermodynamic properties both hard to determine and a matter of dispute. In this paper, we show how recent work can be used to calculate more reliable values of AfHi(Co3+,aq), AfG :(Co3',aq), Sz(Co3+,aq), and E Q(C03+-Co2+) than were previously obtainable. We also argue that these data are consistent with theoretical expectations arising from the magnetic and spectroscopic properties of the complex [CO(H,O),]~+, and discuss the stabilization of the complex with respect to the high-spin state.Auxiliary Data Unless otherwise stated, thermodynamic data were taken from ref. 5. For data on Co2+(aq), Fe2+(aq), and Fe3+(aq), see Table 1. The Value of A,Hz(Co3+,aq).-Three values '-* exist for AH, of reaction (l), but the best is that of Mowforth et aL8 who worked at 25°C in 2.7 mol dmP3 HClO,, and improved the methods of correcting for dilution effects. Co3'(aq) + Fe2+(aq) + Co"(aq) + Fe3'(aq) (1) Other data' suggest that at these acidities, the correction for hydrolysis is very small, and because of similarities in the constitution of the reactants and products we assume a zero correction to infinite dilution.This gives AHZ(1) = -94 f: 6 kJ mol-' and thus AfHZ(Co3',aq) = 79 7 kJ mol-'. An almost identical value (77 kJ mol-I) is obtained from the temperature variation of the formal potential" of the electrode Co3 +-Co2 + in 3 mol dm-3 HCIO,, but as the data were minimal we have not weighted this in our assessment. The Values of AfGg(Co3+,aq) and E"(CO~+-CO~+).- The value of E"(CO~+--CO~+) has been a matter of dispute for some time, and figures as different as 1.30 and 1.95 V have been proposed. We first ignore those data7," which are incon- sistent with the oxidizing strength6 of Co"(aq). We then confine ourselves to measurements made in perchlorate media where complexing by the anion is small. Zingales and co- Table 1.Thermodynamic properties of the aqueous dipositive and tripositive ions of iron and cobalt AfH," AfG I s,- kJ mol- ' kJ mol- ' J K-' mol-' Fe2+(aq) -91.2 k 1.5" -89.8 & 1.5' -108 k 4' Fe3+(aq) -48.5 k 3d -15.4 k 1.5" -280 k lob Co2+(aq)s -58.2 & 2 -55.6 & 2 -109 k 9 c o 3 + (aq)g 79 _+ 7 131 k 4 -341 25 " A value was obtained by combining the heat of formation of FeCl, given by M. F. Koehler and J. P. Coughlin, J. Phys. Chem., 1959,63,605, with the average of the heats of solution of J. C. M. Li and N. W. Gregory, J. Am. Chem. SOC., 1952,74,4670, and of P. J. Cerutti and L. G. Hepler, Thermochim. Acta, 1977,20,309, corrected in the former case to infinite dilution using data on NiCI,. This was doubly weighted against the values of J. W. Larson, P.Cerutti, H. K. Garber, and L. G. Hepler, J. Phys. Chem., 1968,72,2902 and V. P. Vasil'ev, N. G. Dmitrieva, P. N. Vorob'ev, V. N. Vasil'eva, and I. I. Nechaeva, Zh. Neorg. Khim, 1985,30, 1681. Calculated from the other two entries in this row. The average of the value of J. W. Larson et al., ref. a, and that obtained by combining our first value of A,H,"(Fe'+,aq), with the E"(Fe2+-Fe) of P. R. Tremaine and J. C. Le Blanc, J. Solution Chem., 1980,9,415. From the AH," value for Fe3+-Fez+ of D. 0. Whitteniore and D. Langmuir, J. Chem. Eng. Data, 1972,17,288. From E*(Fe3+-Fe2+) = 0.771 V, the average of the values reviewed in ref. d. R. N. Goldberg, R. G. Riddell, M. R. Wingard, H. P. Hopkins, C. A. Wulff, and L. G. Hepler, J. Phys. Chem., 1966,70,706. See text. workers', determined the formal potential Ef(Co3 +-Co2 ') = 1.841 k 0.002 V in 5.1 mol dm-3 HClO, at 268.15 K using a cell without a liquid junction.[equation (2)]. Using our Co3+(aq) + $H,(g) --+ Co2+(aq) + H+(aq) (2) recommended value AHz(2) = -137 kJ mol-' in the Gibbs- Helmholtz equation then yields Ef(Co3+-Co2+) = 1.888 V at 298.15 K in 5.1 mol dmP3 HClO,. To convert this into a value at infinite dilution, we combine our value E" (Fe3 +-Fe2 ') = 0.771 V with the corresponding formal potential, obtained by correcting Fe3 +-Fe2 + potentials in various concentrations of perchloric acidi3-16 for the hydrolysis of Fe3+(aq)17 and extrapolating them to 5.1 mol dmP3 HClO,. The correction is 0.035 V and it gives E"(Co3+-Co2+) = 1.923 V. A similar correction can be made to the formal potentials obtained by WarnqvistloT1* in 3 and 4 mol dm-3 HClO, using cells with a liquid junction.The resulting values are 1.90 and 1.96 V respectively. Weighting the three figures as 2: 1: 1, we obtain2 J. CHEM. SOC. DALTON TRANS. 1990 A r Figure. Potential-energy curves for the ' A , , and 5T2g states of ICO(H~OM~+ Ee(Co3+-Co2+) = 1.93 -+_ 0.03 V and AfGz(Co3+,aq) = 131 4 kJ molt'. The Value of Sz(Co3+,aq).-This can be calculated from our AfHz(Co3+,aq) and AfG z(Co3+,aq): the result is Sz(Co3+,aq) = -341 k 25 J K-' mol-'. Discussion The thermodynamic properties of Co3+(aq) appear in Table 1; their most striking feature is the low, negative value of S z(Co3 +,as). This quantity is about 60 J K-' mol-' smaller than S:(Fe",aq) even though the entropies of the dipositive aqueous ions are very similar.Such a difference is precisely what is predicted by empirical equations which relate the entropies of aqueous ions to charge, relative atomic mass, and crystal r a d i u ~ . ' ~ ~ ~ ~ When in its d6 low-spin state, Co3+ forms unusually short bonds. This is explained by ligand field theory,2 and is apparent, for example, in metal-oxygen distances in the complexes [M(H20),]3+ which are found in alums with the formula type C S M ( S O ~ ) ~ . ~ ~ H ~ O . ~ ' Likewise, the ionic radius of low-spin Co3+ in octahedral co-ordination is smaller than that of any other tripositive ion of the first transition series. Shannon22 recommends radii of 64.5 and 54.5 pm for high-spin Fe3 + and low-spin Co3 + respectively.The best- known empirical equation for the entropies of aqueous ions is that of Powell and Latimer." It should include a magnetic entropy term, Sm,mag. For aqueous transition-metal ions, is often uncertain, but both high-spin Fe3+ and low-spin Co3+ have ground states which are not split by a ligand field, so the Sm,mag values can be specified precisely as Rln 6 and Rln 1 respectively. With Shannon's ionic radii, the Powell-Latimer equation then gives Sz(Fe3+,aq) - Sz(Co3+,aq) = 53 J K-' mol-' in good agreement with our experimental value. A more recent equation proposed by Morss2' yields a very similar figure (52 J K-' mol-'). The Stabilization of Low-spin [Co(H,0),I3+(aq) with res- pect to the High-spin State.-The ion [Co(H,O>,] + ex- changes water molecules with the solvent, and electrons with [CO(H~O>,]~+, at unexpectedly high rates.As [CO(H,O),]~ + (as) is very labile, and is always present in solutions of [Co(H,O),] +, an explanation of the fast electron-transfer reaction could account for both anomalies. The origin of such an explanation might lie in a spin equilibrium, if the labile, high- spin 'T2, state lies <20 kJ mol-' above the ' A , , ground However, Johnson and Sharpe' assigned bands in the spectrum of CsCo(S0,),-12H20 and calculated a value of 60 kJ mol-' for the ' A l g - T2, transition energy. They noted that the transition occurs at fixed internuclear distance, a fact which by itself suggests that its energy provides an upper limit for the thermodynamic stablization, but they preferred to adopt 6& 85 kJ mol-' as a best estimate of the stabilization because of other approximations involved in the calculation of the transition energy. took Johnson and Sharpe's transition to be the energy AC in the Figure, and tried to correct it by calculating the energy CD.They obtained E,, = 43 kJ mo1-', and thus EAD = 17 kJ mol-' for the thermodynamic stabilization energy, a value low enough to be consistent with spin-equilibrium involvement in the anomalous reaction kinetics. The calculation required the symmetric Co-0 stretching frequency in the 5T2, state, and a figure of 357 cm-' was estimated by reducing the value observed in the ground state of a Tutton salt of cobalt(I1) by 10%. It is now known that compounds containing the tripositive ions [M(H20)6]3 + show symmetric stretching frequencies in the range 5 1Ck540 ~ m - ' .~ ~ If Winkler, et al.'s calculation is repeated with such values, then E,, exceeds 60 kJ mol-', and contradicts the experimental evidence'.2 by making 5T2g the ground state. Clearly their method substantially overestimates the correction. An alternative approach relies on the valence force-field appr~ximation,~~ and writes the potential energy for the totally symmetric vibration of the 'T2, state in the form (3), where Ar More recently, Winkler et V = 6[ik(Ar)2] (3) is the displacement from the equilibrium internuclear distance during the vibration and k is the force constant. Then, we can write equation (4) where v is the stretching frequency, m(H,O) EcD = 3l~(Ar,)~ = 12.n2v2rn(H20)(Are)2 (4) is the mass of a water molecule, and Are is the difference between the equilibrium internuclear distances in the ' A , , and 'T2, states.There are several possible ways of estimating Are. Shannon's value for the difference in the ionic radii of high-spin and low- spin Co3+ is 6.5 pm.22 An estimate for [CO(H,O>,]~' made by using data from the alums is 7.0 pm.21 We have estimated the cobalt-oxygen distance in the high-spin complex from a linear plot of the ionic radii of the high-spin ions in octahedral co- ordination22 against the metal-oxygen distances2' in the caesium alums of Ti, V, Cr, Mn, Fe, and Ga. This yields 194.3 pm and Are = 7.0 pm. If, instead of ionic radii, one uses the metal-fluorine distances in the high-spin trifluorides,26-28 the Are value becomes 7.7 pm.We take Are = 7.0 1.5 pm. To obtain v, we substitute our estimated cobalt-oxygen distance in high-spin CsCo(S04),~12H20 into Figure 5 of ref. 24. This yields v z 530 cm-', and studies of [A1(H20)6]3' suggest that the value will be lowered to about 510 cm-' in aqueous solution.24 Equation (4) then yields E,, = 24 k 10 kJ mol-', and hence EA, = 36 +_ > 10 kJ mol-' for the thermodynamic stabilization energy. This estimate relies on Johnson and Sharpe's calculation2 of the energy of the two-electron tran- sition from those of one-electron transitions using the strong- field approximation. Because of possible sources of error in the calculation, other than the one corrected for,* we have tried to * An obvious source of error is the neglect of vibrational changes, but these are largely expected to cancel one another out.More difficult to assess are the errors in ligand-field theory which is used to obtain AC.J. CHEM. SOC. DALTON TRANS. 1990 3 Table 2. Thermodynamic properties of some tripositive aqueous ions AfH," Af G," s," kJ mol-' kJ mol-' J K - l mol-' Sc3 +(as) - 647" -255' v3 + (aq) -281' - 242d - 299" Cr3 +(as)' - 236 Mn3+(aq) - 104" - 74a - 267" Fe3 +(aq)h. - 48.5 - 280 Ga3 '(as)' -218 a The results of J. M. Stuve, U.S. Bur. Mines. Rep. Invest. 6705,1965, and of E. J. Huber, G. C. Fitzgibbon, E. L. Head, and C. E. Holley, J. Phys. Chem., 1963,67,1731 give A,H,"(ScCl,,s) = -942.2 5 2 kJ mol-'. This was combined with the heat of solution of J. Burgess and J. Kijowski, J. Inorg. Nucl. Chem., 1981, 43, 2389, corrected to infinite dilution using data for LaCl,.' Ref. 5. Calculated from the two other figures in this row. From the E"(V02+-V3+) value of G. Jones and J. H. Colvin, J. Am. Chem. Soc., 1944, 66, 1563, and A,G,"(V02+,aq) of ref. 5. " Estimated; see text. Only reactions of well established stoicheiometry were considered. The enthalpy of oxidation of Fe2+(aq) by H+(aq) in 0.5 mol dm-, HClO, is 40.8 & 1.5 kJ mol-' according to R. Connick and W. H. McVey, J. Am. Chem. Soc., 1951,73,1798, and the work of B. J. Fonatana cited therein. This was used to calculate two values for the enthalpy of reduction of HCr0,-(aq) by hydrogen in 0.5 mol dm-, HC104 from the results of M. W. Evans, 'The Transuranium Elements,' eds. G. T. Seaborg, J. T. Katz, and W. M.Manning, McGraw-Hill, New York, 1949, part I, pp. 282-294, and of 1. Dellien and L. G. Hepler, Cun. J. Chem., 1976, 54, 1383 with a correction in the former case for the presence of H,CrO,(aq) based upon the data of I. Dellien, F. M. Hall, and L. G. Hepler, Chem. Rev., 1976, 76, 283. The average gives the datum in Table 2 when the extrapolation to infinite dilution is neglected. A value 15 kJ mol-' more negative has been obtained by I. Dellien et al. from heats of oxidation of Cr2+ by Fe3+ and Cu2+ in 0.5 mol dm-, HClO,, but there is a considerable uncertainty in AfH;(Cr2', as), and some oxidation of Cr2+ by ClO, may have occurred (cf. G. Biedermann and V. Romano, Acta Chem. Scand., Ser. A, 1975, 29, 615). From AfGz(MnZf,aq) of ref. 5 and E"(Mn3'-Mn2+) = 1.60 V given by G.Biedermann and R. Palombari, Actu Chem. Scand., Ser. A , 1978, 32, 381. See Table 1. 'The heat of solution of GaC1, given by W. A. Roch and A. Buchner, %. Electrochem., 1934, 40, 87, was corrected to 25 "C and infinite dilution using AC; data for GdCl,, and heat of dilution data for LaCl,. It was then combined with A,H;(GaCl,,s) from W. Klemm and H. Jacobi, 2. Anorg. Allg. Chem., 1932,207,177. obtain an independent value for the stabilization energy using the empirical methods of K a r a p e t ' y a n t ~ . ~ ~ , ~ ' This calls for experimental figures for A,Hg(M3+,aq), and those which are required and available are shown in Table 2. The values for V3+(aq) and Mn3 '(as) rely on estimated entropies obtained by interpolation. Interpolation is quite difficult because it is hard to separate the magnetic entropy from the remainder which includes a detectable ligand-field contribution. As the magnetic entropy of d 5 ions is well established, we have assumed that the magnetic entropies of M2+(aq) and M3+(aq) increase approximately uniformly from zero at do to Rln 6 at d ', and then drop back approximately uniformly to zero at dl'.When these magnetic entropies are subtracted, the plot of the entropies of d" tripositiveionsagainst thoseofdnf5 dipositiveions(n = b 5 ) was assumed to be approximately linear. The values of S:(V",aq) and Sz(Mn3+,aq) in Table 2 were established by using this plot and then adding the required magnetic entropy. The values of A,Hg(M3+,aq) in Table 2 and other data allow calculations of AH at 298.1 5 K for reactions (5)-(7).The results are shown in Table 3. When AHz(5) is plotted, first against A H G(6) and then against AH37), two straight Table 3. Values of AH,"(kJ mol-') for reactions (5)-(7)" at 298.15 K AH,"(5)b AH,"(6)c AH,"(7)d M s c -5 299 -8 001 -5 516 V -5 706 -8 463 Cr -5 884 -8 638 -6 042 Mn - 5 880 -8 638 -6 047 Fe -5 761 -8 532 -5 939 c o -8 729 -6 108 Ga -6 035 -8 830 -6 214 " All values were obtained using AfH,"(M3+,g) of ref. 5. bAfH,"(M3f,aq) -AfH:(M3+,g). ' From equation (1) of ref. 31, using the data recommended there, except for A,Hz(M3+,g). From the values of A,Hz(MF,,s) and A,H,"(F-,g) recommended in ref. 31. Table 4. Thermodynamic properties of high-spin and low-spin Co3 + (as) AfH," AfG," sz kJ mol-I kJ mol-I J K - ' mol-' c o 3 +(as, low spin) 79 131 - 341 Co3+(aq, high spin) 125 169 -314 M3+(g) + %H2(g) + M3+(aq) + 3Hf(aq) + 3e-(g) ( 5 ) M3+(g) + 3K+(g) + 6F-(g) - K3MF6(s) (6) M3+(g) + 3F-(g) - MF3(s) (7) lines are obtained.On the K3MF6 plot, there are six points, the correlation factor is 0.999, and deviations from linearity are < 10 kJ mol-'. On the MF, plot, there are five points, the correlation factor is 0.999, and deviations from linearity are < 15 kJ mol-'. Now the hexafluorometalate(II1) compounds and the trifluo- rides are all high spin or very nearly so the experimental AHZ(6) and A H 3 7 ) for cobalt can be inserted into the plots to obtain two values of AHz(5) for high-spin Co3+(aq). These yield 131 and 143 kJ mol-I respectively for AfHZ- (Co3 +,aq,high spin), figures which when combined with the low-spin datum of Table 1 give thermodynamic stabilization energies for the low-spin state of 52 and 64 kJ mol-l. Because the K3MF, plot contains more points, and has better linearity, we doubly weight the first of these figures against the second, and then take the mean of the result (56 kJ mol-') and the spectroscopic value given earlier. This yields 46 & 20 kJ molt' which is our preferred value for the enthalpy of stabilization.Table 4 contains the resulting data for both the high-spin and the low-spin aqueous ion. The entropy of Co3+(aq,high spin) was estimated by assuming that it differs from S;(Fe2+,aq) by the amount that S;(Fe3 ',as) differs from S:(Mn2+,aq). As expected, for reaction (8), AS; is positive because of an increase in ionic radius and magnetic entropy.The estimate Co3+(aq,low spin) Co3+(aq,high spin) (8) for AGZ(8) is 38 & 25 kJ mol-I which corresponds to an equilibrium constant of 2 x lop7 at 298.15 K. It lies above the upper threshold which is usually specified by those who invoke a spin-state equilibrium to account for the high rates of the electron-transfer reactions of Co3 +(as), and in that respect, it favours other32 explanations.4 J. CHEM. SOC. DALTON TRANS. 1990 Finally, our recommended values for AG i ( 8 ) and A H z ( 8 ) are in agreement with the n.m.r. measurements of N a ~ o n ~ ~ who concluded that AG 3 8 ) > 23 kJ mol-l, a lower threshold which is converted by our A S 3 8 ) into AH38) > 31 kJ mol-l. Clack and Smith34 obtained A H z ( 8 ) = 223 kJ mol-I by a quantum- mechanical calculation, but this was for an isolated [Co(H,0)J3 + ion, and used an approximate method (INDO).References 1 2 3 4 5 6 7 8 9 10 11 12 13 H. L. Friedman, J. P. Hunt, R. A. Plane, and H. Taube, J. Am. Chem. Soc., 1951, 73,4028. D. A. Johnson and A. G. Sharpe, J. Chem. Soc. A, 1966,798. N. Sutin, Prog. Inorg. Chem., 1983,30,441. D. H. Macartney and N. Sutin, Inorg. Chem., 1985,24,3403. D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churney, and R. L. Nuttall, ‘The NBS Tables of Chemical Thermodynamic Properties,’ American Chemical Society, Washington D.C., 1982. D. A. Johnson and A. G. Sharpe, J. Chem. Soc., 1964,3490. A. L. Rotinyan, L. M. Borisova, and R. W. Boldin, Electrochim.Acta, 1974, 19,43 and refs. therein. C. W. Mowforth, D. R. Rosseinsky, and K. Stead, J. Chem. Soc., Faraday Trans. 1,1979, 1268. G. Davies and B. Warnqvist, Coord. Chem. Rev., 1970,5,349. B. 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