THERMODYNAMIC PROPERTIES OF THE LANTHANIDE(III) HALIDES
3. Low-temperature heat capacity and standard entropy of the solid trihalides
154 R.J.M. KONINGS AND A. KOVÁCS
THERMODYNAMIC PROPERTIES OF THE LANTHANIDE(III) HALIDES 155
the partitioning functionQ, which is described by the Maxwell–Boltzmann distribution law:
(2) Qexs=
n
i=0
gie−εi/RT,
whereεi is the energy andgi the degeneracy of leveli,Ris the universal gas constant andT is the absolute temperature. The excess heat capacity is then calculated from the equations:
(3) Cexs=R d
dT
T2d(lnQ) dT
.
The electronic states of the lanthanide 3+ions are characterised by(2J+1)-fold degeneracy which is removed by the crystalline electric field. For the ground state this results in a set of energy levels generally below 1000 cm−1. These crystal-field states have been identified for most of the lanthanide trifluorides, and are summarised in table 2. Examples ofCexsthus calculated are shown in fig. 5 for PrF3, NdF3, DyF3, and ErF3.
In LaF3, GdF3and LuF3the excess contribution is zero which is due to the fact that these lanthanide ions have an empty, half-filled and completely filled f-shell, respectively. This is evident from a plot of the Cp values at 298.15 K as a function of the atomic number as shown in fig. 6 which indicates that the values for these three compounds form a straight line, is spite of their different crystal structures. Flotow et al. resolved the lattice and excess contribution by assuming the heat capacity of LaF3to represent the lattice component of the other hexagonal earth trifluorides (PrF3, NdF3) and GdF3and LuF3that of the orthorhombic rare earth trifluorides. Flotow et al. obtained the values in between GdF3 and LuF3 by an interpolation using weighing factorf based on the molar volume:
(4) Clat=(1−f )Clat(GdF3)+f Clat(LuF3).
Fig. 5. The excess heat capacity of PrF3, NdF3, DyF3and ErF3as calculated from the crystal field energies.
156 R.J.M. KONINGS AND A. KOVÁCS
Fig. 6. The variation of Cp (298.15 K) in the lanthanide trifluoride series (◦). The bro- ken line connects the values for LaF3, GdF3 and LuF3for whichCexsis zero at 298.15 K.
The solid circles show the val- ues ofClat(298.15 K) obtained by subtractingCexs(298.15 K) from the experimental values.
Fig. 7. The excess heat capac- ity of DyF3; curve shows the values calculated from the crys- tal field energies, symbols are the values derived from the ex- perimental data by subtracting Clat, as explained in the text.
The experimental excess heat capacity thus obtained as the difference between measuredCp
andClat can then be compared to the values calculated from the crystal field levels. As an example, fig. 7 shows the good agreement of the experimental and calculated excess heat capacity of DyF3.
The standard molar entropies at 298.15 K derived from the low-temperature heat capacity measurements are summarized in table 3. Similar to the heat capacity, the entropy can be described as the sum of the lattice and excess components (Konings, 2001, 2002):
(5) S=Slat+Sexs.
The excess entropy is then calculated from the following equation:
Sexs=RlnQexs. (6)
Combining eqs. (2) and (6) gives:
Sexs=Rln(g0)+Rln (7) n
i=1
gie−εi/RT
.
The first term of eq. (7) represents the temperature independent contribution of the ground state, the second term the contribution of the excited energy levels.
The lattice contribution in the lanthanide fluorides is only known with sufficient accuracy when the f shell of the metal ions is empty (4f0) or completely filled (4f14). In these casesSexs is zero and the experimental entropy corresponds toSlat. Also in case the f-shell of the metal
THERMODYNAMIC PROPERTIES OF THE LANTHANIDE(III) HALIDES 157 Table 3
The entropies of the solid lanthanide(III) fluorides and chlorides at 298.15 K, in J·K−1·mol−1
Calculated Experimentala
Slat Sexs Stot Sexp References
LaF3 105.84 0.00 105.84 106.98±0.11 1
CeF3 104.97 13.73 118.70 115.23 2
PrF3 104.10 16.75 120.85 121.22±0.12 3
NdF3 103.23 17.24 120.47 120.79±0.12 4
PmF3 102.37 18.27 120.64 –
SmF3 101.50 15.00 116.50 –
EuF3 100.63 9.44 110.07 –
GdF3 99.76 17.29 117.05 114.77±0.22 5
TbF3 98.90 20.07 118.97 –
DyF3 98.03 21.83 119.86 118.07±0.12 6
HoF3 97.16 23.18 120.34 –
ErF3 96.29 22.62 118.91 116.86±0.12 6
TmF3 95.42 19.56 114.98 –
YbF3 94.55 17.29 111.84 –
LuF3 93.69 0.00 93.69 94.83±0.09 5
LaCl3 137.57 0.00 137.57 137.57 7
CeCl3 136.71 14.71 151.42 –
PrCl3 136.28 17.87 154.15 153.30 7
NdCl3 135.85 18.30 154.15 153.43 7
PmCl3 135.42 17.89 153.31 –
SmCl3 134.99 15.27 150.26 150.12 8
EuCl3 134.56 9.32 143.88 144.06 8
GdCl3 134.13 17.29 151.42 151.42 8
TbCl3 133.70 21.15 154.85 –
DyCl3 155.15 22.83 177.98 175.4 9
HoCl3 154.72 23.16 177.88 177.1 10
ErCl3 154.29 22.60 176.89 175.1 11
TmCl3 153.86 20.84 174.70 173.5 12
YbCl3 153.43 15.80 169.23 169.3 13
LuCl3 153.00 0.00 153.0 153.0 9
aThe uncertainty for the standard entropies derived from the calorimetric measurements has not been given in some cases.
References 1. Lyon et al. (1978)
2. Westrum Jr. and Beale Jr. (1961) 3. Lyon et al. (1979a)
4. Lyon et al. (1979b) 5. Flotow and O’Hare (1981)
6. Flotow and O’Hare (1984) 7. Sommers and Westrum Jr. (1976) 8. Sommers and Westrum Jr. (1977) 9. Tolmach et al. (1987)
10. Tolmach et al. (1990a)
11. Tolmach et al. (1990b) 12. Tolmach et al. (1990c) 13. Gorbunov et al. (1986)
ion is half filled (4f7),Slat can be derived easily from the experimental entropy since only a correction for the temperature independent term in eq. (7) needs to be made in the absence of a significant crystal-field splitting of the ground state.
In fig. 8 the experimental entropies for the lanthanide trifluorides are plotted as a function of the atomic number. The figure shows thatSlat for LaF3, GdF3and LuF3form approximately
158 R.J.M. KONINGS AND A. KOVÁCS
Fig. 8. The variation of the Sexp(◦) andSlat(•) in the lan- thanide trifluorides. The broken line shows Slat derived from the values for LaF3, GdF3and LuF3, the solid line showsSexs derived from the experimental data for the other configura- tions.
a straight line, as was the case for the heat capacities at 298.15 K.Slatfor the other lanthanide compounds is then obtained by inter- or extrapolation of the data (which is a simpler, but essentially identical approximation as the weighing factors based on the molar volume, as indicated by eq. (4)). These numbers are listed in table 3. The excess contribution is calcu- lated from eq. (7) using the crystal field energies listed in table 2, which are mainly based on spectroscopic studies of the lanthanide ions in transparent host crystals. In case of PmF3the spectroscopic data are missing, andSexsis calculated from the degeneracy of the ground state of the lanthanide ion. This neglect of the crystal energy splitting leads to a small overestima- tion ofSexs at 298.15 K, which increases when the energy gap of the crystal-field splitting becomes larger. For example, we obtainSexs=17.24 J·K−1·mol−1 at 298.15 K for NdF3
from the known crystal field levels, whereas we obtainSexs=Rln(10)=19.14 J·K−1·mol−1 at 298.15 K from the approximation.
The total entropy values thus obtained for CeF3, PrF3, NdF3, DyF3and ErF3compare well with the experimental values by Flotow et al. and the differenceSexp−Stot is in the order of 1–2%. The recommended entropy values are the experimental values of Flotow et al., and the calculated values for those compounds for which no experimental data are available. An uncertainty of±3.0 J·K−1·mol−1has been assigned to the calculated values.
3.2. LnCl3
Heat capacity measurements in the 10 to 350 K temperature range have been reported by Sommers and Westrum Jr. (1976, 1977) for the hexagonal lanthanide trichlorides and Tolmach et al. for the monoclinic ones (Gorbunov et al., 1986; Tolmach et al., 1987, 1990a, 1990b, 1990c). No compound was measured in parallel by both groups but remarkable differences are observed between the results of the two groups. This is shown in fig. 9 in which the heat capacity curves of GdCl3measured by Sommers and Westrum and LuCl3by Tolmach et al.
are plotted. It can be seen that the heat capacity curve of LuCl3is significantly higher than that of GdCl3up to about 150 K above which the heat capacity curve approach each other.
This is significantly different from the situation for the trifluorides where the curves of LaF3, GdF3and LuF3have the same shape and the heat capacity slightly decreases with increasing atomic number.
The differences in the entropy values derived for the two groups of compounds are also significant: the lattice values at 298.15 K derived from the work of Tolmach et al. are about 20 J·K−1·mol−1 higher than the extrapolation of results for the hexagonal compounds, as
THERMODYNAMIC PROPERTIES OF THE LANTHANIDE(III) HALIDES 159
Fig. 9. The heat capacity of GdCl3() and LuCl3( ).
Fig. 10. The variation ofSexp (◦, hexagonal;, monoclinic) and Slat (•) in the lanthanide trichloride series at 298.15 K;
the broken line shows the lat- tice contribution (see text).
Fig. 11. The molar volumes in the LnF3(◦) and LnCl3series () at 298.15 K.
shown in fig. 10. This suggests a distinct difference between entropies of hexagonal and mon- oclinic crystallographic modifications. This difference is also evident in the molar volumes (fig. 11):Vmof the lanthanide trifluorides, calculated from the lattice constants, decreases lin-
160 R.J.M. KONINGS AND A. KOVÁCS Table 4
The corrected standard entropy values at 298.15 K for the monoclinic lanthanide(III) chlorides (in J·K−1·mol−1) as derived from the work of Tolmach et al. (Tolmach et al., 1987, 1990a, 1990b, 1990c; Gorbunov et al., 1986);Tminis
lower temperature limit of the measurements
Tmin Sexp(298.15 K−Tmin) Slat(Tmin) Sexs(Tmin) S◦(298.15 K)
DyCl3 0 169.6 0.00 5.76 175.4
HoCl3 6.61 170.1 0.25 6.74 177.1
ErCl3 9.86 168.1 1.25 5.76 175.1
TmCl3 15.34 166.5 3.35 3.68 173.5
YbCl3 0 163.5 0.00 5.76 169.3
Table 5
The crystal field energy levels for the lanthanide trichlorides (Morrison and Leavitt, 1982; Carnall, 1982)
compound state εi/cm−1
LaCl3 1S0 0
CeCl3 2F5/2 0, 37.5, 110
2F5/2 2166, 2208.6, 2282.6, 2399.5
PrCl3 3H4 0, 33.1, 96.4, 130.2, 137.0, 199.1
NdCl3 4I9/2 0, 115.4, 123.2, 244.4, 249.4
4I11/2 1973.85, 2012.58, 2026.90, 2044.19, 2051.60, 2058.90 PmCl3 2F7/2 0, 66.6, 84.5, 110.1, 127.0, 240.0
SmCl3 6H5/2 0, 40.7, 66.1
6H7/2 992.8, 1051.2, 1104.7, 1172.6
EuCl3 7F0 0
7F1 355,05, 405.27
7F2 1022.54, 1027.52, 1084.33
GdCl3 8F7/2 0
TbCl3 7F6 0, 56.8, 90.6, 97.2, 99.3, 104.6, 112.8, 118.0 DyCl3 6H15/2 0, 9.82, 9.97, 15.65, 40.8, 80.5, 121.7, 140.5
HoCl3 5I8 0, 12.5, 43.8, 66.4, 89.9, 104.1, 118.4, 154.2, 155.4, 203.7, 212.8 ErCl3 4F15/2 0, 37.9, 64.3, 96.5, 113.7, 141.6, 181.0, 229.43
TmCl3 3H6 0, 29, 92, 121, 127, 181, 193, 195, 207
YbCl3 3F7/2 0, 50, 185, 401
LuCl3 1S0 0
early along the series due to the lanthanide contraction with almost no difference between the two crystal structures, whereasVmof the lanthanide trichlorides shows a distinct difference between the hexagonal and the monoclinic compounds.
The data shown in fig. 10 are not the values reported by Gorbunov et al. (1986) and Tolmach et al. (1987, 1990a, 1990b, 1990c), because they did not extrapolate their mea- surements to 0 K in all cases. To deriveS◦(298.15 K) we have assumed that the heat ca- pacity of LuCl3 represents the lattice component, and Slat at the lower temperature limit is derived from the results for this compound. The excess contribution at this temperature is calculated from the crystal field energies (see table 5) derived from spectroscopic stud- ies of the ions in transparent host crystals (Dieke et al., 1968; Morrison and Leavitt, 1982;
THERMODYNAMIC PROPERTIES OF THE LANTHANIDE(III) HALIDES 161
Carnall, 1982). The ‘experimental’ standard entropy values thus obtained for the monoclinic lanthanide trichlorides are listed in table 4.
The entropies for those compounds for which no experimental data have been reported (CeCl3, PmCl3, TbCl3) are calculated according to the method outlined for the trifluorides.
The lattice component has been derived from linear interpolation between LaCl3and GdCl3
for the hexagonal compounds and a parallel relation fitted to the LuCl3value for the mon- oclinic compounds. The excess entropies have been calculated from the energy levels of the Ln3+ions. The calculated values for the standard entropy at 298.15 K (Stot) are in good agree- ment with the calculated values obtained from the sum of the lattice and excess contribution at 298.15 K, as shown in table 3. Because there is only one ‘reference’ point for the lattice contribution in the monoclinic series (LuCl3), the trend along this series is assumed to have the same slope as the hexagonal trichlorides.
3.3. LnBr3and LnI3
Only a few low-temperature heat capacity measurements have been reported for the lanthanide tribromides and triiodides. Deline et al. (1975) measured the low-temperature heat capacity of EuBr3, Gavrichev et al. (1992) of LuI3. These data are insufficient to derive the lattice component as was done for the trifluorides and the trichlorides. In addition, few data on the crystal field levels for the bromides and iodides are available (Morrison and Leavitt, 1982).
A number of assumptions thus had to be made. Figure 12 shows that the trends in the molar volume of the LnBr3and LnI3series indicate two groups, as is the case for the trichlorides.
We therefore conclude that the entropies of the two groups must be derived separately. The lattice entropies of the tribromides in the La–Eu series are obtained by subtractingSexsfrom the measurements for EuBr3 and these values are used to approximateSlat, assuming that this quantity will vary regularly through the orthorhombic and hexagonal series. An estimated slope for the relation ofSlatversus atomic number dependence was used. The same was done for rhombohedral triiodides using the experimental value for LuI3. The lattice entropy of the rhombohedral tribromides and the orthorhombic triiodides have been derived by an extrapo- lation of the experimental values for the lanthanum (including the value for LaBr3obtained indirectly from the experimental value of EuBr3) and lutetium trihalides, as shown in fig. 13.
The entropies of these compounds correlate perfectly with the logarithm of the molecular weight of the halide ion. The crystal field levels in the tribromides and triiodides are assumed to be the same as in the trichlorides, which can be justified by the experimental data for PrBr3,
Fig. 12. The molar volumes in the LnBr3() and LnI3series () at 298.15 K.
162 R.J.M. KONINGS AND A. KOVÁCS
Fig. 13. The standard molar entropy as a function of the logarithm of the molecular weight of the halide atom ln(M); LaX3(◦), LuX3() and EuX3( ); es- timated values are indicated byand.
NdBr3and ErBr3doped in LaBr3as given by Morrison and Leavitt (1982). The entropy data thus obtained are summarised in table 6.
4. High-temperature heat capacity of the solid trihalides