J. Phys. Chem. 1996, 100, 2353-2359
2353
Calorimetric Study of Phase Transitions in SeF6 and WF6 Crystals† Tomoko Ohta, Osamu Yamamuro, Takasuke Matsuo,* and Hiroshi Suga‡ Department of Chemistry and Microcalorimetry Research Center, Faculty of Science, Osaka UniVersity, Toyonaka, Osaka 560, Japan ReceiVed: July 17, 1995; In Final Form: October 19, 1995X
The heat capacities of SeF6 and WF6 were measured with an adiabatic calorimeter in the temperature range between 6 and 290 K. SeF6 and WF6 transformed from the crystalline form to the plastically crystalline and finally to the liquid form. Their temperatures and entropies are as follows: SeF6, Ttrs ) 134.50 ( 0.02 K, ∆trsS ) 28.12 ( 0.01 J K-1 mol-1, Tfus ) 239.24 ( 0.01 K, ∆fusS ) 21.87 ( 0.01 J K-1 mol-1; WF6, Ttrs ) 264.95 ( 0.02 K, ∆trsS ) 36.50 ( 0.01 J K-1 mol-1, Tfus ) 275.00 ( 0.02 K, ∆fusS ) 14.88 ( 0.01 J K-1 mol-1. The sublimation enthalpy of SeF6 was also measured at 205.10 K. The molar enthalpy and entropy of sublimation at 205.10 K were 24.96 ( 0.04 kJ mol-1 and 121.7 ( 0.2 J K-1 mol-1, respectively. The third-law entropy of SeF6 at 298.15 K and 105 Pa was calculated from the present results to be 312.9 ( 0.5 J K-1 mol-1. This value agrees well with the statistical value (313.6 ( 0.4 J K-1 mol-1) calculated from the spectroscopic data, indicating that SeF6 becomes completely ordered at 0 K. The transition entropies of SF6, SeF6, and WF6 were reproduced by a model considering the entropy associated with the excited levels of the rotational vibration. It was found that the large transition entropy of the MF6 family is due to the broad potential surface caused by the orientational frustration inherent to the bcc structure of the high-temperature phase.
1. Introduction One of the basic problems about plastic crystals1 is the relationship between the entropy S and molecular and crystal structures. In most of the transitions from an ordered crystalline phase to a plastic phase, the thermodynamic and structural properties are not related simply by classical Boltzmann’s law (∆S ) k ln(W1/W2), where W1 and W2 are the numbers of molecular orientations in the high- and low-temperature phases), even though this formula is frequently used for interpretation of the transition entropies of order-disorder transitions. An illustrative example is the plastic phases of MF6 (M: chalcogen, W, Mo, etc.). The molecules are octahedral in shape and crystallize in a bcc structure (space group Im3m), in which M atoms are located at m3m positions and the average directions of all the M-F bonds are parallel to the 〈100〉 axes.2-7 The crystal structure of the low-temperature phase of SF6 is monoclinic (space group C2/m),5,8 and those of other MF6 compounds are orthorhombic.9-14 The symmetry of the hightemperature phase could mean that the plastic phases are orientationally ordered, and so the transition entropies should be 0 from the classical Boltzmann’s law. This conjecture is irreparably contradicted by the experimentally observed large entropy of transition (23-33 J K-1 mol-1).15-20 It also has to be explained that the transition entropies depend on the compounds. Dove and Pawley21 suggested that the disorder in the SF6 crystal originates from the frustration caused by the attractive F‚‚‚F interaction between the nearest-neighbor molecules and the competing repulsive one between the next-nearest neighbors. The orientational distribution function of M-F bonds is actually much broader than that expected from their thermal vibration.2-7 In the present study, we have measured the heat capacities of SeF6 and WF6 and enthalpy of sublimation of SeF6. The †
Contribution No. 118 from the Microcalorimetry Research Center. Present address: Research Institute for Science and Technology, Kinki University, Kowakae, Higashi-Osaka 577, Japan. X Abstract published in AdVance ACS Abstracts, January 1, 1996. ‡
0022-3654/96/20100-2353$12.00/0
purposes of this study are to obtain precise thermodynamic properties on the phase transitions and to clarify the origin of the transition entropies of the MF6 family. SeF6 and WF6 were chosen since WF6 has a much larger M-F bond length and intermolecular distance than SF6, and SeF6 is intermediate between SF6 and WF6; the heat capacity and sublimation enthalpy of SF6, which has the shortest intermolecular distance in the MF6 family, have already been reported.15 The intermolecular distance may be an important factor that dominates the orientation frustration. A number of physicochemical studies (e.g., X-ray and neutron diffraction,2-6,8-12,22,23 NMR,24-27 neutron inelastic scattering,28,29 molecular dynamics29-37) have been performed for SF6 and WF6. The heat capacity of WF6 has also been measured by Weinstock et al.,17 but the numerical data of the heat capacity and the detail of the analysis on the transition (e.g., base-line determination) were not stated in their paper. For SeF6, only X-ray9 and NMR24,25 studies have been performed, but the transition temperatures determined by them were different from each other: 173 K for the former and 127 K for the latter. 2. Experimental Section Liquefied SeF6 and WF6 contained in gas cylinders were purchased from Tomoe-Shoukai Co., Ltd., and Kanto-DenkaKogyo Co., Ltd., respectively. Their purities were claimed to be 99.99% and 99.999%, respectively. From the fractionalmelting experiment carried out during the heat capacity measurement, the purities of the samples were confirmed to be better than 99.99% and 99.95%, respectively. The heat capacities were measured with a low-temperature calorimeter on the basis of the adiabatic principle. The sample is heated with an accurately measured electrical energy, and the resultant temperature increment is measured by the use of the precision thermometer. The heat capacity is given by the ratio of the energy input to the temperature increment. Further details are described elsewhere.38 For SeF6 a special type of calorimeter cell,39 which could hold the inner pressure up to 10 © 1996 American Chemical Society
2354 J. Phys. Chem., Vol. 100, No. 6, 1996
Ohta et al.
TABLE 1: Experimental Molar Heat Capacities of SeF6 (M ) 192.96 g mol-1, R ) 8.314 51 J K-1 mol-1) T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
5.85 6.19 6.54 6.90 7.26 7.62 7.98 8.35 8.71 9.07 9.43 9.79 10.18 10.57 10.98 11.41 11.86 12.32 12.81 13.31 13.83 14.36 14.91 15.48 16.05 16.63 17.21 17.77 18.40 19.07 19.75 20.46 21.21 22.03 22.91 23.81 24.72 25.63
0.11154 0.13510 0.16452 0.19660 0.23334 0.27324 0.31660 0.36434 0.41645 0.46820 0.52549 0.58685 0.65484 0.72550 0.80225 0.88628 0.97598 1.0725 1.1722 1.2789 1.3908 1.5033 1.6223 1.7449 1.8674 1.9876 2.1100 2.2230 2.3530 2.4842 2.6189 2.7509 2.8900 3.0311 3.1765 3.3197 3.4501 3.5767
26.53 27.43 28.33 30.17 31.12 32.10 33.09 34.09 35.10 35.90 37.19 38.35 39.46 40.56 41.64 42.71 43.77 44.81 45.86 46.90 47.94 48.98 50.02 51.07 52.13 53.21 54.30 55.41 56.50 57.58 58.66 59.73 60.80 61.87 62.94 64.01 65.07 66.13
3.6952 3.8173 3.9480 4.1577 4.2538 4.3627 4.4599 4.5450 4.6378 4.7001 4.8021 4.8862 4.9537 5.0321 5.1080 5.1762 5.2328 5.2900 5.3466 5.4101 5.4571 5.5231 5.5768 5.6304 5.6867 5.7429 5.8007 5.8554 5.9149 5.9744 6.0271 6.0883 6.1403 6.1988 6.2539 6.3083 6.3610 6.4207
67.19 68.25 69.30 70.37 71.42 72.46 73.51 74.55 75.60 76.04 77.42 78.81 80.38 81.59 82.80 84.02 85.24 86.46 87.69 88.92 90.15 91.39 92.63 93.87 95.12 96.37 97.63 98.89 100.16 101.42 102.70 103.97 105.25 106.54 107.83 109.12 110.42 111.72
6.4774 6.5355 6.5930 6.6549 6.7093 6.7718 6.8328 6.8888 6.9491 6.9794 7.0568 7.1343 7.2213 7.2952 7.3653 7.4486 7.5290 7.6136 7.6898 7.7585 7.8489 7.9400 8.0098 8.0978 8.2015 8.2840 8.3840 8.4616 8.5406 8.6447 8.7537 8.8304 8.9405 9.0326 9.1510 9.2307 9.3453 9.4559
113.03 114.34 115.65 117.03 118.46 119.90 121.35 122.80 124.26 125.73 127.20 128.67 130.15 131.64 133.13 134.20 134.53 134.55 134.60 134.74 135.50 137.00 138.66 140.41 142.27 144.18 146.11 148.04 149.99 151.96 153.96 155.97 157.99 160.02 162.06 164.11 166.18 168.25
9.5795 9.6913 9.8195 9.9459 10.076 10.227 10.352 10.497 10.650 10.797 10.976 11.132 11.339 11.520 11.938 73.048 3555.4 2330.1 1220.1 306.09 21.465 9.9713 9.9180 10.030 10.113 10.191 10.322 10.381 10.470 10.568 10.650 10.786 10.893 10.991 11.091 11.197 11.318 11.427
170.34 172.43 174.54 176.66 178.79 180.93 183.08 185.24 187.41 189.59 191.78 193.98 196.18 198.40 200.63 202.86 205.11 207.36 210.97 213.44 215.92 218.40 220.88 223.36 225.85 228.34 230.82 233.31 235.80 238.05 239.16 239.21 239.23 239.24 239.46 240.70
11.537 11.653 11.779 11.888 12.024 12.136 12.267 12.401 12.535 12.648 12.791 12.945 13.056 13.218 13.359 13.483 13.629 13.762 13.971 14.136 14.297 14.438 14.569 14.735 14.896 15.058 15.214 15.365 15.529 25.750 1490.2 6779.2 10644 11008 248.69 18.205
MPa, was used because the vapor pressure of SeF6 is 1.3 MPa at room temperature. For WF6, we used a normal type cell40 containing a inside copper vessel which is gold-plated and sealed without indium to avoid reaction between the sample and the cell. The sample was introduced into the calorimeter cell (for SeF6) and the inside container (WF6) by vacuum distillation after degassing, and then the transfer tube was pinched-off by soldering at the tip. The amount of SeF6 and WF6 loaded in the sample cell was 11.387 g (0.059 013 mol) and 39.593 g (0.132 93 mol), respectively. The dead space of the sample cell was filled with helium gas at 0.1 MPa and 77 K to enhance thermal equilibration at low temperature. The heat capacity measurement was carried out in the temperature range between 6 and 290 K by the standard intermittent heating method, as described elsewhere.41 For SeF6 the heat due to the vaporization of sample inside the calorimetric cell was corrected for by the use of the vapor-pressure42 and diffraction9 data in the manner described for SF6.15 The accuracy of the heat capacity measurement was better than 1% at T < 20 K, 0.3% at 20 < T < 30 K, and 0.1% at T > 30 K. The temperature measurement was performed by using RhFe resistance thermometers calibrated on the temperature scale EPT76 (T < 30 K) and IPTS68 (T > 30 K). The heat capacity difference caused by the conversion to the new temperature scale ITS9043 was estimated to be smaller than 0.05% over the temperature range 12-300 K. The sublimation enthalpy of SeF6 was measured at 205.10 K and 22.0 kPa using an adiabatic apparatus described elsewhere.15,44 The sample was sublimed in a thermally isolated condition in the cryostat. The amount of electrical energy
needed to keep the sample temperature constant is equal to the sublimation enthalpy. The amount of the sample sublimed was determined by collecting the sample vapor through a capillary tube connecting the cell to an external reservoir. The inaccuracy of the measurement was 0.1% of the total sublimation enthalpy. The temperature, 205.10 K, of the experiment was chosen because the sublimation pressure, 22.0 kPa, at this temperature was convenient for the measurement. In the previous measurement on SF6 the sublimation pressure was 18.6 kPa.15 Since WF6 is quite corrosive, the sublimation enthalpy of WF6 could not be measured with our apparatus. 3. Results and Discussion A. Heat Capacity. The molar heat capacities of SeF6 and WF6 are collected in Tables 1 and 2, respectively. They are plotted in Figure 1 together with the heat capacities of SF6.15 The heat capacities of SeF6 and WF6 are each shifted upward by 50 J K-1 mol-1 for the sake of clarity. For SeF6, a solidsolid phase transition and fusion occurred at 134.50 ( 0.02 and 239.24 ( 0.01 K, respectively. These temperatures are different from those determined by X-ray diffraction9 (173 and 238 K) and NMR24,25 (127 and 238 K) studies. For WF6, a solidsolid phase transition and fusion appeared at 264.95 ( 0.02 and 275.00 ( 0.02 K, respectively. These agree with the transition temperatures obtained by the adiabatic calorimetry by Weinstock et al. (264.7 and 275.2 K).17 To extract the excess enthalpy and entropy of the fusion, the normal heat capacity (base line) was determined by fitting two straight lines to the experimental heat capacities below and
Phase Transitions in SeF6 and WF6 Crystals
J. Phys. Chem., Vol. 100, No. 6, 1996 2355
TABLE 2: Experimental Molar Heat Capacities of WF6 (M ) 297.84 g mol-1, R ) 8.314 51 J K-1 mol-1) T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
T/K
Cs,m/R
6.15 6.48 6.90 7.35 7.82 8.30 8.80 9.31 9.81 10.33 10.89 11.52 12.20 12.93 13.73 14.63 15.58 16.55 17.58 18.70 19.90 21.14 22.39 23.64 24.88 26.17 27.51 28.87 30.28 31.75 33.22 34.74 36.27 37.75 39.25 40.76 42.26 43.74 45.20 46.66
0.10769 0.12852 0.15792 0.19431 0.23620 0.28492 0.33915 0.39883 0.46279 0.53218 0.61274 0.70505 0.81135 0.92904 1.0602 1.2107 1.3739 1.5423 1.7222 1.9173 2.1231 2.3329 2.5401 2.7381 2.9305 3.1245 3.3233 3.5361 3.7379 3.9325 4.1067 4.2926 4.4874 4.6711 4.8257 4.9881 5.1552 5.3187 5.4757 5.6288
48.11 49.55 51.00 52.44 53.87 55.31 56.74 60.39 61.53 62.67 63.83 65.01 66.19 67.39 68.60 69.82 71.06 72.30 73.55 74.82 78.28 79.77 81.28 82.78 84.30 85.83 87.36 88.90 90.46 92.02 93.58 95.16 96.75 98.34 99.95 101.56 103.22 104.91 106.62 108.33
5.7789 5.9327 6.0798 6.2376 6.3838 6.5350 6.6878 7.0214 7.1318 7.2362 7.3477 7.4627 7.5705 7.6784 7.7862 7.8870 7.9866 8.0812 8.1720 8.2614 8.5235 8.6306 8.7479 8.8752 8.9960 9.1062 9.2312 9.3573 9.4705 9.5784 9.6924 9.8099 9.9241 10.036 10.151 10.264 10.370 10.489 10.607 10.720
110.06 111.80 113.55 115.31 117.08 118.87 120.66 122.47 124.28 126.11 127.95 129.81 131.67 133.55 135.43 137.33 139.24 141.16 143.09 145.04 146.99 148.95 150.94 152.94 154.95 156.98 159.03 161.09 163.16 165.24 167.34 169.45 171.58 173.72 175.86 178.03 180.20 182.38 184.58 186.79
10.834 10.944 11.048 11.160 11.275 11.385 11.497 11.606 11.718 11.829 11.934 12.032 12.142 12.253 12.365 12.478 12.579 12.673 12.779 12.895 13.004 13.096 13.201 13.312 13.426 13.531 13.635 13.738 13.859 13.970 14.069 14.185 14.300 14.421 14.540 14.690 14.830 14.903 15.039 15.086
189.02 191.26 193.51 195.78 198.05 200.34 202.64 204.95 207.27 209.66 212.02 214.41 216.81 219.23 221.66 224.12 226.59 229.07 231.57 234.09 236.62 239.17 241.73 244.30 246.89 249.48 252.06 254.64 257.21 259.78 262.33 264.11 264.70 264.82 264.89 264.93 264.95 264.95 264.95 264.95
15.207 15.319 15.437 15.553 15.681 15.801 15.925 16.065 16.188 16.320 16.440 16.571 16.709 16.844 16.980 17.128 17.268 17.418 17.566 17.720 17.891 18.074 18.257 18.442 18.673 18.901 19.145 19.427 19.747 20.189 21.033 75.180 565.05 1223.0 1970.6 2761.5 52515 52818 52860 53150
264.98 265.17 265.91 266.59 267.35 268.05 268.81 269.52 270.28 271.01 272.51 273.81 274.45 274.60 274.70 274.78 274.83 274.87 274.90 274.93 274.95 274.97 274.99 275.01 275.04 275.06 275.09 275.12 275.14 275.25 276.30 278.21 280.13 282.06 283.99 286.22 288.63 291.04
2104.1 273.89 18.541 18.566 18.618 18.672 18.712 18.750 18.802 18.829 18.953 32.699 130.03 221.11 320.26 429.83 569.33 833.73 1031.8 1194.9 1395.4 1588.6 1688.9 1043.6 1080.1 1091.4 990.88 1027.9 1035.4 135.53 20.276 20.315 20.352 20.396 20.429 20.513 20.574 20.661
Figure 1. Molar heat capacities of SF615 (triangles), SeF6 (squares), and WF6 (circles). Those of SeF6 and WF6 are each shifted upward by 50 J K-1 mol-1 for the sake of clarity.
above the fusion, allowing a gap of the base line at the fusion temperature. The temperature regions used for the fitting were 230.9-234.8 and 240.6-244.3 K for SeF6 and 266.6-272.5
and 276.3-286.2 K for WF6. The molar enthalpies and entropies of the fusion thus determined were 5.232 ( 0.002 kJ mol-1 and 21.87 ( 0.01 J K-1 mol-1 for SeF6 and 4.091 ( 0.002 kJ mol-1 and 14.88 ( 0.01 J K-1 mol-1 for WF6. The values for WF6 agree with those obtained by the previous calorimetry.17 B. Enthalpy of Sublimation. All of the results concerning the sublimation experiment of SeF6 are summarized in Table 3. The average value of the molar enthalpy and entropy of sublimation at 205.10 K were 24.96 ( 0.04 kJ mol-1 and 121.7 ( 0.2 J K-1 mol-1, respectively. C. Third-Law Entropy of SeF6. The process of calculation of the third-law standard molar entropy of SeF6 gas is summarized in Table 4. The first line gives the entropy increment between 0 and 5 K. The heat capacity down to 0 K was calculated by the model function used for the base line in the determination of the transition enthalpy and entropy as described in the next section. Lines 2 to 5 are self-explanatory. The sixth line is the correction for gas imperfection at temperature Tsub and pressure psub, made by using the Berthelot equation and critical parameters reported elsewhere.45 The seventh line was calculated by assuming the harmonic oscillation for the intramolecular vibration and using the reported spectroscopic data.46,47 The eighth line, the molar entropy change for compression of the ideal gas to the standard pressure 105 Pa, was calculated by R ln{psub/(105 Pa)}. The third-law standard molar entropy of ideal gas SeF6 at 298.15 K and 105 Pa was determined to be 312.9 ( 0.5 J K-1 mol-1.
2356 J. Phys. Chem., Vol. 100, No. 6, 1996
Ohta et al.
TABLE 3: Experimental Sublimation Enthalpy of SeF6a Tsub/K 205.096 205.111 205.107
Q/J
∆Ecorr/J
V/mg min-1
∆t/min
m/g
638.550 747.952 749.077
-0.184 -0.899 -1.321
31.8 22.5 24.5
155 257 236
4.933 5.784 5.774
average values at 205.10 K
∆subHm/kJ mol-1
∆subSm/J K-1 mol-1
24.970 24.922 24.989
121.75 121.51 121.83
24.96 ( 0.04
121.7 ( 0.2
Tsub, sublimation temperature; Q, supplied electric energy; ∆Ecorr, correction term for heat leakage; V, sublimation rate; ∆t, total time of the measurement; m, mass of the sample collected in the ampule; ∆subHm, molar enthalpy of sublimation; ∆subSm, molar entropy of sublimation. a
TABLE 4: Standard Molar Entropy of SeF6 (g) ∆S/J K-1 mol-1 0-5.00 K (extrapolation) 5.00-130.15 K (integration of Cs/T) 130.15-139.50 K (transition) 139.50-205.10 K (integration of Cs/T) 205.10 K, 2.20 × 104 Pa (sublimation) 205.10 K, 2.20 × 104 Pa (correction to ideal gas) 205.10-298.15 K, 2.20 × 104 Pa (integration of Cp/T) 298.15 K, 2.20 × 104-105 Pa (compression)
0.182 103.076 26.471 37.132 121.7 0.197 36.741 -12.589
S°298 (this study)
312.9 ( 0.5
S°298 (spectroscopic study)48
313.6 ( 0.4
This value agrees well with the statistical entropy calculated from the spectroscopic data48 within experimental error. This indicates that the residual entropy of SeF6 is 0 and the lowtemperature phase of SeF6 is completely ordered at 0 K, as in the case of SF6. D. Base-Line Determination. To evaluate the excess heat capacity due to the transition, a model function was fitted to the experimental heat capacities for temperatures where they were considered to be free from the effect of the transition. It was assumed to have the following form:
Cp ) Clat(3) + Clib(3) + Cvib(15) + ∆Ccorr
Figure 2. Molar heat capacity of SeF6 and the contributions from the vibrational degrees of freedom determined from the least squares fitting (see text for the details): (A) intramolecular vibration, (B) libration, (C) lattice vibration, (D) Cp - CV correction.
(1)
where Clat, Clib, and Cvib are the heat capacities due to the translational lattice vibration, rotational vibration (libration), and intramolecular vibration, respectively. The numbers in the parentheses denote the degrees of freedom included in each term. The last term in eq 1 gives the correction for the difference between Cp and CV, which is represented as usual by
∆Ccorr ) ACp2T
(2)
where A is a constant and T the temperature. Clat was approximated by a Debye function and Clib by an Einstein function, each having 3 degrees of freedom. Cvib due to intramolecular vibrations was represented by a combination of 15 Einstein functions. All of the frequencies of the intramolecular vibrations were known from Raman and infrared spectroscopic data for the gaseous state.46,47,49 The Debye temperature θD for the lattice vibration, the Einstein temperature θE for the libration, and the correction coefficient A were determined by least-squares fitting to the experimental heat capacities in the temperature ranges 5.9-28.3 and 170.3-205.1 K for SeF6 and 6.2-55.3 and 265.9-272.5 K for WF6. The present analysis assumes that the base line is continuous at the transition temperature. The optimum values were θD ) 72.0 K, θE ) 77.4 K, and A ) 4.26 × 10-6 mol J-1 for SeF6 and θD ) 74.6 K, θE ) 103.0 K, and A ) 3.70 × 10-6 mol J-1 for WF6. The fitting was satisfactory for both samples. Figure 2 shows the experimental heat capacities of SeF6 and the base line determined by the fitting. The contribution from each part is also shown by A (intramolecular vibration), B (libration), C (lattice vibration),
Figure 3. Excess entropies due to the transitions of SF615 (triangles), SeF6 (squares), and WF6 (circles).
and D (Cp - CV correction). It is noteworthy that the heat capacity due to the lattice vibration and libration are saturated to the classical values below the transition temperature. This result supports the validity of our assumption of a continuous base line at the transition temperature because the remaining contribution (intramolecular vibration) is not affected much by the transition. E. Transition Entropy. Figure 3 shows the temperature dependence of the excess molar entropy due to the transitions of SF6,15 SeF6, and WF6. The saturated values, the total transition entropies, of SeF6 and WF6 are 28.12 ( 0.01 and 36.50 ( 0.01 J K-1 mol-1, respectively. The transition entropy for WF6 are about 10% larger than that reported by Weinstock et al. (32.7 J K-1 mol-1).17 This is probably due to the difference in the base line employed.
Phase Transitions in SeF6 and WF6 Crystals
J. Phys. Chem., Vol. 100, No. 6, 1996 2357
F. Origin of the Transition Entropy of the MF6 Family. As shown above, SF6, SeF6, and WF6 undergo phase transitions with large transition entropies, even though, as pointed out in the Introduction, the high temperature phases may arguably be ordered from the structural point of view. The experimental transition entropy depends on the central atom of the octahedron molecule. These observations can not be explained by the classical Boltzmann’s law that considers only the number of distinct molecular orientations. In this section, we will describe a calculation of the transition entropy of SF6 using a quantum mechanical method and investigate the origin of the transition entropy of the MF6 family. SF6 was chosen because the potential energy for the rotational motion as well as the required thermodynamic data has been reported. As the rotational potential in the high-temperature phase, we employed the mean single-molecule potential determined by Dove and Pawley21 in a molecular dynamic simulation, in which Lennard-Jones type F‚‚‚F interaction was assumed for the nearest- and next-nearest-neighbor molecules on the bcc lattice. The mean potential function (Figure 7a of ref 21) has a minimum at the orientation of the molecule in which the S-F bond is oriented along the 〈100〉 direction. The potential surface is quite broad around the minimum. This is consistent with the result of the neutron diffraction study2-5 and reflects the orientational frustration.21 We approximate the three-dimensional rotation by the combination of three degrees of onedimensional rotation about the 4-fold axis for simplicity of the calculation. The 4-fold axis was taken since the potential barrier about the 4-fold axis (ca. 6 kJ mol-1) was considerably smaller than those about the 2- and 3-fold axes. Numerical values of the potential as a function of the rotational angle were read off from Figure 7a of ref 21 and fitted to the following polynomial function: -1
V(θ)/J mol
) aθ + bθ + cθ + dθ 8
6
4
2
(3)
where θ is the rotation angle in radians. The coefficients determined are a ) 1.2402 × 105, b ) -1.9476 × 105, c ) 8.7530 × 104, and d ) 9.9333 × 102. It has to be pointed out that the angular potential fluctuates significantly about the mean function as a result of the orientational frustration.21 However, its use for the calculation of the single-molecule rotational energy levels may be acceptable, since our purpose is to evaluate the thermodynamic quantities averaged over the energy levels. The energy levels En are obtained as eigenvalues of the following Schro¨dinger equation: 2 h2 d ψ(θ) + {En - V(θ)}ψ(θ) ) 0 8π2I dθ2
(4)
where ψ(θ) is the wave function; h, Planck’s constant; and I ()3.086 × 10-45 kg m2), the moment of inertia of a SF6 molecule.50 This differential equation was solved by numerical integration using the Runge-Kutta method.51 The energy levels above the potential barrier were assumed to be those of the free rotation with the same origin of energy as that of V(θ); that is,
En )
h2n2 8π2I
(5)
Figure 4. Potential functions and calculated energy levels for the rotational motion of SF6 about the 4-fold axis: (a) low-temperature phase, (b) high-temperature phase.
The rotational entropy in the high-temperature phase was then calculated by
[
Srot,h(T) ) 3R ln{∑ exp(-En/kT)} +
]
∑(En/kT) exp(-En/kT) ∑ exp(-En/kT)
(6)
where T is the temperature; R, the gas constant; and k, the Boltzmann constant. The rotational potential in the low-temperature phase was approximated to be that of the harmonic vibration,
V(θ) )
2π2Ik2θE2 2
h
θ2
(7)
Here the Einstein temperature θE is equal to 77.5 K, as described elsewhere.15 The rotational entropy in the low-temperature phase was calculated by
[{
Srot,1(T) ) 3R ln
}
]
(θE/T) exp(-θE/T) 1 + 1 - exp(-θE/T) 1 - exp(-θE/T)
(8) This assumption does not seriously affect the result of the following calculation since the potential barrier, 6 kJ mol-1 ()720 K), is much higher than the transition temperature, 95 K.
The potentials and the energy levels in the high- and lowtemperature phases are shown in Figure 4. The potential function in the high-temperature phase is much broader than
2358 J. Phys. Chem., Vol. 100, No. 6, 1996
Ohta et al. TABLE 6: Molar Thermodynamic Functions of WF6 (M ) 297.84 g mol-1, R ) 8.314 51 J K-1 mol-1, Φ°m ) ∆0TS°m ∆0TH°m/T) T/K
Figure 5. Temperature dependence of the calculated rotational entropy of SF6 in the low-temperature phase (dashed line) and the hightemperature phase (solid line). The difference at Ttrs represents the rotational part of the transition entropy.
TABLE 5: Molar Thermodynamic Functions of SeF6 (M ) 192.96 g mol-1, R ) 8.314 51 J K-1 mol-1, Φ°m ) ∆0TS°m ∆0TH°m/T) T/K
C°p,m/R
∆T0 H°m/RT
5 0.0674 0.0164 10 0.6233 0.1528 15 1.641 0.4740 20 2.665 0.8967 25 3.489 1.337 30 4.139 1.752 40 4.992 2.465 50 5.576 3.032 60 6.101 3.499 70 6.633 3.908 80 7.200 4.284 90 7.838 4.643 100 8.531 4.998 110 9.308 5.354 120 10.24 5.721 130 11.32 6.107 phase transition at 134.50 K 140 10.00 8.780 150 10.47 8.878 160 10.99 8.993 170 11.52 9.126 180 12.09 9.275 190 12.67 9.438 200 13.32 9.616 210 13.91 9.807 220 14.52 10.01 230 15.16 10.22 fusion at 239.24 K
∆T0 S°m/R
Φ°m/R
0.0218 0.2006 0.6392 1.255 1.943 2.638 3.957 5.137 6.199 7.179 8.102 8.986 9.848 10.70 11.55 12.41
0.00542 0.04782 0.1652 0.3579 0.6053 0.8860 1.492 2.105 2.700 3.271 3.818 4.343 4.851 5.344 5.825 6.298
15.64 16.34 17.04 17.72 18.39 19.06 19.73 20.39 21.06 21.72
6.858 7.467 8.044 8.593 9.118 9.624 10.11 10.59 11.05 11.50
that in the low-temperature phase, and separations between the energy levels in the high-temperature phase are much narrower than those in the low-temperature phase. Figure 5 shows the temperature dependence of rotational entropies in the high- and low-temperature phases. The difference between rotational entropies at the transition temperature (95 K) corresponds to the transition entropy ∆Srot of the rotational part. ∆Srot thus calculated is 26.5 J K-1 mol-1. In first-order transitions with a large volume change, as in the present case, the entropy due to the volume change, ∆Sv, should be considered. This part can be evaluated by the equation52
∆Sv ) (R/κ)∆trsV
(9)
where R is the cubic expansion coefficient; κ, the compress-
C°p,m/R
∆T0 H°m/RT
5 0.0704 10 0.4937 15 1.276 20 2.136 25 2.958 30 3.690 40 4.908 50 5.984 60 6.995 70 7.886 80 8.672 90 9.422 100 10.15 110 10.83 120 11.45 130 12.05 140 12.62 150 13.16 160 13.68 170 14.22 180 14.75 190 15.25 200 15.79 210 16.34 220 16.89 230 17.47 240 18.13 250 18.95 260 20.26 phase transition at 264.95 K 270 18.78 fusion at 275.00 K 280 20.35 290 20.62
∆T0 S°m/R
Φ°m/R
0.0176 0.1307 0.3774 0.7095 1.078 1.454 2.170 2.827 3.438 4.012 4.546 5.046 5.520 5.973 6.404 6.815 7.210 7.589 7.953 8.306 8.649 8.984 9.310 9.632 9.949 10.26 10.58 10.89 11.23
0.0235 0.1765 0.5188 1.004 1.570 2.175 3.410 4.622 5.803 6.950 8.055 9.120 10.15 11.15 12.12 13.06 13.98 14.86 15.73 16.58 17.40 18.21 19.01 19.79 20.57 21.33 22.09 22.84 23.61
0.00588 0.04575 0.1414 0.2941 0.4917 0.7215 1.239 1.795 2.365 2.939 3.510 4.074 4.631 5.178 5.717 6.246 6.765 7.276 7.777 8.270 8.754 9.231 9.700 10.16 10.62 11.07 11.51 11.95 12.38
15.32
28.20
12.89
17.23 17.34
30.70 31.42
13.48 14.08
ibility; and ∆trsV, the volume change at the transition. Using the compressibility53 and the diffraction data5,8,9 reported, ∆Sv of SF6 was calculated to be 5.60 J K-1 mol-1. The total calculated entropy is therefore 32.1 J K-1 mol-1. This value is consistent with the observed one (23.37 J K-1 mol-1), taking the effect of the rough approximations employed into consideration. To investigate the effect of the central atoms on the transition entropy, we calculated the transition entropies of SeF6 and WF6 using the potential function of SF6 and neglecting the effect of the volume entropy; the molecular identities were represented by the moments of inertia54,55 (3.569 × 10-45 and 4.251 × 10-45 kg m2 for SeF6 and WF6, respectively) and Einstein temperatures in the low-temperature phase (77.4 and 103.0 K, respectively). The rotational entropies for SeF6 and WF6 were determined to be 27.8 and 37.5 J K-1 mol-1, respectively. These values roughly agree with the observed ones and are consistent with the relation between the transition entropy and the size of the central atom; the transition entropy is larger for the larger central atom. To estimate the effect of the approximation that the rotational potential of SF6 is used for SeF6 and WF6, we made a similar calculation using the rotational potential of SF6 where the rotational angle was scaled by a factor dS-F/dW-F; θW ) (dS-F/dW-F)θS. The rotational transition entropy of WF6 thus calculated was 33.3 J K-1 mol-1. Thus, the elongation of the M-F bond decreases the calculated transition entropy considerably. Nevertheless, this does not change our conclusion that a larger central atom gives a larger value for the transition entropy. The present calculation may be improved with respect to the dimensionality of the rotation. The one-dimensional model used in the present calculation exaggerates the contribution from the low-lying energy levels. This results in a larger transition
Phase Transitions in SeF6 and WF6 Crystals entropy and may explain the discrepancy between the experiment and calculation. Three-dimensional hindered rotation presents a formidable problem, requiring a full theoretical consideration. This has not been solved yet. The present calculations have shown that the large transition entropies of the MF6 family are not the configurational entropy but have a dynamical (rotational) origin. In other words, the transition entropy is not associated with the multiplicity of the ground levels but with a large number of narrowly separated excited levels. This situation is caused by the orientation frustration and resultant broad potential function around the potential minima. The large moment of inertia of the MF6 molecules is also responsible for the dense distribution of the levels. The significance of the present calculation lies no more in the exact comparison between the experimental and theoretical results than in the possibility raised by this model for the dynamical origin of the transition entropy. The importance of the dynamical entropy is clearly evident for the present substances on account of the crystal symmetry. But it may be equally important in other systems, including those classified as plastically crystalline substances, and constitute, along with the discrete orientational entropy, a significant part of the transition entropies of orientationally disordered crystals. Appendix The Standard Thermodynamics Functions. The molar heat capacities, enthalpies, entropies, and Giauque functions of SeF6 and WF6 were calculated from the smoothed heat capacity data and are summarized in Tables 5 and 6, respectively. Extrapolation of the heat capacity down to 0 K was performed by using the base line for the phase transition. References and Notes (1) For example: Sherwood, J. N. The Plastically Crystalline State, Orientationally-disordered Crystals; John-Wiley & Sons: Chichester, 1979. Parsonage, N. G.; Staveley, L. A. K. Disorder in Crystals; Clarendon Press: Oxford, 1978. Press, W. Single-Particle Rotation in Molecular Crystals; Springer-Verlag: Berlin, 1981. (2) Dolling, G.; Powell, B. M.; Sears, V. F. Mol. Phys. 1979, 37, 1859. (3) Powell, B. M.; Dolling, G. Mol. Cryst. Liq. Cryst. 1979, 52, 27. (4) Taylor, J. C.; Waugh, A. B. J. Solid State Chem. 1976, 18, 241. (5) Cockcroft, J. K; Fitch, A. N. Z. Kristallogr. 1988, 184, 123. (6) Levy, J. H.; Taylor, J. C.; Wilson, P. W. J. Less-Common Met. 1976, 45, 155. (7) Levy, J. H.; Sanger, P. L.; Taylor, J. C.; Wilson, P. W. Acta Crystallogr. 1975, B31, 1065. (8) Dove, M. T.; Powell, B. M.; Pawley, G. S.; Bartell, L. S. Mol. Phys. 1988, 65, 353. (9) Michel, J.; Drifford, M.; Rigny, P. J. Chim. Phys. 1970, 67, 31. (10) Siegel, S.; Northrop, D. A. Inorg. Chem. 1966, 5, 2187. (11) Levy, J. H.; Taylor, J. C.; Wilson, P. W. J. Solid State Chem. 1975, 15, 360. (12) Levy, J. H.; Taylor, J. C.; Waugh, A. B. J. Fluorine Chem. 1983, 23, 29. (13) Levy, J. H.; Taylor, J. C.; Wilson. P. W. Acta Crystallogr. 1975, B31, 398.
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