Thermodynamic studies of disorder in inorganic solids

One of the best methods of investigating disorder in crystals is the thermodynamic method, in which the calorimetric entropy calculated from a study of the heat capacity from low temperatures is compared with the entropy obtained by an independent method, such as statistical calculation or a study of some equilibrium involving the compound under consideration. In this work, the thermodynamic method has been used to investigate three distinct problems concerned with disorder.
The first problem is concerned with the size of the entropy difference between two forms of the same compound, having different crystal structures. The interest in this topic lies in its connection with the entropy change in an order-disorder transition. A disordered phase of a substance possesses a configurational entropy of R ln W, where W is the number of energetically equivalent configurations that each formula unit can adopt, and a large part of the entropy change accompanying the transition to an ordered form of the same substance will be the configurational term, R ln W. The ordered and disordered forms will, however, differ slightly in their crystal structure, and consequently the lattice-vibrational frequency spectrum will not be the same for the two polymorphs. The configurational entropy term will therefore be accompanied by an entropy term arising from changes in the lattice-vibrational frequency spectrum. The easiest way of assessing this last term would be tomeasure the entropy difference between two polymorphs of the smae substance where questions of disorder do not arise, and any entropy difference is caused solely by differences in the lattice vibrations. Two such polymorphs are calcite and argonite which are respectively the trigonal and orthorhombic forms of calcium carbonate, CaCo3. The results and conclusions of this study are set out later in the abstract.
The second problem is one that has interested mathematicians, physicists and chemists for some time, and is to find the number of ways that dimers , occupying two lattice cells, can be arranged on a three-dimensional lattice. This problem has a bearing on theories of gas adsorption on solids and theories of the liquid state. The two dimensional case has been solved rigidly by topological algebra (refs 1 and 2) and by random walk methods (ref 3), the number of configurations being 1.791623 per dimer; an approximate solution has been found for the three dimensional case (ref 4) which suggests that the number of configurations should be about 2.38 per dimer, but no rigid mathematical solution has yet been put forward. A rather unusual chemical compound, fusible white precipitate, Hg(Nh3)2Cl2 has a structure in which 'rods' of NH3-Hg2+-NH3 (corresponding to the dimers in the classical case) are randomly orientated parallel to three mutually perpendicular axes, on a cubic lattice of chloride ions. On cooling this compound to low temperatures, there are in principle two things that can happen; either the structure will remain disordered, in which case comparison of the calorimetric entropy with that obtained by an equilibrium study will reveal residual entropy, or there will be an order-disorder transition, with an accompanying entropy change. From the entropy change, or residual entropy, the number of configurations can be obtained. The result of this investigation is given later in the appendix.
The final problem with which this work was concerned was part of an investigation into whether or not the hydrates of sodium carbonate were disordered. Comparison of the calorimetric entropies, and the entropies obtained from an equilibrium study, of Na2CO3.H20 and Na2CO3.10H20 measured by a previous worker (ref 5) appeared to show that the figure for S° for anhydrous Na2CO3, on which the equilibrium study entropies of the hydrates depended, was too low, if the anhydrous salt was in fact completely ordered at room temperature. The calorimetric entropy of Na2CO3 was therefore remeasured and the results are summarised later in this abstract.
In order to obtain an accurate value of the calorimetric standard entropies of the substances mentioned above, namely calcite, aragonite, diammino mercuric chloride and anhydrous sodium carbonate, precise measurements of the low temperature heat capacities of these substances had to be carried out. To enable this to be done, an adiabatic, low-temperature calorimeter capable of measuring heat capacities from a little above the boiling point of liquid helium to room temperature, was constructed. The construction, with the exception of some early stages in the wiring, was entirely carried out in the first year of this work.
The calorimeter is similar to those recently constructed in Oxford and elsewhere (no commercial model being, as yet, obtainable) but certain features are worthy of mention. The sample container is made of platinum-rhodium alloy, rather than the more usual copper, to enable compounds that attack copper to be investigated. Adiabatic conditions are maintained by a metal shield around the sample container, the temperature being controlled by automatic, rather than manual, control of the shield heater. It was found that this system functioned best when the shield was of light weight and consequently of small wall thickness. This precluded bringing the leads to the sample container into thermal equilibrium with the shield by carrying them in groves in the shield wall, and instead they were carried in commercially available, alumina packed, metal sheathed tubes, soldered to the inside of the shield. The calorimeter is designed for use with liquid nitrogen, liquid hydrogen and liquid helium, and the whole of the temperature range is covered without the need to pump on the refrigerants.
The heat capacity of the empty sample container was measured, and also the heat capacity and standard entropy of benzoic acid. The latter values agreed with those found by other workers (refs 6, 7, 8) to within 0.2% above 30°K and to 0.75% below 30°K. S° was found to be 40.05 e.u. compared with 40.054 e.u. (ref 6) and 40.04 e.u. (ref 7), showing that there does not seem to be any systematic error in the measurements.
The heat capacities, and standard calorimetric entropies of calcite and aragonite were measured and the values confirmed the early, rather imprecise measurements of Anderson (ref 9) who measured the heat capacities from 56°K upwards. The measurements in this work extended from 10°K in the case of calcite, agreeing with other early measurements (refs 10, 11), and from 20°K for aragonite giving values that differed considerably from those of Güntler (ref 12), whose results were also inconsistent with those of Anderson. The standard caloriraetiic entropies were found to be
Aragonite   S° =   21.03 e.u.
Calcite   S° =   21.92 e.u.
showing an entropy difference between the two polymorphs, of 0.89 e.u. This agrees within experimental error with the difference found by the early calorimetric workers , and also with the value of 0.74 &pm; 0.2 e.u. found from an equilibrium study (ref 13), showing that the polymorphs are equally ordered.
The entropy difference between the polymorphs arises from differences in their lattice-vibrational spectra. An attempt was made to calculate the heat capacities of calcite and aragonite from spectral data, but this met with only partial success for several reasons. Although the lattice frequencies for calcite are well assigned, and indeed one spectroscopist has said that 'calcite is the touchstone in the application of modern theories of spectra to the solid state' , even here, three frequencies are inactive in both the Raman and the i/r and use has had to be made of estimated values. For aragonite, only some of the modes of vibration have had frequencies assigned, and the information on lattice frequencies is too sketchy to form the basis of any calculations. The optical modes of vibration of calcite were treated as Einstein functions and the acoustic modes as Debye Functions, and although agreement with experimental heat capacities was satisfactory over most of the range, the calculated values were too low at very low temperatures and also above 200°K.
It is felt that the discrepancy at high temperatures arises from neglecting: anharmonicity, and the low values at low temperatures are thought to arise because pure Einstein functions do not truely represent the relation between vibrational modes and heat capacity; Einstein functions very rapidly fall to low values as the temperature drops.
Although no values could be calculated for aragonite, to compare with the calculated calcite values, it is felt that such calculations would reveal that the modes of torsional oscillation of the carbonate ion in aragonite had higher frequencies and contributed less to the heat capacity than in calcite. In calcite, the carbonate ions lie in planes midway between adjacent planes of calcium ions, and the CO32− oxygens do not lie directly over or under adjacent Ca2+ ions, whereas in aragonite, the carbonate ions are near the lower plane of Ca2+ ions, and the CO32− oxygens lie directly under the calcium ions in the upper plane.
The heat capacity of Hg(NH3)2Cl2 was measured from 3°K to room temperature, measurements below 12°K being carried out in a calorimeter built by another worker. No transition or thermal anomaly was observed which agrees with the behaviour predicted from the Ising model (ref 14). Hence disorder of the dimer units would appear to persist to the absolute zero, and no co-operative transition involving ammonia molecules would seem to have occurred. Here Hg(NH3)2Cl2 is at variance with certain other ammines, the heat capacities of which have been studied; Ni(NH3)6I2 for example has recently been shown (ref 15) to undergo a co-operative transition involving the ammonia molecules at 19.5°K, and also displays two other, superimposed transitions at 0.35°K, one an anti-ferromagnetic transition involving the unpaired nickel d electrons and the other a Schottky effect arising from quantum mechanical tunnelling through the potential energy barrier to hindered rotation. The latter transition could in principle occur at a very low temperature in Hg(NH3)2Cl2, in which case presumably each ammonia molecule would contribute R ln 2 to the transitional entropy, as is the case for the nickel salt.
If this transition does not occur, then then the standard calorimetric entropy, found in this work to be 57.31 e.u. should be lower than the value for the standard entropy obtained by an equilibrium study by R ln W, where W is the number of configurations of the NH3-Hg-NH3 dimers on a three-dimensional lattice. Another worker is determining S° from the temperature coefficient of the e.m.f. of the cell
&ominus;   Hg(NH3)2Cl2(c) ⊕
Hg(l)   | NH4Cl(aq)   |
|
|
|
|   AgCl(c)   |   Ag(c)
NH3(aq) |
|
the cell reaction being
Hg(l) + 2AgCl(S) + 2NH3 (aq) &longrightarrow; Hg(NH3)2Cl2(S) + 2Ag(c)
and the standard entropies of the other reactants are known. These measurements are not yet complete.
Although many studies of this type, where S° obtained from heat capacity measurements is compared with S° obtained from an equilibrium study, have been carried out on hydrate systems, this is believed to be the first study of this type on an ammoniate.
The calorimetric standard entropy of anhydrous sodium carbonate was found to be 32.26 e.u. Taken in conjunction with Waterfield's values (ref 5) for the standard entropies of the mono- and deca- hydrate, this determination shows that the monohydrate is completely ordered at the absolute zero, and that the decahydrate possesses residual entropy of 1.50 &pm; 0.5 e.u. The results are best summarised in a table.
Compound   Standard Entropy   Calorimetric Entropy
Na2CO3   32.26 &pm; 0.2 e.u.
Na2CO3.H2O   40.16 &pm; 0.2 e.u.*   40.19 &pm; 0.2 a.u.
Na2CO3.10H2O   134.93 &pm; 0.3 e.u.*   133.43 &pm; 0.2 e.u.
* Obtained from measuring the enthalpy changes and dissociation pressures of water in the reactions
Na2CO3.H2O(c) + 9H2O(g) &longrightarrow; Na2CO3.10H2O(c)
Na2CO3(c) + H2O(g) &longrightarrow; Na2CO3.H2O(c)
The residual entropy in the decahydrate is interpreted in terms of disorder in some of the hydrogen bonding. The thesis concludes with a brief summary of studies of residual entropy in hydrates.