Title:

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 orderdisorder 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 latticevibrational 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 latticevibrational 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, CaCo_{3}. 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 threedimensional 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(Nh_{3})_{2}Cl_{2} has a structure in which 'rods' of NH_{3}Hg^{2+}NH_{3} (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 orderdisorder 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 Na_{2}CO_{3}.H_{2}0 and Na_{2}CO_{3}.10H_{2}0 measured by a previous worker (ref 5) appeared to show that the figure for S° for anhydrous Na_{2}CO_{3}, 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 Na_{2}CO_{3} 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, lowtemperature calorimeter capable of measuring heat capacities from a little above the boiling point of liquid helium to room temperature, was constructed.
