The effective volumes of waters of crystallization: general organic solids.

Using a list of compatible hydrate/anhydrate pairs prepared by van de Streek and Motherwell [CrystEngComm (2007), 9, 55-64], we have examined the effective volume per water of crystallization for 179 pairs of organic solids using current data from the Cambridge Crystallographic Structural Database (CSD). The effective volume is the difference per water molecule between the asymmetric unit volumes of the hydrate and parent anhydrate, and has the mean value 24 Å3. The conformational changes in the reference molecule between the hydrate and its anhydrate are shown in two figures: one for a relatively rigid standard organic molecule and (in the supplementary file) one for a more flexible linear molecule. Using data from Nyman and Day [Phys. Chem. Chem. Phys. (2016), 18, 31132-31143], we have also established a generic volumetric coefficient of thermal expansion of organic solids with a value of 147 ± 56 × 10-6 K-1. There is a significant number of outliers to the data, negative, near zero, and large and positive. Some explanation for the existence of these outliers is attempted.

Apatite Thermochemistry: The Simple Salt Approximation.

Drouet has recently comprehensively reviewed the thermochemistry of phosphate apatites, M10(PO4)6X2. On the basis of the data assembled, he has established an optimized additive set of data values based upon the component ions, M2+, P2O5, or PO43-, and halide or OH- ions, which yield phosphoapatite enthalpy and Gibbs energy values generally within 1% of their experimental values. In this paper, we introduce and compare the Simple Salt Approximation (SSA) for generation of the same thermodynamic values by addition of values for the component salts of the apatite. SSA is widely applicable to ionic systems so that familiarity with this procedure is worthwhile. Drouet's additive values focused on phosphoapatites yield better results (at 1%) than does the more general SSA (at ∼3%), but the SSA provides an alternative to data prediction when the Drouet additive factors are not available. In addition, we here use the SSA to generate approximate entropy values and estimate the Debye temperature of phosphoapatites under ambient conditions, yielding insight into the extent of activation of their vibrational modes.

The effective volumes of waters of crystallization: non-ionic pharmaceutical systems.

The physical properties of organic solids are altered when hydrated (and, more generally, when solvated) and this is of particular significance for pharmaceuticals in application; for instance, the solubility of a hydrate is less than that of its parent. The effective volumes of waters of crystallization for non-ionic pharmaceuticals (where the `effective' volume is the difference per water molecule between the hydrate volume and the volume of the anhydrous parent) are here examined. This investigation contrasts with our earlier study of effective volumes of waters of crystallization for ionic materials where the coulombic forces are paramount. Volumetric properties are significant since they correlate strongly with many thermodynamic properties. Twenty-nine hydrate/parent systems have been identified, and their volumetric properties are reported and analysed (apart from aspartame and ephedrine for which the structural data are inconsistent). Among these systems, the data for paracetamol are extensive and it is possible to differentiate among the volumetric properties of its three polymorphs and to quantify the effect of temperature on their volumes. The effective volumes in both ionic and non-ionic systems are similar, with a median effective volume of 22.8 Å3 for the non-ionic systems compared with 24.2 Å3 for the ionic systems, and both are smaller than the molecular volume of 30 Å3 of ambient liquid water - which appears to be an upper limit to the effective volumes of waters of crystallization under ambient conditions. These results will be supportive in checking and confirmation of hydrated crystal structures and in assessing their thermodynamic properties.

Systematic Thermodynamics of Layered Perovskites: Ruddlesden-Popper Phases.

Perovskite, CaTiO3, is the prototype of an extensive group of materials. They are capable of considerable chemical modification, with the further capability of undergoing structural modification by the intercalation of thin sheets of intrusive materials (both inorganic and organic) between the cubic perovskite layers, to form a range of "layered" perovskites. These changes bring about alterations in their electronic, structural, and other properties, permitting some "tuning" toward specific ends. This paper collects the limited known thermodynamic data for layered perovskites of various chemical compositions and demonstrates by example that the thermodynamic layer values are substantially additive. This additivity may be exploited by summing properties of the constituent oxides, by adding differences between adjacent compositions within a series, or even by substitution of oxides for one another, thus permitting prediction beyond the known range of compositions. Strict additivity implies full reversibility so that the additive product may be unstable and may undergo structural changes, producing materials with new and potentially useful properties such as ferroelectricity, polarity, giant magnetoresistance, and superconductivity.

Cohesive Energies and Enthalpies: Complexities, Confusions, and Corrections.

The cohesive or atomization energy of an ionic solid is the energy required to decompose the solid into its constituent independent gaseous atoms at 0 K, while its lattice energy, Upot, is the energy required to decompose the solid into its constituent independent gaseous ions at 0 K. These energies may be converted into enthalpies at a given temperature by the addition of the small energies corresponding to integration of the heat capacity of each of the constituents. While cohesive energies/enthalpies are readily calculated by thermodynamic summing of the formation energies/enthalpies of the constituents, they are also currently intensively studied by computational procedures for the resulting insight on the interactions within the solid. In supporting confirmation of their computational results, authors generally quote "experimental" cohesive energies which are, in fact, simply the thermodynamic sums. However, these "experimental" cohesive energies are quoted in many different units, atom-based or calorimetric, and on different bases such as per atom, per formula unit, per oxide ion, and so forth. This makes comparisons among materials very awkward. Additionally, some of the quoted values are, in fact, lattice energies which are distinctly different from cohesive energies. We list large numbers of reported cohesive energies for binary halides, chalcogenides, pnictogenides, and Laves phase compounds which we bring to the same basis, and identify a number as incorrectly reported lattice energies. We also propose that cohesive energies of higher-order ionic solids may also be estimated as thermodynamic enthalpy sums.

Predictive thermodynamics for ionic solids and liquids.

The application of thermodynamics is simple, even if the theory may appear intimidating. We describe tools, developed over recent years, which make it easy to estimate often elusive thermodynamic parameter values, generally (but not exclusively) for ionic materials, both solid and liquid, as well as for their solid hydrates and solvates. The tools are termed volume-based thermodynamics (VBT) and thermodynamic difference rules (TDR), supplemented by the simple salt approximation (SSA) and single-ion values for volume, Vm, heat capacity, , entropy, , formation enthalpy, ΔfH°, and Gibbs formation energy, ΔfG°. These tools can be applied to provide values of thermodynamic and thermomechanical properties such as standard enthalpy of formation, ΔfH°, standard entropy, , heat capacity, Cp, Gibbs function of formation, ΔfG°, lattice potential energy, UPOT, isothermal expansion coefficient, α, and isothermal compressibility, ß, and used to suggest the thermodynamic feasibility of reactions among condensed ionic phases. Because many of these methods yield results largely independent of crystal structure, they have been successfully extended to the important and developing class of ionic liquids as well as to new and hypothesised materials. Finally, these predictive methods are illustrated by application to K2SnCl6, for which known experimental results are available for comparison. A selection of applications of VBT and TDR is presented which have enabled input, usually in the form of thermodynamics, to be brought to bear on a range of topical problems. Perhaps the most significant advantage of VBT and TDR methods is their inherent simplicity in that they do not require a high level of computational expertise nor expensive high-performance computation tools - a spreadsheet will usually suffice - yet the techniques are extremely powerful and accessible to non-experts. The connection between formula unit volume, Vm, and standard thermodynamic parameters represents a major advance exploited by these techniques.

Thermodynamic consistencies and anomalies among end-member silicate garnets.

Materials with the garnet crystal structure include silicate minerals of importance both in geology, on account of their use in geothermobarometry, and industrially as abrasives. As a consequence of the former, there is considerable published thermodynamic information concerning them. We here examine this thermodynamic information for end-member silicate garnets (some of which are synthetic since not all occur in nature) for consistencies and anomalies, using thermodynamic relations between thermodynamic properties that we have established over recent years. The principal properties of interest are formula volume, heat capacity, entropy, formation enthalpy (from which the Gibbs energy may be obtained), and isothermal compressibility. A significant observation is that the ambient-temperature heat capacities of the silicate garnets are rather similar, whereas their ambient-temperature entropies are roughly proportional to their formula volumes. Evaluation of their Debye temperatures implies that their vibrational contributions to heat capacity are fully excited at ambient temperatures. The relatively small isothermal compressibilities of these garnets is related to the rigidity of their constituent silicate tetrahedra. We here establish additive single-ion values for each of the thermodynamic properties, which may be applied in estimating corresponding values for related materials.

Ambient heat capacities and entropies of ionic solids: a unique view using the Debye equation.

Entropies of solids are obtained experimentally as integrals of measured heat capacities over the temperature range from zero to ambient. Correspondingly, the Debye phonon distribution equation for solids provides a theoretical connection between these two chemical thermodynamic measures. We examine how the widely applicable Debye equation illuminates the relation between the corresponding experimental measures using more than 250 ionic solids. Estimation of heat capacities for simple ionic solids by the Dulong-Petit heat capacity limit, by the Neumann-Kopp elemental sum, and by the ion sum method is examined in relation to the Debye equation. We note that, and explain why, the ambient temperature heat capacities and entropies of ionic solids are found to be approximately equal, and how deviations from equality may be related to the Debye temperature, ΘD, which characterizes the Debye equation. It is also demonstrated that Debye temperatures may be readily estimated from the experimental ratio of ambient heat capacity to entropy, C(p)/S(p), rather than requiring resort to elaborate theoretical or experimental procedures for their determination. Correspondingly, ambient mineral entropies and heat capacities are linearly correlated and may thus be readily estimated from one another.

Single-ion values for ionic solids of both formation enthalpies, Δ(f)H(298)(ion), and Gibbs formation energies, Δ(f)G(298)(ion).

Formation enthalpies, Δ(f)H(298), are essential thermodynamic descriptors of the stability of materials, with many available from the numerous thermodynamic databases. However, there is a need for predictive methods to supplement these databases with missing values for known and even hypothetical materials, and also as an independent check on the not-always reliable published values. In this paper, we present 34 additive single-ion values, Δ(f)H(298)(ion), from the formation enthalpies of 124 ionic solids, including an extensive group of silicates. In addition, we have also developed an additive set of 29 single-ion formation Gibbs energies, Δ(f)G(298)(ion), for a smaller group of 42 materials from within the full set, constrained by the limited availability of the corresponding experimental data. Such single-ion values may be extended among related materials using simple differences from known thermodynamic values, but always with critical consideration of the results. Using the excellent available data for silicates, we propose that the solid-state silicate ion formation enthalpies can be estimated as -Δ(f)H(298)(silicate)/kJ mol(-1)= -252[n(Si) + n(O)] - 27, where n(X) represents the number of species X in the silicate. More speculatively, we estimate the contribution per silicon and oxygen species as -490 and -184 kJ mol(-1), respectively. Similarly, -Δ(f)G(298)(silicate)/kJ mol(-1)= -266[n(Si) + n(O)] - 7, with the contribution per silicon and oxygen species being -140 and -300 kJ mol(-1), respectively. We compare and contrast these results with the extensive collection of "modified lattice energy" (MLE) ion parameters from the M.S. thesis of C. D. Ratkey. Our single-ion formation enthalpies and the MLE parameters may be used in complementary predictions. While lattice energies, U(POT), entropies, S(o)(298), and heat capacities, C(p,298), of ionic solids are reliably estimated as proportional to their formula volumes (using our Volume-Based Thermodynamic, VBT, procedures), this is not the case in general for thermodynamic formation properties, other than within select groups of related materials.

Simple route to lattice energies in the presence of complex ions.

Lattice energies for ionic materials which separate into independent gaseous ions can be calculated by standard Born-Haber-Fajans thermochemical cycle procedures, based on the energies of formation of those ions. However, if complex ions (such as sulfates) occur in the material, then a sophisticated calculation procedure must be invoked which requires allocation of the total ion charge among the atom components of the complex ion and evaluation of the attractive and repulsive energy terms. If, instead, the total ion charge is allocated to the central atom of the complex ion (with zero charge on the coordinated atoms), to create a "condensed charge ion" (having zero self-energy), then a straightforward calculation of the electrostatic (Madelung) energy, E(M)', correlates well with published lattice energies: U(POT)/kJ mol(-1) = 0.963E(M)', with a correlation coefficient, R(2) = 0.976. E(M)' is here termed the "condensed charge" electrostatic (Madelung) energy. Thus, using the condensed charge ion model, we observe that a roughly constant proportion (∼96%) of the corresponding lattice energy arises from the electrostatic interaction terms. The above equation permits ready evaluation of lattice energies for ionic crystal structures containing complex ions, without the necessity to estimate any of the problematic nonelectrostatic interaction terms. A commentary by Prof. H. D. B. Jenkins substantiating this analysis is appended.

Single-ion heat capacities, C(p)(298)ion, of solids: with a novel route to heat-capacity estimation of complex anions.

Single-ion heat capacities, C(p)(298)(ion), are additive values for the estimation of room-temperature (298 K) heat capacities of ionic solids. They may be used for inferring the heat capacities of ionic solids for which values are unavailable and for checking reported values, thus complementing our independent method of estimation from formula unit volumes (termed volume-based thermodynamics, VBT). Analysis of the reported heat-capacity data presented here provides a new self-consistent set of heat capacities for both cations and anions that is compatible (and thus may be combined) with an extensive set developed by Spencer. The addition of a large range of silicate species permits the estimation of the heat capacities of many silicate minerals. The single-ion heat capacities of individual silicate anions are observed to be strictly proportional to the total number of atoms (Si plus O), n, contained within the silicate anion complex itself (e.g., for the anion Si(2)O(7)(2-), n = 9, for SiO(4)(2-), n = 5), C(p)(silicate anion)/J K(-1) mol(-1) = 13.8n, in a new rule that is an extension of the Neumann-Kopp relationship. The same linear relationship applies to other homologous anion series (for example, oxygenated heavy-metal anion complexes such as niobates, bismuthates, and tantalates), although with a different proportionality constant. A similar proportionality, C(p)(complex anion)/J K(-1) mol(-1) ≈ 17.5n, which may be regarded as a convenient "rule of thumb", also applies, although less strictly, to complex anions in general. The proportionality constants reflect the rigidity of the complex anion, being always less than the Dulong-Petit value of 25 J K(-1) mol(-1). An emergent feature of our VBT and single-ion approaches to an estimation of the thermodynamic properties is the identification of anomalies in measured values, as is illustrated in this paper.

Solid-state energetics and electrostatics: Madelung constants and Madelung energies.

The Madelung constants of ionic solids relate to their geometry and electrostatic interactions. Furthermore, because of issues in their evaluation, they are also of considerable mathematical interest. The corresponding Madelung (electrostatic, coulomb) energy is the principal contributor to the lattice energies of ionic systems, and these energies largely influence many of their physical properties. The Madelung constants are here defined and their properties considered. A difficulty with their application is that they may be defined relative to various lattice distances, and with various conventions for inclusion of the charges, leading to possible confusion in their use. Instead, the unambiguous Madelung energy, E(M), is to be preferred in chemistry. An extensive list of Madelung energies is presented. From this data set, it is observed that there is a strong linear correlation between the lattice energies of ionic solids, U(POT), and their Madelung energies: U(POT)/kJ mol(-1) = 0.8519E(M) + 293.9. This correlation establishes that the lattice energy, U(POT), for ionic solids is about 15% smaller than the attractive Madelung energy, the difference arising from the repulsions unaccounted for by the solely coulombic Madelung energy calculation. Correlations of U(POT) against E(M) for alkali metal hydrides and transition metal compounds, each having considerable covalency, show much reduced Madelung contributions to the lattice energy. These correlations permit ready estimation of lattice energies, and are the first to be based on actual data rather than a broad analysis. The independent volume-based thermodynamic (VBT) method, which relies on a separate correlation with the formula unit volume of the ionic material, complements these correlations.

Ambient isobaric heat capacities, C(p,m), for ionic solids and liquids: an application of volume-based thermodynamics (VBT).

Thermodynamic properties, such as standard entropy, among others, have been shown to correlate well with formula volume, V(m), thus permitting prediction of these properties on the basis of chemical formula and density alone, with no structural detail required. We have termed these procedures "volume-based thermodynamics" (VBT). We here extend these studies to ambient isobaric heat capacities, C(p,m), of a wide range of materials. We show that heat capacity is strongly linearly correlated with formula volume for large sets of minerals, for ionic solids in general, and for ionic liquids and that the results demonstrate that the Neumann-Kopp rule (additivity of heat capacity contributions per atom) is widely valid for ionic materials, but the smaller heat capacity contribution per unit volume for ionic liquids is noted and discussed. Using these correlations, it is possible to predict values of ambient (298 K) heat capacities quite simply. We also show that the heat capacity contribution of water molecules of crystallization is remarkably constant, at 41.3 ± 4.7 J K(-1) (mol of water)(-1), so that the heat capacities of various hydrates may be reliably estimated from the values of their chemical formula neighbors. This result complements similar observations that we have reported for other thermodynamic differences of hydrates.

Volume-based thermoelasticity: consequences of the (near) proportionality of isothermal compressibility to formula-unit volume.

Groups of structurally related materials, including the alkali halides, exhibit a proportionality of isothermal compressibility to formula-unit volume. The relationship has recently been explored by Glasser and by Recio et al. In this paper, we present the consequences of such proportionality on the relationships of Born-Landé and Born-Mayer parameters to the formula-unit volume. These relationships have then been tested separately on (i) alkali (excluding cesium) halides and (ii) cesium halides. We conclude that the equations fit the NaCl-type materials satisfactorily, but less well for the CsCl-type materials, and that the Born-Mayer equation is more applicable. These results confirm the conclusion that volume is intimately linked to thermodynamic quantities, as already demonstrated by our development of volume-based thermodynamics (VBT).

Volume-based thermoelasticity: compressibility of inorganic solids.

Thermodynamic properties such as entropy, among others, have been shown to correlate well with formula volume, thus permitting prediction of these properties on the basis of chemical formula and density alone, with no structural detail required. We here extend these studies to the thermoelastic property of isothermal compressibility, beta. We show that compressibility is strongly linearly correlated with formula volume per atom pair, V(pr), for binary solids, with the alkali halides having a proportionality constant of 0.908 GPa(-1) V(pr)(-1) while 1:1 monoxides, monochalcogenides, monopnictides, and chalcopyrites (ABX(2), which may be considered as AX plus BX) have a common compressibility proportionality constant of 0.317 GPa(-1) V(pr)(-1). Oxides with closely packed oxygen lattices (such as Al(2)O(3)), garnets (such as Y(3)Fe(5)O(12) = 4M(2)O(3)), spinels (MgAl(2)O(4) = MgO.Al(2)O(3)), and other oxides (e.g., FeTiO(3) = FeO.TiO(2)) have compressibilities which are only slightly dependent on volume, at about 0.108 GPa(-1) V(pr)(-1) + 0.003 GPa(-1).

Systematic thermodynamics of Magneli-phase and other transition metal oxides.

Both the molar enthalpies of formation and the absolute entropies of eight transition metal oxides are found to correlate very strongly with their formula unit volumes at room temperature. The metals are Ti, V, Cr, Nb, Mo, Ce, Pr, and Tb. In particular, the thermodynamic values of additive entities (such as TiO(2) in Ti(n)O(2n-1)) in Magneli phases (that is, recombination phases based on rebuilding after shear) are very close to those of the entity as a pure compound. Thus, reliable values of these thermodynamic properties can readily be predicted for unmeasured or even unsynthesized examples, and literature values can be checked. These assertions are checked against published results for which incomplete data is available. The contributions of the disordered regions which form between the added entities is tentatively estimated.

Single-ion entropies, S(ion)(o), of solids--a route to standard entropy estimation.

Single-ion standard entropies, S(ion)(o), are additive values for estimation of the room-temperature (298 K) entropies of ionic solids. They may be used for inferring the entropies of ionic solids for which values are unavailable and for checking reported values, thus complementing the independent method of estimation from molar volumes (termed volume-based thermodynamics). Current single-anion entropies depend on the charge of the countercation, and so are difficult to apply to complex materials, such as minerals. The analysis of reported data here presented provides a self-consistent set of entropies for cations and charge-independent values for anions. Although the S(ion)(o) values presented encompass only a limited set of ions, the retrieval of values for ions not listed is straightforward and is described. An unexpected and significant observation is that cation entropies are related to the molar volumes of the corresponding (neutral) condensed-phase metals.

Systematic thermodynamics of hydration (and of solvation) of inorganic solids.

The thermodynamics of the formation of solid and liquid inorganic hydrates and ammoniates is examined. In earlier studies, average values of the Gibbs energy of reaction, Delta(r)G, assuming a constant additivity per mole of bound water, have been obtained and have suggested that hydration is always marginally thermodynamically favorable. More detailed consideration now demonstrates that the mean value of Delta(r)G per mole of water, from anhydrous parent to hydrate within a sequence, increases consistently toward zero, becoming progressively less favorable as the degree of hydration, n, increases, and broadly independent of any structural features of the materials. Furthermore, the consistent behavior suggests that missing intermediate hydrates in hydrate sequences are likely to be thermodynamically stable, even if difficult to prepare, isolate, or measure.The behavior of ammoniates is similar but less regular, the irregularity being ascribed to a wider range of interactions within the solid ammoniates than in the hydrates. The "Ostwald Rule of Stages" suggests that the first precipitate from a supersaturated solution is usually a metastable phase, having an intermediate value of the Gibbs energy between that of the anhydrous parent and of the thermodynamically stable phase, then progressing to the stable phase with the lowest Gibbs energy, implying kinetic rather than thermodynamic control of the sequence of precipitation. The implications for hydrate formation are briefly considered.