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Predictive Thermodynamics for Condensed Phases and Nanotechnology

Thermodynamic relations may be regarded as the motor driving chemical processes (while chemical kinetics represents the brakes). In order to operate, the relations need to be powered with the fuel of values for the thermodynamic quantities concerned, such as enthalpies, entropies, heat capacities, Gibbs and Helmholtz energies, and so forth. The following discussion considers how these important values may be obtained.

The ultimate source of the requisite values is experiment, in procedures such as calorimetry, electrochemistry and equilibrium studies and, latterly, even computational modelling of systems and processes. Resulting therefrom, there are available extensive databases from which necessary data can often be obtained. Unfortunately, however, these are insufficient because data for the desired materials may be unavailable (or conflicting) or the materials themselves may even be hypothetical, never having been synthesised.

Furthermore, experimental thermodynamics is a demanding and complex discipline requiring specialist equipment and expertise. Consequently, this is currently unfashionable science and unlikely to yield much data in times to come. Computational modelling, although certainly fashionable, itself relies on experimental bases for confirmation of its results.

Consequently, there is considerable incentive to establish relations among currently known experimental quantities, which will then permit prediction of desired values. Indeed, there is a long history of seminal predictions which have, themselves, developed our understanding of the properties of materials. Consider, for example, the Dulong-Petit Rule which states that the molar heat capacity, Cp, of monatomic metals approximates 25 J K-1 mol-1, from which, in due course, there developed the Einstein and Debye heat capacity relations.

Empirical predictive methods have been classified by us1 in terms of increasing number of the contributing units as:

  • Zero order: predictive methods independent of any molecular parameters, and so possibly quite rough: e.g., Dulong and Petit’s Rule for the molar heat capacity of monatomic metals (~25 J K-1 mol-1); Trouton’s Rule for the entropy of vaporisation of organic molecules (~85 J K-1 mol-1); Richards's Rule for the entropy of fusion of rigid spherical molecules (~7-14 J K-1 mol-1), Walden's Rule for the entropy of fusion of rigid non-spherical molecules (~20-60 J K-1mol-1).
  • First order: thermodynamic properties predicted from molecular units, depending then only on chemical formula, molar volume (or density) and charge: e. g., entropy/volume relations.
  • Second order: additivity of atomic or ionic properties: e.g., molar volume, magnetic susceptibility.
  • Third order: additivity of bond properties: e.g., enthalpies, entropies, heat capacities, oxide components of a mineral.
  • Fourth order: additivity of group properties: e.g., heat capacities, energies (and enthalpies) of formation, absolute entropies, enthalpies and entropies of fusion, vaporization and dissolution, densities (and, hence, molar volumes), and many others.

An increasing number of empirical (fitted) parameters is required in this sequence, as one proceeds from zero order upwards: thus, zero order requires only a single parameter value for each given class of materials, first order requires a relation to a single property for a group of materials, second order for each atom type, third order for each bond type, and fourth order for each type of group constituting the material.

Predictive First Order Relations

Volume-Based Themodynamics (VBT)1-3

Over the last dozen or so years, colleagues and I have developed a number of first order empirical relations based on formula unit volumes (or, equivalently, density) which have proven to be exceptionally useful in terms of their simplicity and broad generality, and are now quite widely applied to bulk condensed phases. These relations (termed Volume-Based Thermodynamics – VBT) are here introduced, illustrated and discussed, followed by a brief consideration of their possible application to nanomaterials. A feature of the relations is that they are generally independent of crystallographic structure which means that they can often be applied knowing only the chemical formula of the relevant material.

The simplest of our VBT relations is a linear correlation of entropies of condensed phases to formula volumes (Fig. 1 for organic liquids4 and Fig. 2 for ionic solids,5 with parameters listed in Table 1).

So298 / J K-1 mol-1 = k (Vm / nm3 molecule-1) + c = k’ ([M/ρ] / cm3 mol-1) + c

where ρ represents density.

Entropy,So298, versus molar volume,Vm, for 1495 organic liquids.

Figure 1. Entropy,So298, versus molar volume,Vm, for 1495 organic liquids.

Entropy,So298, versus molar volume,Vm, for 132 anhydrous and hydrated ionic solids.

Figure 2. Entropy,So298, versus molar volume,Vm, for 132 anhydrous and hydrated ionic solids.

We have also considered6 the relation of heat capacity, Cp(298 K), to Vm. In this case, if the materials are separated into suitable classes (silicate minerals, other inorganics including hydrates, ionic liquids) then good linear relations are observed.

We have recently extended these ideas to thermoelasticity. We have observed good linear relations between isothermal compressibility, β, and Vm for many materials7-8 - provided that they are grouped into appropriate classes, essentially based upon the tightness of their ion packings.

Table 1: Regression coefficients for linear relations between thermodynamic property (entropy, heat capacity and isothermal compressibility) and formula unit volume for a variety of materials.



Material Class Linear Regression Coefficients Correlation coefficient, R2

/ J K-1 mol-1 nm-3

/ J K-1 mol-1

Entropy,5 S298o

ionic inorganic

1360 ± 56

15 ± 6


ionic hydrates

1579 ± 30

6 ± 6



1262 ± 28

13 ± 5


Entropy,4 S298o

organic liquids

1133 ± 7

44 ± 2


organic solids

774 ± 21

57 ± 6


Heat capacity,6 Cp(298)

silicate minerals

(no frameworks)




ionic inorganic








ionic liquids




H2O (crystal)

41.3 ± 4.7

m / GPa-1 nm-3 per ion pair

c / GPa-1

Compressibility,7-8 β

alkali halides


MO, MS, MN, MP, MAs, MSb


close-packed oxides




structure-related values

Charge Effects

Lattice energy, UPOT, which is the energy required to remove the ionic constituents of a condensed phase into its independent gaseous ions, is the energetic property most amenable to correlational analysis since it is independent of other considerations (in contrast to formation enthalpy, which depends on the properties of the forming elements). In this case, the lattice energy has an inverse dependence on Vm1/3 because the energy increases as the ions approach one another. This Vm-1/3 dependence was first reported, for 1:1 binary ionic materials only, by Mallouk, et al.9 In order to develop the relations further, account must be taken for the fact that the energy will increase as the charges increase.

Glasser10 found that charge effects may be catered for by using the ionic strength factor, I = ½Σnizi2, which allows the correlations to be extended to ionic materials of essentially any complexity. A complication is that there is an alteration in behaviour as the lattice energy increases. At lower values (< 5000 kJ mol-1),UPOT has an inverse linear relation with Vm1/3 but approaches a limiting relation containing no adjustable parameters at larger lattice energies (see Fig. 3 and Table 2).11

It is interesting to note that the same charge dependency, using a related ionic strength factor, S (= 2I), was earlier noted by Templeton12 - but the combination of charge with volume effects was not pursued.

Figure 3: Lattice energy relations for 125 ionic solids and minerals (applicable to both solids and liquids).
Ionic strength

Lattice energy relations for 125 ionic solids and minerals (applicable to both solids and liquids). Ionic strength,

Table 2: Ionic strength factor, I, and coefficients, α and β, for various stoichiometries in the lattice energy equation: UPOT = 2IVm-1/3 + β) ( < 5 000 kJ mol-1).13

salt (charge ratio)

ionic strength factor, I

α / kJ mol-1 nm

β / kJ mol-1

correlation coefficient, R2

MX (1:1)





MX2 (2:1)





M2X (1:2)





MX (2:2)





MpXq (q:p)

½(pq2 + qp2)




Thermodynamic Difference Rules (TDR),14 Single Ion Sums, and Isomegethic Rules15

Thermodynamic Difference Rules are a form of additive relations whereby solvated materials are related by the addition or subtraction of solvate molecules; for example, the contribution of water of crystallization to hydrates may be treated as additive.16-18 As an alternative to correlational analysis of thermodynamic properties, it is also possible to use simple ion additive relations: that is, of ion volumes19 and of ion entropies.20

Isomegethic Rules15 rely on the substitution of ions whereby related materials are compared with one another on the basis that their formula volumes are similar; this implies that thermodynamic properties, such as entropies and lattice energies, are also likely to be similar.

Predictive Thermodynamics applied to Nanomaterials

It is inevitable that the energetics of nanomaterials differ from those of their bulk counterparts because of the prominent effects of surfaces in nanomaterials. There is as yet very little in the way of experimental thermodynamic values.21 In the following, we suggest a new approach to estimating nanoparticle lattice energies and make some suggestions as to entropy estimation.

It has very recently been observed22 that lattice energies of ionic solids are about 15% smaller than their electrostatic (Madelung) energies.

While it is not possible to determine the lattice energies of nanoparticles directly, it is possible to calculate their electrostatic energies quite simply, by direct summation, on the assumption that the crystal structure at the surface remains unaltered from that of the bulk.23-25 While that seems a large assumption, it is probable that deviations will have only small effects on the overall electrostatic energy. If we now further assume that the 15% difference between lattice and Madelung energies remains (approximately) true even for nanoparticles, we have a first estimate of their lattice energies.

Estimation of entropies is more problematic. At the deepest level, entropy is a bulk property of matter, and so the entropy of an individual nanoparticle may well be undefined. However, our interest will rather be in a collection of nanoparticles, and it is the relationships between such particles which will largely determine the entropy of that collection. Perhaps the only current conclusion will be that the entropy will be rather larger than that of the bulk material but not nearly as large as that of the free-flowing liquid.


We have shown that there are methods available whereby the thermodynamic properties of condensed phases may be quite reliably estimated. The extension to nanomaterials has not yet been developed, but there are indications as to how that extension may be approached.


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