Thermochemistry of Monocharged Cation Substitutions in Ionic Solids

There is considerable interest in the synthesis of new materials with properties adapted to particular technological purposes such as for electronics, catalysis, energy storage, and biomaterials. A rather straightforward synthetic procedure is substitution of principal ions in the crystal for one another and some general principles relating to structure have been developed which guide the material designer as to which substitutions are likely to lead to stable materials. These are brie�y reported. We here consider the thermochemical consequences, rather than the structural implications, of unit-charged cation substitutions, namely of NH 4+ , Li + , Na + , K + , Rb + , Cs + , Tl + , Ag + and Cu + , among a large group of ionic solids. It is observed that the formation enthalpies hardly differ among these materials while their absolute entropies, heat capacities and formula unit volumes follow similar linear trends but with some relative displacements.


Introduction
The connection between chemical composition and crystal structure of ionic solids is enshrined in a number of well-established empirical relationships, commencing with Mitscherlich's [1] isomorphism "law" (~ 1820) that substances with the same crystal structure often have similar chemical formulae.Obvious examples are the alkali halides with common face-centred crystal structures (apart from the simple cubic CsCl, CsBr and CsI).Goldschmidt [2] developed three rules relating to the possibility of ions replacing one another based on the relative sizes of the ions, their charges and the strength of the interactions between cations and anions.[3] Ringwood [4,5] has further elaborated these rules.In addition, Hume-Rothery[6] established relations among metals in the formation of alloys as solid solutions and these relations have recently been further developed.[7] Structure maps basically provide a two-dimensional coordinate system (based on what are termed Mendeleev numbers -that is, down the groups of the Periodic Table : Li + to Cs + ) whereby materials with similar properties are likely to be grouped.[8] These empirical rules have aided chemists by providing guidance towards the synthesis of materials with interesting physical and electrical properties.[9,10] For example, Hautier, et al., (2010) [11] have used machine learning methods based on the ICSD experimental crystal structure database [12] to predict novel ternary compositions that are most likely to form a compound and their most-probable crystal structures, leading to a structure map (their Fig. 2).In a further step, Hautier, et al. (2011), [13] have developed a quantitative probabilistic method for the discovery of new compounds by substitution of cations within known materials.These authors provide in their Supplementary Information an extensive list of pair correlation factors, gAB, where larger values imply greater likelihood of substitution of one cation with another with unchanged crystal structure.This capability of an atom to replace another in a particular crystal structure is a process [14] sometimes referred to as "diadochy".

Thermochemistry of Cation Substitution
The studies noted above have focussed on the equivalence of the crystal structures of the substituted materials and on their stability.Iin an alternative approach, Liebman and colleagues have considered the changes in thermochemical values when potassium and ammonium ions are substituted for each other.[19][20][21][22] Using data from the Wagman NBS tables [23] for ions in both solid and aqueous phases, they observed that the heat capacities of 20 materials were related by the tted linear relation: + ) − 45.749 with R 2 = 0.992, while the entropies of 47 similar materials were related by S(K + ) /J K − 1 mol − 1 = 0.998×Cp(NH 4 + ) -12.635 with R 2 = 0.997 (cf.their Figs. 1 and 2).These relations are little altered if the materials are separated into solid state and aqueous medium as indicated by the large values of R 2 .The remarkable feature of these results is that the slopes are nearly equal to one while the intercepts alter greatly from zero -implying that the structures are largely undisturbed except for the introduction of an ion of different thermodynamics.
In support and extension of this work, we here consider the relations between monocharged ion-substituted materials based on a comparison of their thermochemistries; namely, their enthalpies, entropies, heat capacities and formula volumes.Data from the literature for materials containing the unit-charged cations NH 4 + , Li + , Na + , K + , Rb + , Cs + , Tl + , Ag + and Cu + were extracted from the extensive database of "HSC Chemistry" [24] and is supplied in the Supporting Information le.The anions of the materials cover a wide range which also includes some hydrates.
Figure 1 plots the formation enthalpies of up to 26 materials (some data is missing) containing the above-listed cations against the formation enthalpy of the sodium-substituted versions.Note that the choice of the sodium-substituted materials as reference is purely arbitrary but convenient because it comprises a very wide-ranging set.It may be observed that there is quite remarkable linear alignment of the formation enthalpies of the alkali metal-substituted materials against one another with slopes close to unity while the formation enthalpies of the remaining monocharged cation-substituted materials (Tl + , Cu + and Ag + ) are somewhat elevated to less negative values, implying lesser energetic stability.This has the surprising implication that the formation enthalpies are effectively independent of the nature of the substituted alkali-metal cation.
Fig. 2 plots the standard absolute entropies of the same materials.In general, the entropies are linearly related to each other with slopes approximating unity, but there is much scatter, particularly of the data for the ammonium-substituted materials.Fig. 3 provides a similar plot of the heat capacities at 298.15 K of these materials, again with much scatter in the data for the ammonium-substituted materials.Because of the close connection between entropy and heat capacity, [25] it is no surprise that the heat capacity behaviour mirrors that of the entropy plot.
Finally, Fig. 4 plots the formula unit volumes of the materials against those of the sodium-substituted materials.The data again exhibits linear relationships but with wider separation of the data groups, re ecting the different volumes of the substituting cations, with Li + -materials having the smallest volumes and Cs + -and NH 4 + -materials the largest.

Discussion and Conclusions
By examining the thermochemistry of ion substitution among a group of similar materials we have established that the formation enthalpies among the alkali metal cations and ammonium are very closely related, while their entropies, heat capacities and formula unit volumes follow similar linear patterns without reference to their crystal structures.
The fact that the slopes of the thermochemical functions of each cation versus that of the sodium cation are close to unity demonstrates that it is the constant charge which is signi cant in these comparisons of the substitution while the different intercept values re ect differences between each cation and the sodium cation.This provides useful information relative to the synthesis of related materials within the group, as well as other diverse and desirable attributes.
Table 1 lists the parameters of the tted linear trendlines for each of H, S, C p and V m at 298 K for each monocharged cation against the sodium cation.These values may be useful when considering materials not listed in the Supporting Information le.2011), [13] have provided online a list of pair correlation factors having 6 436 entries in numerical sequence of these gAB values.In Table 2, we list the extracted entries for the pairs relating to NH 4 + , Li + , Na + , K + , Rb + , Cs + , Tl + , Ag + and Cu + .The larger the gAB value, the more likely it is that the ions may substitute for one another.Thus Cs + and Rb + are very interchangeable while Cs + and Na + are unlikely to substitute for one another.The listed reference should be consulted for more detailed analysis.The choice of ionA vs ionB from the published list is unclear since gAB is symmetrical while the list items do not follow the expected Mendeleev sequence down the columns in the Periodic Table .For example, an early entry is for Li + vs Na + while a later entry has the reversed pair K + vs Li + .The standard formation enthalpies, Δ f H° / kJ mol -1 , of materials containing the unit-charged cations NH 4 + , Li + , Na + , K + , Rb + , Cs + , Tl + , Ag + and Cu + with an extensive range of anions together with some hydrates.The symbols for the alkali metal-related materials are lled while those for the non-alkali metal-containing materials (Tl + , Ag + and Cu + ) are un lled.As an arbitrary example, the tted trendline for Li + materials versus Na + materials is shown.It has the formula Δ f H°(Li + )= 1.0153×Δ f H°(Na + ) -10.707 with R 2 = 0.99.The Tl + vs Na + trendline has the equation Δ f H°(Tl + )= 0.745×Δ f H°(Na + ) + 101.95 with R 2 = 0.99.
The standard formation entropies, S° / kJ mol -1 , of materials containing the unit-charged cations NH 4 + , Li + , Na + , K + , Rb + , Cs + , Tl + , Ag + and Cu + with an extensive range of anions together with some hydrates.The symbols for the alkali metal-related materials are lled while those for the non-alkali metal-containing materials (Tl + , Ag + and Cu + ) are un lled.The lowest tted trendline for Li + materials versus Na + materials is shown, having the formula S°(Li + )= 0.8685×S°(Na + ) -3.63 with R 2 = 0.87.

Figures Figure 1
Figures

Table 1
Slopes, intercepts and squared correlation coe cients for linear least-squares trends between monocharged cation-containing solids against the corresponding sodium cation-containing solids for enthalpy, H, entropy, S, formula volume, V m , and heat capacity, C p , all at 298 K, as plotted in the Supporting Information le.?
Based on their probabilistic analysis of ion substitutions in ionic solids,Hautier, et al. (

Table 2 :
[13]pair correlation factors, gAB, between ions substituting the same crystal structure[13]with larger values implying increased probability of successful substitution.Substitution between ions of different charge (in red italics) requires accompanying charges for charge balance.