The Fundamental Excess-Property Relation

This is the fundamental excess property relation, analogous to the fundamental residual property relation (eq. 169). 

The fundamental excess-property relation is derived in exactly the same as the fundamental residual-property relation and leads to analogous rest Equation (13.12), written for the special case of an ideal solution, is subtra from Eq. (13.12) itself, yielding ... 

Given an equation for Ge/RT as a function of T, P, and composition, the fundamental excess-property relation, Eq. (13.21) or (13.22), provides complete excess-property information. However, this formulation represents less-complete property information than does the residual-property formulation, because it tells us nothing about the properties of the pure constituent chemical species. 

Whereas the fundamental residual property relation derives its usefulness from its direct relation to experimental PVT data and equations of state, the excess property formulation is useful because and are all experimentally accessible. Activity coefficients are found from vapor—Hquid... 

The last relation is Eq. (11.62), which demonstrates the partial property re ship that In y, bears to GE/RT. These equations are analogous to Eqs. through (13.17). Whereas the fundamental residual-property relation de usefulness from its direct relation to experimental PVT data and equati state, the excess-property formulation is useful because VE, HB, and yt experimentally accessible. Activity coefficients are found from VLE d... 

Just as the fundamental property relation of Eq. (11.50) provides complete property information from a canonical equation of state expressing G/RT as a function of T, P, and composition, so the fundamental residual-property relation, Eq. (11.51) or (11.52), provides complete residual-property information from a PVT equation of state, from PVT data, or from generalized P VT correlations. However, for complete property information, one needs in addition to PVT data the ideal-gas-state heat capacities of tile species tliat comprise tlie system. In complete analogy, thefundamentalexcess-property relation, Eq. (11.86) or (11.89), provides complete excess-property information, given an equation for G /RT as a function of its canonical variables, T, P, and composition. However, tliis formulation represents less-complete property information tlian does the residual-propertyfonmilation, because it tells us no tiling about the properties of the pure constituent chemical species. 

Fundamental Excess-Property Relation Equations for excess properties are developed in much the same way as those for residual properties. For the special case of an ideal solution, Eq. (4-191)... 

We have already seen that the excess properties relate to each other in exactly the same way as do the usual thermodynamic functions. The same thing now applies to the Marguies parameters, since they are simply another way of writing excess properties. Thus, as we showed with the excess functions, we can now write relations such as the fundamental equation (5.11) to interrelate Marguies parameters ... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

It is recognised that the fundamental inter-relations between structures, lattice, interactions, microscopic and macroscopic properties would be (and indeed is) a challenging task for basic research and an indispensable pre-requisite for a broad introduction of ICPs in commercial applications. Conversely, in view of the excessively one-sided strategic orientation of research work and the resulting less time, there is now an urgent need to broaden the first significant commercial application if ICP research is to receive further substantial financial and intellectual support and not be condemned to lead a life of exotic isolation. 

Since the definition (5.2.1) is a linear combination of thermodynamic properties, all the usual relations for extensive properties (see Chapter 3) can be expressed in terms of excess properties. Those relations include the Legendre transforms, the four forms of the fundamental equation, the response functions, and the Maxwell relations. Such relations reduce the amount of information needed to compute values for excess properties. 

Primitive models have been very useful to resolve many of the fundamental questions related to ionic systems. The MSA in particular leads to relatively simple analytical expressions for the Helmholtz energy and pair distribution functions however, compared to experiment, a PM is limited in its ability to model electrolyte solutions at experimentally relevant conditions. Consider, for example, that an aqueous solution of NaCl of concentration 6 mol dm (a high concentration, close to the precipitation boundary for this solution) corresponds to a mole fraction of salt of just 0.1 i.e. such a solution is mostly water. Thus, we see that to estimate the density of such solutions accurately the solvent must be treated explicitly, and the same applies for many other thermodynamic properties, particularly those that are not excess properties. The success of the Triolo et approach can be attributed to the incor-... 

In summary, our DFT + U calculations showed that the two excess electrons brought by the occurrence of an O vacancy are in fact separately distributed on Ce02(lll), which may be also a fundamental characteristics of rare earth materials with f electrons. The physical origin of the finding is discussed. The cerium s characteristic highly localized 4f orbital that can take a whole electron as well as the surface relaxation with multiple configurations were formd to be responsible for such a feature. We expect that these results can help us rmderstand many properties of ceria. They may also have s pif-icant implications in the application performance of ceria and other related materials. 

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