Continuous thermodynamics, principles

The history of CALPHAD is a chronology of what can he achieved in the field of phase equilibria by combining basic thermodynamic principles with mathematical formulations to describe the various thermodynamic properties of phases. The models and formulations have gone through a series of continuous improvements and, what has become known as the CALPHAD approach, is a good example of what can be seen as a somewhat difficult and academic subject area put into real practice. It is indeed the art of the possible in action and its applications are wide and numerous. 

This is the second of a two-volume series in which we continue the description of chemical thermodynamics. The first volume, titled Chemical Thermodynamics Principles and Applications, contained ten chapters and four appendices, and presented the basic thermodynamic principles and applied these principles to systems of chemical interest. We will refer to that volume in this chapter as Principles and Applications. We begin this second volume that we have titled Chemical Thermodynamics Advanced Applications, with Chapter 11 where we summarize and review the thermodynamic principles developed in the first volume, and then focus in subsequent chapters on a discussion of a variety of chemical processes in which we use thermodynamics as the basis of the description. 

We are destined to be in a world where disorder or Entropy has to be increased every moment. Free Energy has to be dissipated out continuously to areas, which can absorb it. The Universe was apparently formed by a Big - Bang . The fact that the distant stars and the constellations continuously move away from each other, as also from our world, gives further weight to the thermodynamic principle that the disorder (Entropy) in the universe is constantly increasing. 

Rouquerol et al. (11, 12) have recently described the experimental determination of entropies of adsorption by applying thermodynamic principles to reversible gas-solid interactions. Theoretically, the entropy change associated with the adsorption process can only be measured in the case of reversible heat exchange. The authors showed how isothermal adsorption microcalorimetry can be used to obtain directly and continuously the integral entropy of adsorption by a slow and constant introduction of adsorbate under quasi-equilibrium conditions (11) or by discontinuous introduction of the adsorbate in an open system (12). 

In subsequent sections we will continually apply various mathematical procedures that are listed below. These operations must be properly mastered before one can undertake the unified description of thermodynamic principles. 

Phase Transitions in Lipid Assemblies. The rich polymorphism of amphiphilic systems, of which the multilamellar and the Hn phases are only two structures, was made evident from the seminal work of Luzzati and co-workers. Since that early work, an immense variety of water-induced phase transitions have been observed and rationalized in terms of an apparently systematic connection between water content and polar group molecular area. Therefore, the recent observation of a double transition—Hn to lamellar back to Hn—from continual hydration of dioleoylphosphatidyl-ethanolamine (40) was a surprise. Furthermore, an estimate of the cost of uncurling the monolayer in the formation of bilayers based on the previously described bending modulus far exceeds the osmotic work that actually produced the transition. Although this transition sequence can successfully be accounted for by simple thermodynamical principles, it, in fact, contains many geometry-dependent free energy contributions that we simply do not yet understand (41). 

Typically, the segregated phase has a smaller characteristic length scale than the continuous phase. In a monomer-flooded emulsion polymerization, the aqueous continuous phase will contain monomer drops and polymer particles, although large monomer drops may also contain smaller water droplets or polymer particles (if crosslinked or insoluble). This is the consequence of a thermodynamic principle that acts in the direction of a constant chemical potential for all species, throughout the whole system. In other words, there is a driving force that pushes all of the components of a system to be present in different proportions in all of its phases. This principle has been proven in spontaneous emulsification experiments, where droplet formation is observed on either side of the liquid-liquid interface [7]. Moreover, the chemical potential is size-dependent at the colloidal scale and hence, particles of different size will possess different compositions. 

As this subchapter is devoted to solvent activities, only the monodisperse case will be taken into account here. However, the user has to be aware of the fact that most LLE-data were measured with polydisperse polymers. How to handle LLE-results of polydisperse polymers is the task of continuous thermodynamics, Refs. Nevertheless, also solutions of monodisperse polymers or copolymers show a strong dependence of LLE on molar mass of the polymer, or on chemical composition of a copolymer. The strong dependence on molar mass can be explained in principle within the simple Flory-Huggins %-function approach, please see Equation [4.4.61]. 

For certain reasons the lattice model will be chosen here to give some examples. Following Koningsveld et al. [71], we extend the random mixing assumption and the surface contact statistics to a system consisting of two random polydisperse copolymer, each built up by two different units, B(a, P) + C(y5). Applying the principles of continuous thermodynamics in calculating the contact probabilities in such mixtures [34,40], we obtain for [66]... 

In the first equation of eq 9.4, p(Af) is the continuous version of the partial molar chemical potential and corresponds to the usual thermodynamic expression provided as the second equation of eq 9.4. The integral relates to the total range of the characterization variable M. In principle, knowing G and calculating 6G according to eq 9.3 the comparison of eqs 9.3 and 9.4 leads to the partial molar quantities p M). Based on the previously outlined principles the well-known fundamental equations of usual thermodynamics may be translated into continuous thermodynamics to give the basic equations... 

In this treatment, presented previously, all chemical species of the mixtures are considered to be similar. Thus, they all are described by one distribution function. In a generalized version the occurrence of several ensembles of very similar species (e.g. paralBnic and aromatic hydrocarbons or polymer blends) in the mixture may be accounted for by describing each ensemble with its own distribution function. Furthermore, some individual components may also be present e.g. in a polymer solution), that need to be included into the formalism too. Polydisperse mixtures of this kind are often called semicontinuous mixtures. There are no fundamental difficulties in generalizing the simple version of continuous thermodynamics discussed previously, but the equations become somewhat more complex. Here, the principles of generalization are to be presented only by some examples (for a more detailed treatment the reader should refer to ref. 48 for instance). 

For petroleum fractions or similar systems the treatment of phase equilibria will now be discussed briefly. The basic principles are the same as those outlined for polymer systems without recourse to segment-molar quantities. For petroleum fractions the phase-equilibrium problem of importance is the so-called flash calculation that is analogous to the calculation of coexistence curves for a polydisperse polymer solution and in the simplest case a single distribution function is required. For example, the system may contain many alkanes characterized by their normal boiling-point temperatures Tb that in this work will be denoted by x. At moderate pressures the equilibrium condition is given by the continuous thermodynamics form of Raoult s law ... 

In many cases distribution functions are determined experimentally the characterization of petroleum fractions by true-boiling-point distillation or gas-chromatographically simulated distillation, and the characterization of polymers by gel-permeation chromatography. In principle, the integrals of continuous thermodynamics may be directly solved based on these experimentally determined distribution functions. However, this approach delicate numerical analyses and the assumption the complete distribution function has been obtained by experiment clearly this is no the case, for example, for some polymers only molar-mass averages are determined. Thus, there are numerous cases where smoothed or analytical distribution function provides more reliable phase equilibrium calculation than those obtained by use of the experimentally determined distribution function. When the integrals of continuous thermodynamics possess analytical solutions considerably numerical simplification is afforded and this is one motive for the desire to have analytical expressions for the distribution function. 

The parameters which characterize the thermodynamic equilibrium of the gel, viz. the swelling degree, swelling pressure, as well as other characteristics of the gel like the elastic modulus, can be substantially changed due to changes in external conditions, i.e., temperature, composition of the solution, pressure and some other factors. The changes in the state of the gel which are visually observed as volume changes can be both continuous and discontinuous [96], In principle, the latter is a transition between the phases of different concentration of the network polymer one of which corresponds to the swollen gel and the other to the collapsed one. 

Deviations from ideality in real solutions have been discussed in some detail to provide an experimental and theoretical basis for precise calculations of changes in the Gibbs function for transformations involving solutions. We shall continue our discussions of the principles of chemical thermodynamics with a consideration of some typical calculations of changes in Gibbs function in real solutions. 

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