Thermodynamic Consideration

Thermodynamically, the oxidation of hydrocarbons to carbon dioxide and water is preferred to any partial oxidation reaction. The possibility of forming partial oxidation products is thus entirely dependent on the kinetics of the oxidation process. The oxidation of hydrocarbons, is in general, a stepwise process. One way to confine the depth of oxidation, therefore, is to apply a low oxygen to hydrocarbon ratio and a short reaction time. However, to avoid a multitude of products with different oxidation depths, the use of a catalyst is obviously required. In that case, the above two factors (oxygen deficient conditions and short reaction time) may loose their importance. 

Free energy for the transition of a higher to a lower oxide (kcal per mole of liberated oxygen), calculated from ref. 362 

With inorganic compounds, there can also be a selectivity problem, as illustrated by the oxidation of ammonia to nitrogen. Deep oxidation leads to nitrogen oxides. With sulphur dioxide, no selectivity problem rises. 

In the following section, the metal—oxygen bonds will be treated in more detail. 

A thermodynamic study was done using HSC Chemistry software [9] and data provided by the literature [15-18]. The calculations aimed to determine the limiting factors of the technique and thus to set the best conditions to perform the back-extraction. The selection of the most suitable chloride salt for the process is of first importance. 

The thermodynamic approach led to the first conclusion that the optimal working temperature should not exceed 700 °C. The two main reasons are a significant loss of back-extraction efficiency and an increase of AICI3 volatility. 

Moreover, the study highlighted the fact that uranium is expected to be the most difficult actinide to be back-extracted from the A1 matrix. An excess of AICI3 should be needed to perform a quantitative uranium extraction in a single batch. As a consequence, the most suitable conditions for the U back-extraction will automatically be suitable for the back-extraction of the other actinides. This conclusion justified a first experimental study focussed on U back-extraction. 

This antagonist phenomenon justified the study of different chloride salts or salt mixtures. Some (low bases) were selected for their extraction efficiency while the others were selected in order to decrease AICI3 volatility. 

The first experiments of oxidative liquid-liquid back-extraction in chloride melts were performed for uranium extraction in order to define the best chloride salt and AICI3/U ratio. Once these parameters were optimised, a second set of experiments was performed in order to sffidy the feasibility of a grouped back-extraction of U, Pu and Am. 

3 Thermodynamic Considerations 2.3.1 Enthalpy, Entropy, and Free Energy 

The enthalpy change associated with such a reaction is given by the difference between the enthalpy of formation of the products and the enthalpy of formation of the reactants  

It must be noted that AH is a function of temperature. Tabulations of AH generally refer to AH298.15K- The temperature dependence of AH is given by 

Entropy may also influence the spontaneity of chemical reactions. In general terms the entropy represents the disorder or randomness associated with a particular process. The Gibbs free energy of reaction includes both the enthalpy and entropy associated with chemical processes, and is defined as 

In the case of an elementary reaction (i.e., reaction without energy barrier associated with the formation of an intermediate complex), if AG is negative, then the reaction can proceed spontaneously at the temperature under consideration. Values of AG°, AH0, S° and C° for most atmospheric species can be found in chemical handbooks. 

From the point of view of thermodynamics, a polymer is a valuable low-entropy product that is produced by dissipating heat to the surrounding. Any breaking of the bonds within the recycled polymer will be costly in terms of energy. As a result, plastic recycling ideally should not involve the breaking of chemical bonds. 

Furthermore, the second law of thermodynamics (disorder is the preferred state) implies that the separation (creating order) of plastic materials from a mixture of plastics must involve supplying heat and work. Recycling plastics is, by nature, a costly business and in the extreme case, total recycling requires infinite work and heat. 

In the above systems AAa TAS and AAy TAS and hence AG 0. This implies thermodynamic instability and the production of suspension or emulsions by the dispersion process is nonspontaneous, i.e. energy is required to produce the smaller particles or droplets from the larger ones. In the absence of any stabilization mechanism (that will be discussed below), the smaller particles or droplets tend to aggregate and/or coalesce to reduce the total interfacial area, hence reducing the total surface energy of the system. Prevention of aggregation and/or coalescence of suspensions or emulsions requires fundamental understanding of the various interaction forces between the particles or droplets and these will be discussed in subsequent sections. 

4 MORPHOLOGY DEVELOPMENT IN LATEX PARTICLES 8.4.1 Thermodynamic Considerations 

Before reviewing work on the characterization of species formed by sulfur poisoning of ceria-supported catalysts, it is helpful to examine what is known from thermodynamic investigations of sulfur-containing Ce compounds. Cerium, in a 

An important part of the optimisation process is the stabilisation of the monomer-template assemblies by thermodynamic considerations (Fig. 5.17). The 

Diffraction experiments at high pressures provide information concerning the compression-induced changes of lattice parameters and, thus, sample volume. In pure phases of constant chemical composition and in the absence of external fields, the thermodynamic parameters volume V, temperature T and pressure P are related by equations of state, i.e. each value of a state variable can be defined as a function of the other two parameters. Some macroscopic quantities are partial differentials of these equations of state, e.g. the frequently used isothermal bulk modulus Bq of a phase at a defined temperature and zero pressure 5q = — Fq (9P/9F) for T= constant and P = 0, with the reciprocal of Bq V) being the isothermal compressibility k. Equations of state can also be formulated as derivatives of thermodynamic functions like the internal energy U or the Helmholtz free-energy F. However, for practical use the macroscopic properties of solids are often described by means of semi-empirical equations, some of which will be discussed in more detail. 

For small volume changes, the bulk modulus can be expanded as a power series of the pressure at P = 0  

Pq describes the pressure-dependent change of the isothermal bulk modulus Pq, i.e. the stiffening of a material upon application of pressure. If only the first two terms of the expansion series are taken into consideration, integration for constant temperature leads to the well-known Murnaghan equation of 

Solving the equation for P results in the frequently-used inverse Murnaghan equation  

An independent approach for the derivation of the relation between pressure and volume was developed by Birch. Under the assumption of isotropic deformation the strain tensor gy can be treated as a scalar quantity Then, the Helmholtz free energy is expanded as a power series of this isostatic strain e 

It is clear that as [A] approaches [A]sat, x approaches 1, and the surface-adsorbed layer thickness Eq. 11.60 goes to infinity that is, there is an infinite reservoir of liquid in equilibrium with the vapor. This is the desired limiting behavior for the model. 

Now consider the form of the BET adsorption isotherm written in Eq. 11.59. If multilayer adsorption were not possible, then Km would be zero. The adsorbed site fraction from Eq. 11.59 becomes 

In this section we consider the thermodynamics of heterogeneous processes, particularly, adsoption processes. Entropy losses upon converting a gas-phase species to a surface-adsorbed species are very important in such cases. The heat of adsorption must counterbalance the entropy decrease for the process to occur spontaneously. These thermodynamic quantities are considered in Sections 11.5.1 and 11.5.2 

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

By far the largest contribution to a gas-phase species entropy comes from translational motion. Equation 8.98 provided a means to calculate this contribution  

Solubilities and Aqueous Activity Coefficients of Organic Liquids 

Let us first imagine an experiment in which we bring a pure, water-immiscible organic liquid into contact with pure water at a given temperature and ask what will happen. Intuitively, we know that some organic molecules will leave the organic phase and 

To describe this process thermodynamically, at any instant in time during our experiment, we can express the chemical potentials of the organic compound i in each of the two phases (Chapter 3). For the compound in the organic liquid phase, we have  

In the beginning of our experiment, /r, is much larger than (xiw is near zero). Therefore, a net flux of organic molecules from the organic phase (higher chemical potential) to the aqueous phase (lower chemical potential) occurs. This process continues and xiw increases until the chemical potentials (or the fugacities) become equal in both phases. At this point, equilibrium is reached and we may say yw xiw = ya xiL and fiw = fa Once at equilibrium, we obtain  

For the majority of the compounds of interest to us, we can now make two important simplifying assumptions. First, in the organic liquid, the mole fraction of water is small compared with the mole fraction of the compound itself that is, x,L remains nearly 1 (see Table 5.1). Also, we may assume that the compound shows ideal behavior in its water-saturated liquid phase that is, we set yiL = 1. With these assumptions, after some rearrangement, Eq. 5-4 simplifies to  

Numerous physical transformations can be considered as a system changes from one energy state to another. Chemical reactions also involve reactants and products that have different energies. As a result, it is important to understand the relationship between equilibrium and energy. 

The desired unsaturated hydrocarbons only appear to be stable in relation to the saturated structures from which they are derived at relatively elevated temperatures. This fact is illustrated by Fig. 2.1, which shows the variation of the free enthalpy of formation as a function of temperature, related to a carbon atom, of a number 

Acetylene only becomes stable in relation to the simplest paraffins at temperatures substantially above 1000. The situation is more favorable for unsaturated hydrocarbons with lower energy content, such as ethylene, which is stable in relation to ethane above 750 C, and benzene, which is favored in relation to normal hexane above 350 to 400 C. 

Given the extreme simplicity of the chemical structure of a saturated hydrocarbon, thermal activation can only cause the scission ofaC-CorC—H bond. In the former case, the random scission of a C C bond of the carbon chain — the cracking reaction — produces a paraffin and an olefin  

The scission of a C—H bond gives rise to the formation of an olefin by dehydrogenation, 

These conversions are highly endothermic and take place with an increase in the number of molecules, which are therefore favored in terms of thermodynamics at high temperature and low pressure. 

Let s also define free energy terms for the surfaces (a and ot, per unit area, because now we re considering a surface). These are actually excess terms, related to the amount by which the free energy of a segment at the surface exceeds that of a segment in the lattice. We can then write an expression for the free energy of this primary nucleus in terms of the difference between the surface and bulk terms (Equation 10-23)  

FIGURE 10-27 Electron micrograph of annealed PE single crystals (Courtesy Professor Ian R. Harrison, Penn State). 

All this begs the question If extended chain crystals are the thermodynamically 

Interfacial stractures in thermal equilibrium with a bulk electrode and a bulk electrolyte have the lowest interfacial free energy defined as  

The equation is dependent on the contact area, A, the Gibb s free energy of the interface, G, the number of atoms (or molecules) of the /th-species. A, and the corresponding electrochemical potential of the reservoir at temperature T, activity a, and electrostatic potential The sum over i involves 

the first term on the right side denotes the standard chemical potential at temperature T and a water activity = 1- This expression allows us to evaluate the interfacial free energies from DFT calculations since all relevant quantities can be deduced from first principles. From these, we can obtain the electrochemical phase diagram. 

For any reaction system, the chemical and phase composition of the final product depends on the green mixture composition, gas pressure, reactive volume, and initial temperature. As shown in Section I, CS reactions can be represented in the following general form  

Thermodynamic calculations can identify the adiabatic combustion temperature, as well as the equilibrium phases and compounds present at that temperature. The composition of the equilibrium final products is determined by minimizing the thermodynamic potential. For a system with gas and A solid number of components, at constant pressure, this may be expressed as 

Thermodynamic calculations of adiabatic combustion temperatures and their comparison with experimentally measured values have been made for a variety of systems (cf Holt and Munir, 1986 Calcote et al, 1990 Glassman et al, 1992). Under conditions that lead to full conversion, a good agreement between theoretical and experimental values has generally been obtained (Table XX). 

Adiabatic and Measured Combustion Temperatures for Various Reaction Systems 

System Adiabatic Combustion Temperature, Tt, (K) Measured Combustion Temperature, Tc (K) Lowest Melting Point On the Phase Diagram (K) 

Standard Gibbs functions of various reactions of the W-O system are given below for the temperature range of 700 to 1000 °C [3.23]  

Based on the Gibbs functions, the temperatures for the invariant three-phase equilibria can be calculated  

Vapor pressures above WO3 and WO2 are listed as a function of temperature elsewhere [3.19,3.25]. 

The stability regions of the oxides are determined by the temperature and the PH20/PH2 ratio, as demonstrated in Fig. 3.1 [3.17]. Any of the equilibrium oxides can be produced by annealing metallic tungsten or WO3 under the respective equilibrium-temperature and humidity conditions. 

In one of the older equilibrium studies [3.20] an additional oxide was described (W3O) to occur at low-temperature/humidity conditions. However, this compound was not confirmed in any of the later investigations [3.22-3.24], and it is well accepted today that it (W3O/P-W) is a metastable transient and not a stable equilibrium phase. 

The central quantity in the context of this chapter is the Joule-Thomson coefficient, which we define by analogy with its bulk counterpart [sec Eq. (5.123)] 

It is positive if the confined fluid is cooled (dJ 0) upon transverse compression (dT 0) and negative if the fluid is heated instead. According to the assertions at the beginning of Section 5.7.1, the key thermodynamic potential in the current context is the enthalpy W, which we obtain as a Legendre transform (see Section 1.5) of the internal energj via 

At this point it is coiivonicnt to define a. specialized isostress (r = con.st) heat capacity 

Because all coefficients in Eq. (5.134) are pasitive definite we obtain as a thermodynamic expression for the inversion temperature (5 = 0) 

For subsequent checks on the MC simulations, from which we seek to determine Tjnvi it will prove convenient to derive a consistency relation that must hold regardless of molecular details of the specific model under r-onsideration. The derivation starts by assuming that an equation of staite t (T, A) (fixed TV, Sji) exists such that 

The desired unsaturated hydrocarbons only appear to be stable in relation to the saturated structures from which they are derived at relatively elevated temperatures. This fact is illustrated by Fig. 21, which shows the variation of the free enthalpy of formation AGj as a function of temperature, related to a carbon atom, nf a number of characteristic hydrocarbon compounds. In this graph, and at a given temperature, a substance is unstable in relation to all the compounds or elements (C + H2), whose representative point remains below its own, since formation from these compounds or elements requires an input energy the substance is stable in the opposite case. Accordingly, hydrocarbons are unstable at all temperatures in relation to their elements, except for methane, which is stable at the low and medium temperatures. 

The enthalpy of formation for h-BN is on the order of 60kcalmol This nitride melts at high pressure to avoid dissociation at temperature 3300 K. The equilibrium partial nitrogen pressure above the BN surface can be calculated as follows IgPjj (mm-Hg) =4.0-6450/T. Because of the product dissociation, the combustion in B-N system can be described as a system of the second-type with adiabatic temperature (T ) equal to the dissociation temperature of the product [20] and the degree of conversion 

Conditions for Combustion Synthesis of Boron Nitride Material 

Initial relative volume (vq) and initial porosity (0g) of the green sample are other important parameters, which, together with dilution (/), define the strategy of the CS of BN ceramics. These parameters can be introduced as follows  

One can also estimate the duration of the synthesis process tj, which depends on the sample size (i) and is on the order of (s) 2-L (cm) for example, for the 10 cm long cylinder, almost regardless of its diameter D L), the synthesis time is about 20 s. 

It was shown in Chapter 1 that the work of adhesion between two phases, x and y, in an inert medium, can be expressed in terms of the surface and interfacial free energies shown in Eq. (3).  

If the inert medium is replaced by a liquid, the work of adhesion becomes Wal, defined in Eq. (4). 

A positive value of or W l indicates that the interface is stable, whereas a negative value indicates a spontaneous tendency for the interface to dissociate. Thus a comparison of the values of and enables a prediction to be made about the stability of the interface in the presence of the liquid, i.e., on the environmental resistance of the interface. 

Epoxy/aluminum oxide Epoxy/ferric oxide Epoxy/silica Epoxy/CFRP PMMA/aluminum oxide PMMA/ferric oxide 

It should be remembered, however, that there is evidence for chemical bonding at some interfaces and that, in general, mechanical interlocking is frequently assumed to make a contribution to joint strength. In these cases, predictions based on thermodynamic considerations will no longer be reliable. Also, such considerations give no indication of the expected service life of the bond in a hostile environment the thermodynamics enables a prediction to be made about the eventual outcome, but gives no indication of the kinetics of the processes involved. 

The simplest (and weakest ) definition of an elastic material is one for which the stress depends only on the current strain these materials are termed Cauchy elastic. A subset of these materials is occupied by those for which the strain energy depends only on the current strain. These are termed Green elastic or hyperelastic and for these the strain energy is a function of the current strain only, and fully defines the material behaviour. For Cauchy 

We shall develop the concept of strain energy using a small strain approach. Because we will wish to follow the phenomenological treatment by one based on statistical mechanics, it is important at the outset to examine the different types of strain energy function that can be defined, depending on experimental conditions. This introduces thermodynamic considerations. 

The stress is nonoinal as the area A relates to the undeformed material. For small strains, the nominal and true stresses can be assumed equal. Then, when all six components of stress are acting, the energy is additive and, expressing the result in differential form, we have 

Now consider a small strain deformation of unit volume of an elastic solid occurring under adiabatic conditions. The first law of thermodynamics gives 

Comparison of this equation with Equation (3.44) leads us to make the identifications 

Using the data of Hemingway and Robie (1977) and Robie, Hemingway and Fisher (1977) and the properties of H O (Fisher and Zen, 1971), we can estimate values for the free energy of hydrous 

Knowing free energy values for fluorphlogopite, HF and H O permits the derivation of free energy for phlogopite at any temperature, 

The low value of (dP/dT) for the reaction (L) is due to the very large volume term involving HO for this reaction. The activity of SiO has a profound effect on the stability of phlogopite as has been pointed out by Luth (1967) and Yoder and Kushiro (1969). A whole series of stability curves exist for phlogopite in which the activity of SiO is successively buffered by quartz, enstatite-forsterite, leucite-sanidine, and leucite-kalsilite. 

In contrast, the maximum stability of tremolite is in the presence of quartz. Thus we see that the stability of phlogopite relative to tremolite is a function of the activities of SiO, KAlSi O, H O, and F. 

At this point it is convenient to define a specialized isostress (t = const) heat capacity 

Yttria-stabilised zirconia also exhibits strong stabilisation on forming solid solutions in the fluorite structure [46]. Pure zirconia has the stable monoclinic phase in which the zirconium ions are coordinated with 7 oxide ions, whereas the cubic phase with 8 coordinates becomes stable only at high temperatures. On doping with Y2O3, the oxide ion vacancies are formed preferentially around the zirconium ions, which leads to stabilisation of zirconia in the cubic phase. 

Perovskite cathodes and yttria-stabilised zirconia (YSZ) electrolyte can react in several ways as discussed below. 

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Tolman [21] concluded from thermodynamic considerations that with sufficiently curved surfaces, the value of the surface tension itsc//should be affected. In reviewing the subject, Melrose [22] gives the equation... 

One can distinguish between a tliennodynamic and kinetic stability to corrosion. C2.8.2.1 THERMODYNAMIC CONSIDERATIONS... 

Reaction (5. EE) is particularly useful for the discussion of thermodynamic considerations because of the way differences in thermodynamic state variables are independent of path. Accordingly, if we know the value of AG for reaction (5. EE), we have characterized the following ... 

Thermochemistry. Thermodynamic considerations ate of utmost importance in fluorinations. Table 1 is based on JANAF data (25) for CH, which indicate an average carbon-hydrogen bond strength of 410.0 kj/mol (98 kcal/mol) based on the atomization energy of CH. ... 

Reaction with Metals. Thermodynamic considerations for the reaction... 

Thermodynamic data are available only for the lower alkylamines, mainly estimates based on a few experimental deterrninations (3,4). Many manufacturing processes appear to be limited by thermodynamic equiUbria. The lack of accurate free energy data for these amines limits the appHcation of thermodynamic considerations, in contrast to the situation in hydrocarbon technology. 

Thermodynamic considerations demand that the energy necessary for biosynthesis of any substance exceed the energy available from its catabolism. Otherwise, organisms could achieve the status of perpetual motion machines A few molecules of substrate whose catabolism yielded more ATP than required for its resynthesis would allow the cell to cycle this substance and harvest an endless supply of energy. 

An important part of the optimization process is the stabilization of the monomer-template assemblies by thermodynamic considerations (Fig. 6-11). The enthalpic and entropic contributions to the association will determine how the association will respond to changes in the polymerization temperature [18]. The change in free volume of interaction will determine how the association will respond to changes in polymerization pressure [82]. Finally, the solvent s interaction with the monomer-template assemblies relative to the free species indicates how well it will stabilize the monomer-template assemblies in solution [16]. Here each system must be optimized individually. Another option is simply to increase the concentration of the monomer or the template. In the former case, a problem is that the crosslinking as well as the potentially nonselective binding will increase simultaneously. In the... 

However, it was found that the effect on the equilibrium formation of aromatics is not substantial due to thermodynamic considerations. A more favorable effect was found for the reaction between ethylene (formed via cracking during aromatization of propane) and hydrogen. The reverse shift reaction consumes hydrogen and decreases the chances for the reduction of ethylene to ethane byproduct. 

It follows from the electrochemical mechanism of corrosion that the rates of the anodic and cathodic reactions are interdependent, and that either or both may control the rate of the corrosion reaction. It is also evident from thermodynamic considerations (Tables 1.9 and 1.10) that for a species in solution to act as an electron acceptor its redox potential must be more positive than that of the M /M equilibrium or of any other equilibrium involving an oxidised form of the metal. 

In the electrochemical series of elements, copper is near the noble end and will not normally displace hydrogen, even from acid solutions. Indeed, if hydrogen is bubbled through a solution of copper salts, copper is slowly deposited (more rapidly if the process is carried out under pressure). (See Section 1.2 for thermodynamic considerations.)... 

The close molecular packing makes diffusion more difficult than with amorphous polymers compared in similar circumstances, i.e. both below Tg or both above (but below of the crystalline polymer). Thermodynamic considerations lead to considerable restriction in the range of solvents available for such polymers. 

It is possible to indicate by thermodynamic considerations 24,25,27>, by spectroscopic methods (IR28), Raman29 , NMR30,31 ), by dielectric 32> and viscosimetric measurements 26), that the mobility of water molecules in the hydration shell differs from the mobility in pure water, so justifying the classification of solutes in the water structure breaker and maker, as mentioned above. 

In addition to the questions of the potentials and capacities of electrodes, which are essentially thermodynamic considerations, practical utilization of alloys as electrodes also requires attractive kinetic properties. 

For example, disproportionation of but-2-yl radicals produces a mixture of butenes as shown (Scheme 1.1 I).138 Thermodynamic considerations suggest thai but-l-ene and but-2-enes should be formed in a ratio of ca 2 98. However, the observed 5 4 ratio of but-1-ene but-2-enes is little different from the 3 2 ratio that is expected on statistical grounds (i.e. ratio of f5-hydrogens in the I- and 3-positions). 

Continuous transition of state is possible only between isotropic states it may thus occur between amorphous glass (i.e., supercooled liquid of great viscosity) and liquid ( sealing-wax type of fusion ), or between liquid and vapour, but probably never between anisotropic forms, or between these and isotropic states. This conclusion, derived from purely thermodynamic considerations, is also supported by molecular theory. 

The investigation above is due initially to Gibbs (Scient. Papers, I., 43—46 100—134), although in many parts we have followed the exposition of P. Saurel Joum. Phys. diem., 1902, 6, 474—491). It is chiefly noteworthy on account of the ease with which it permits of the deduction, from purely thermodynamic considerations, of all the principal properties of the critical point, many of which were rediscovered by van der Waals on the basis of molecular hypotheses. A different treatment is given by Duhem (Traite de Mecanique chimique, II., 129—191), who makes use of the thermodynamic potential. Although this has been introduced in equation (11) a the condition for equilibrium, we could have deduced the second part of that equation directly from the properties of the tangent plane, as was done by Gibbs (cf. 53). 

Basic thermodynamic considerations may be used to establish the phase diagram of a ternary system consisting of liquid components 1 and 3 and supercritical gas 2. In the subsequent discussion we follow the simplified treatment given by Balder (Bl). 

When m < mc, product assemblages are unstable and thermodynamic considerations predict that the product (B) will tend to revert to the reactant unless the local statistical fluctuations of energy are sufficient to achieve... 

Pyrolyses of formates, oxalates and mellitates yield CO and C02 (H2, H20 etc.) as the predominant volatile products and metal or oxide as residue. It is sometimes possible to predict the initial compositions from thermodynamic considerations [94], though secondary reactions, perhaps catalyzed by the solids present, may result in a final product mixture that is very different. The complex mixtures of products (hydrocarbons, aldehydes, ketones, acids and acid anhydrides) given [1109] by reactants containing larger organic groupings makes the collection of meaningful kinetic data more difficult, and this is one reason why there are relatively few rate studies available for the decompositions of these substances. 

It will now be shown from purely thermodynamic considerations that for, adiabatic conditions, supersonic flow cannot develop in a pipe of constant cross-sectional area because the fluid is in a condition of maximum entropy when flowing at the sonic velocity. The condition of the gas at any point in the pipe where the pressure is P is given by the equations ... 

The Feasibility of the Desolvation Hypothesis Can Be Examined with Clear Thermodynamic Considerations... 

I. Metcalfe, Electrochemical Promotion of Catalysis I Thermodynamic considerations, J. Catal. 199,247-258 (2001). 

Hoffman, M. R. (1981). Thermodynamic, kinetic and extra-thermodynamic considerations in the development of equilibrium models for aquatic systems. Environ. Sci. Technol. 15,345-353. 

Hastings, D. and Emerson, S. (1986). Oxidation of manganese by spores of a marine Bacillus Kinetics and thermodynamic considerations. Geochem. Cosmochim. Acta 50,1819-1824. 

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