Local equilibrium state, hypothesis

This is often called the hypothesis of local equilibrium state. ... 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

In nonequilibrium steady system, a flux, such as of energy, matter, electric current exists. The presence of fluxes is related to the presence of a gradient of temperature, concentration, electrical potential, or barycentric velocity. In the local-equilibrium hypothesis, it is assumed that the fluxes have not an essential influence on the thermodynamics of the system the thermodynamic potentials and, consequently, the equations of state of the system, keep their usual equilibrium form but at a local level, namely, for sufficiently small volume elements. 

In a heterogeneous system, the thermodynamic state hence the state parameters are position-dependent. This heterogeneity (hence non-equilibrium) is the driving force which tends to restore the system back to thermodynamic equilibrium However, it is assumed that the (spatial) variation is sufficiently mild so that every elementary particle can be considered as under thermodynamic equilibrium Its state parameters are therefore linked by the state equation expressing this equilibrium requirement. This assumption is called the hypothesis of local equilibrium". This assumption excludes the treatment of fast processes (for example explosions) under the framework of classic irreversible thermodynamics. 

The fundamental hypothesis of CIT is the existence of a local-equilibrium condition. A series of finite volume cells is considered in a material body, in which local variables such as temperature and entropy are uniform and in equilibrium, but time-dependent. The variables can take different values from cell to cell. The majority of textbooks are written using this formulation (see, e.g., Kestin 1979, which refers to this as the principle of local state). The most important result of CIT under the local-equilibrium hypothesis is that, as a natural result of the Second Law of Thermodynamics in the course of a mechano-thermal process, we have the following entropy inequality ... 

As shown above, the classical description has been undoubtedly useful nevertheless, it has some drawbacks both from the fundamental and practical points of view. It is based on the macro- and/or local- equilibrium hypothesis, which may be too restrictive for a wider class of phenomena where other variables, not found in equilibrium, may influence the thermodynamic equations in the situations taking place when we get out of equilibrium. The concept is consistent with the limiting case of linear and instantaneous relations between fluxes and forces, which, however, becomes unsatisfactory under extreme conditions of high frequencies and fast non-linear changes (explosions). Then the need arises to introduce the second derivatives ( ), such as 0 = 0 (T, T T , P,. ..) and the general thermodynamic potential, 0, is can be no longer assumed via a simple linear relation but takes up a new, non-stationary form 0 = - ST + VP + (d0/Fr)T r ,p+ (state variables become dependent on the temperature derivatives. 

Many attempts have been made to quantify SIMS data by using theoretical models of the ionization process. One of the early ones was the local thermal equilibrium model of Andersen and Hinthome [36-38] mentioned in the Introduction. The hypothesis for this model states that the majority of sputtered ions, atoms, molecules, and electrons are in thermal equilibrium with each other and that these equilibrium concentrations can be calculated by using the proper Saha equations. Andersen and Hinthome developed a computer model, C ARISMA, to quantify SIMS data, using these assumptions and the Saha-Eggert ionization equation [39-41]. They reported results within 10% error for most elements with the use of oxygen bombardment on mineralogical samples. Some elements such as zirconium, niobium, and molybdenum, however, were underestimated by factors of 2 to 6. With two internal standards, CARISMA calculated a plasma temperature and electron density to be used in the ionization equation. For similar matrices, temperature and pressure could be entered and the ion intensities quantified without standards. Subsequent research has shown that the temperature and electron densities derived by this method were not realistic and the establishment of a true thermal equilibrium is unlikely under SIMS ion bombardment. With too many failures in other matrices, the method has fallen into disuse. 

In this paper, we have presented and tested a model which allows the calculation of adsorption isotherms for carbonaceous sorbents. The model is largely inspired of the characterization methods based on the Integration Adsorption Equation concept. The parameters which characterize the adsorbent structure are the same whatever the adsorbate. In comparison with the most powerful characterization methods, some reasonable hypothesis were made the pore walls of the adsorbent are assumed to be energetically homogenous the pores are supposed to be slit-like shaped and a simple Lennard-Jones model is used to describe the interactions between the adsorbate molecule and the pore wall the local model is obtained considering both the three-dimension gas phase and the two-dimension adsorbed phase (considered as monolayer) described by the R lich-Kwong equation of state the pore size distribution function is bimodal. All these hypotheses make the model simple to use for the calculation of equilibrium data in adsorption process simulation. Despites the announced simplifications, it was possible to represent in an efficient way adsorption isotherms of four different compounds at three different temperatures on a set of carbonaceous sorbents using a unique pore size distribution function per adsorbent. 

It is well known that these vapors cannot be considered as ideal gases, as the second virial coefficients are different from zero, except in the case of cesium at high temperatures (Nieto de Castro etal. 1990). However, the quasi-chemical equilibrium hypothesis can be used that states that the imperfection has its origin in the atom association, and therefore the mixture of monomers and dimers can be considered a perfect gas mixture (Ewing et al. 1967). Assuming also local chemical equilibrium, Stogryn Hirschfelder (1959) considered that the heat of reaction could affect the reactive contribution and found that the chemical reaction component is given by... 

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