Local thermodynamic equilibrium

Phenomenological laws may describe many common irreversible processes with broken time symmetry. An irreversible phenomenological macroworld and a microworld determined by linear and reversible quantum laws should be related to each other. Prigogine and his colleagues attempted to unify the basic micro and macroscopic descriptions of matter. 

A local thermodynamic state is determined as elementary volumes at individual points for a nonequilibrium system. These volumes are small such that the substance in them can be treated as homogeneous and contain a sufficient number of molecules for the phenomenological laws to be apphcable. This local state shows microscopic reversibility that is the symmetry of all mechanical equations of motion of individual particles with respect to time. In the case of microscopic reversibility for a chemical system, when there are two alternative paths for a simple reversible reaction, and one of these paths is preferred for the backward reaction, the same path must also be preferred for the forward reaction. Onsager s derivation of the reciprocal rules is based on the assumption of microscopic reversibility. 

The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equilibrium, the local states may be in local thermodynamic equilibrium all intensive thermodynamic variables become functions of position and time. The definition of energy and entropy in nonequilibrium systems can be expressed in terms of energy and entropy densities u(T,Nk) and s(T,Nk), which are the functions of the temperature field T(x) and the mole number density Y(x) these densities can be measured. The total energy and entropy of the system is obtained by the following integrations 

Prigogine expanded the molecular distribution function in an infinite series around the equilibrium molecular distribution function f0 

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Local Thermodynamic Equilibrium (LTE). This LTE model is of historical importance only. The idea was that under ion bombardment a near-surface plasma is generated, in which the sputtered atoms are ionized [3.48]. The plasma should be under local equilibrium, so that the Saha-Eggert equation for determination of the ionization probability can be used. The important condition was the plasma temperature, and this could be determined from a knowledge of the concentration of one of the elements present. The theoretical background of the model is not applicable. The reason why it gives semi-quantitative results is that the exponential term of the Saha-Eggert equation also fits quantum-mechanical expressions. 

One would prefer to be able to calculate aU of them by molecular dynamics simulations, exclusively. This is unfortunately not possible at present. In fact, some indices p, v of Eq. (6) refer to electronically excited molecules, which decay through population relaxation on the pico- and nanosecond time scales. The other indices p, v denote molecules that remain in their electronic ground state, and hydrodynamic time scales beyond microseconds intervene. The presence of these long times precludes the exclusive use of molecular dynamics, and a recourse to hydrodynamics of continuous media is inevitable. This concession has a high price. Macroscopic hydrodynamics assume a local thermodynamic equilibrium, which does not exist at times prior to 100 ps. These times are thus excluded from these studies. 

Second, there must be local thermodynamic equilibrium so that the energy of the system does not fluctuate. Finally, the scale of the micro and macro simulations must be significantly different. 

We then use a Feautrier scheme [4] to perform spectral line formation calculations in local thermodynamic equilibrium approximation (LTE) for the species indicated in table 1. At this stage we consider only rays in the vertical direction and a single snapshot per 3D simulation. Abundance corrections are computed differentially by comparing the predictions from 3D models with the ones from ID MARCS model stellar atmospheres ([2]) generated for the same stellar parameters (a microturbulence = 2.0 km s-1 is applied to calculations with ID models). 

Where applications to industrial combustion systems involve a relatively limited set of fuels, fire seeks anything that can bum. With the exception of industrial incineration, the fuels for fire are nearly boundless. Let us first consider fire as combustion in the gas phase, excluding surface oxidation in the following. For liquids, we must first require evaporation to the gas phase and for solids we must have a similar phase transition. In the former, pure evaporation is the change of phase of the substance without changing its composition. Evaporation follows local thermodynamics equilibrium between the gas... 

Up to now we have presented this example without any regard for consistency, i.e. satisfying thermodynamic and conservation principles. This fuel mass flux must exactly equal the mass flux evaporated, which must depend on q and h(g. Furthermore, the concentration at the surface where fuel vapor and liquid coexist must satisfy thermodynamic equilibrium of the saturated state. This latter fact is consistent with the overall approximation that local thermodynamic equilibrium applies during this evaporation process. 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

This situation with thermal equilibrium, where the population of the excited states and hence emission intensity is determined by collisions, is known as local thermodynamic equilibrium (LTE) and holds in the atmosphere up to altitudes of 50-60 km (Lenoble, 1993). Above this altitude, non-LTE models must be used (e.g., see Lopez-Puertas et a.l., 1998a, 1998b). 

The last few years have seen a minor revolution in determining solar and stellar abundances (Asplund, 2005). Much of the previous work assumed that the spectral lines originate in local thermodynamic equilibrium (LTE), and the stellar atmosphere has been modeled in a single dimension. Since 2000, improved computing power has permitted three-dimensional modeling of the Sun s atmosphere and non-LTE treatment of line formation. The result has been significant shifts in inferred solar abundances. 

A and B in the A/AmB /B reaction couple. The (parabolic) reaction rate constant k (if local thermodynamic equilibrium prevails throughout the couple) conforms to Eqn. (6.32) if we disregard stoichiometric factors. The pertinent rate constant is then... 

For spectra corresponding to transitions from excited levels, line intensities depend on the mode of production of the spectra, therefore, in such cases the general expressions for moments cannot be found. These moments become purely atomic quantities if the excited states of the electronic configuration considered are equally populated (level populations are proportional to their statistical weights). This is close to physical conditions in high temperature plasmas, in arcs and sparks, also when levels are populated by the cascade of elementary processes or even by one process obeying non-strict selection rules. The distribution of oscillator strengths is also excitation-independent. In all these cases spectral moments become purely atomic quantities. If, for local thermodynamic equilibrium, the Boltzmann factor can be expanded in a series of powers (AE/kT)n (this means the condition AE < kT), then the spectral moments are also expanded in a series of purely atomic moments. 

ORM assumes that the atmosphere is in local thermodynamic equilibrium this means that the temperature of the Boltzmann distribution is equal to the kinetic temperature and that the source function in Eq. (4) is equal to the Planck function at the local kinetic temperature. This LTE model is expected to be valid at the lower altitudes where kinetic collisions are frequent. In the stratosphere and mesosphere excitation mechanisms such as photochemical processes and solar pumping, combined with the lower collision relaxation rates make possible that many of the vibrational levels of atmospheric constituents responsible for infrared emissions have excitation temperatures which differ from the local kinetic temperature. It has been found [18] that many C02 bands are strongly affected by non-LTE. However, since the handling of Non-LTE would severely increase the retrieval computing time, it was decided to select only microwindows that are in thermodynamic equilibrium to avoid Non-LTE calculations in the forward model. 

Microwave discharge This operates at very high frequencies (e.g., 2.45 GHz) in the range of microwaves, within which only light electrons can follow the oscillations of the electric field. Therefore, this discharge is far from the local thermodynamic equilibrium, and can be operated over a wide pressure range. The performance of a microwave discharge depends heavily on the type of microwave power applicator (detailed information on this subject is available elsewhere [6, 7]). 

Beta values represent the fraction of free atoms present in the hot flame gases of the flame indicated. These values have been taken from various sources and were either experimentally measured or calculated from thermodynamic data using the assumption of local thermodynamic equilibrium in the flame. These values do not have very good agreement within each element however, the values do provide an indication of the probable sensitivity of the particular flame. 

Obviously, plasmas can be used very efficiently within the synthetic approach (i), and all examples given in this paper are assigned to the synthetic approach. It is much less obvious whether plasmas can be used also in the counter-direction. In order to measure a stable and reproducible electromotive force (EMF) the corresponding electrochemical (galvanic) cell must be in (local) thermodynamic equilibrium. Low-temperature plasmas represent non-equilibrium states and are highly inhomogeneous systems from a thermodynamic point of view, often not... 

Eli Ruckenstein Various time scales have to be compared to obtain an answer to this question. If the relaxation time to attain equilibrium is short compared to the time scale determined by the fluid mechanics, local thermodynamic equilibrium can be assumed. Such an assumption can be made for micellar solutions because the surfactant aggregation and the dissociation of aggregates are relatively rapid processes. It is, however, expected that the size of the aggregate will depend on the stress. For concentrated dispersions, the problem may be more complicated because the processes of aggregation and dissociation of aggregates probably are much slower. 

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