Local thermodynamic functions

We conclude thb chapter with a brief assessment of what b meant by a local thermodynamic function. These entities were introduced first in 2.3 and 2.5, and have been used repeatedly in the succeeding diapters. 

We have for example, assumed throughout this chapter that it is meaningful to write p(r) for the equilibrium density at a point r. In a canonical ensemble the formal definition of this function follows from (4.177) 

The local pressure tensor pfi) suffers from ambiguities similar to those of die energy nshy sinoe it can be expressed in terms of the viiial of the intermolecular potential (S 4.3), but again there are the constraints that any consistent definition must lead to invariant values for observaUe quantifies. In a fluid with a gradient only in the z-direction mechanical stability requires that Pn(z) is a constant, and equal to p and p. The zeroth-moment of the difference n (z)-p, z), whidi is a, is likewise invariant, but we have seen that the first moment is apparently not, and have discussed in 4.8 the diflBculties that follow if this moment is identified with the planar limit of the surface of tension that enters into the thermodynamic discussion of the spherical drop. 

Desai (Ado. than. Phys. 46, 279 (1981)) is concerned primarily with the dynamics of the interface. 

143 (1979). A later review by J. K. Percus, to be published in Studia on Statistical Iubthaiiics (ed. E. W. Montndl and J. L. Lebowitz), Norlfa-Hdlland. Amsterdam, covers much of the same material. 

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The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

The comprehension of (electro)chemical forces is based on the macroscopic notion of a (electro(chemical potential difference (AG). As seen in biochemistry textbooks, AG is composed of two mathematical parts a standard-state part (AG°) and a concentration term. The standard-state part (which depends, ultimately, on the local energetic interaction of a particle with its surroundings) of metabolites in channeled systems can be vastly different from that in the bulk solution. For microheterogeneous systems, the local thermodynamic functions cannot be defined by macroscopic rules, and even the Second Law of Thermodynamics may apparently be violated therein (Westerhoff and Welch, 1992). 

A fundamental aspect of our MD treatment is its acceptance or assumption of the existence, in each surface parallel to the electrode, of local thermodynamic functions and variables and of thermodynamic properties for all species whose individual molecules or ions are centered in that surface. This assumption is equivalent to the statement that we consider only systems for which the Gibbs entropy-balance equation holds locally. In some regions of the interphase between the bulk of the metal and the bulk of... 

The above definitions can be extended to other local thermodynamic functions, such as the pressure, entropy density, specific heats, etc. Thus all intensive thermodynamic variables are, by definition, to be functionally related to the mean energy and number densities in the same way as at equilibrimn. It then follows that the various thermodynamic identities derived by equilibrium arguments are still valid for the non-equilibrium state. (It is worth emphasizing that this procedure is quite general, and is not restricted to linear processes.)... 

A further point worth emphasizing is related to the last one, and has already been touched on in the last section, namely that the values of the potentials p, T, and in which enter into the arguments abcrve are those of the bulk phases. We do not have to ask if they have the same value at the surface as they have in the bulk. The question does not arise in Gibbs s treatment it does arise and must be answered, if a third phase is introduced. It is true that Gibbs specifically states that in has the same value at the surface as in the bulk, but he neither uses this result nor provides an operational definition of in at the surface. This and similar questions about local thermodynamic functions are best answered by using the methods of statistical ffiermodynamtcs ( 4.10). 

Local thermodynamic functions can be defined unambiguously for single-molecule properties, e.g. p(r), T(i). 

Computer simulation has been used to obtain sudi local thermodynamic functions as the energy density, < (z), the transverse component of Irving and Kirkwood s definition of the pressure tensor.s" Pr(z), and hence the height of the related surface of tension in a planar interface.s° The essential arbitrariness of these calculations has been discussed in 4.3 and 4.10. 

Local association of the reduced cation and the oxygen vacancy is clearly suggested by the thermodynamics of the hypo-stoichiometric mixed oxides (Ui yPUy)02 x, where the thermodynamic functions do not depend on x and y separately, but rather on a quantity, called plutonium valence , which contains the ratio x/ y73,87) Q sters consisting of this association have been proposed in order to explain the thermodynamics of actinide hypostoichiometric dioxides. 

Now the assumption is dropped that the chemical reaction is a rate-controlled conversion to an invariant product composition, and the composition is permitted to vary with local thermodynamic state. Zel dovich, Brinkley Si Richardson, and Kirkwood Wood pointed out that since in a chemically reactive wave, pressure is a function not only of density and entropy but also of chemical composition, the sound speed for a reacting material should be defined as the frozen sound speed... 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

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. 

Near a critical point, the parent p coexists with another phase that is only slightly different if, as we assume here, the free energy function is smooth, these two phases are separated—in p-space—by a hypothetical phase which has the same chemical potentials but is (locally) thermodynamically unstable. [This is geometrically obvious even in high dimensions between any two minima of f p)—p p, at given p, there must lie a maximum or a saddle point, which is the required unstable phase. ] Now imagine connecting these three phases by a smooth curve in density space p(e). At the critical point, all three phases collapse, and the variation of the chemical potential around p e = 0) = p must therefore obey... 

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... 

The postulate of local thermodynamic equilibrium in a discontinuous system is replaced by the requirement that the intensive properties change very slowly in each part, so that the parts are in thermodynamic equilibrium at every instant. The intensive properties are a function of time only, and they are discontinuous at the interface and may change by jumps. In the following sections, thermomechanical effects and thermoelectricity are summarized. 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

Unlike the density of bulk fluids, which is a function of pressure and temperature only (and composition for a mixture), the average density across the interface between a liquid and its vapor, as well as at the liquid/liquid interface, varies as a function of the distance along the interface normal p(z). Like other local thermodynamic quantities[30], it is defined by a coarse-graining procedure The volume of the system is divided into slabs perpendicular to the interface normal, and the density of each slab is computed in the usual way. The thickness of the slabs is chosen to be small enough so that the density does not vary much... 

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