Equilibrium system, intensive state

For a PVT system of uniform T and P containing N species and 71 phases at thermodynamic equilibrium, the intensive state of the system is hilly determined by the values of T, P, and the (N — 1) independent mole fractions for each of the equilibrium phases. The total number of these variables is then 2 + tt(N — 1). The independent equations defining or constraining the equilibrium state are of three types equations 218 or 219 of phase-equilibrium, N(77 — 1) in number equation 245 of chemical reaction equilibrium, r in number and equations of special constraint, s in number. The total number of these equations is Ar(7r — 1) + r + s. The number of equations of reaction equilibrium r is the number of independent chemical reactions, and may be determined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

The Gibbs phase rule allows /, the number of degrees of freedom of a system, to be determined. / is the number of intensive variables that can and must be specified to define the intensive state of a system at equilibrium. By intensive state is meant the properties of all phases in the system, but not the amounts of these phases. Phase equilibria are determined by chemical potentials, and chemical potentials are intensive properties, which are independent of the amount of the phase that is present. The overall concentration of a system consisting of several phases, however, is not a degree of freedom, because it depends on the amounts of the phases, as well as their concentration. In addition to the intensive variables, we are, in general, allowed to specify one extensive variable for each phase in the system, corresponding to the amount of that phase present. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

Analysis of the last formula shows that in both cases, in principle, we can observe the minimal intensity of radiation or magnetic flow. This is in agreement with the absolute minimal realization of the most probable state in equilibrium system (see fig 3.a and fig 4.a, fig 3.b and fig 4.b). They are in agreement with the values of the observed distribution function observable frequencies and are equal to Im = f(xm) for fluxons and lm = f(xm) for radiating particles. For details of statistical characteristics of observable frequencies see reference (Jumaev, 2004). 

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. 

Consider a material or system that is not at equilibrium. Its extensive state variables (total entropy number of moles of chemical component, i total magnetization volume etc.) will change consistent with the second law of thermodynamics (i.e., with an increase of entropy of all affected systems). At equilibrium, the values of the intensive variables are specified for instance, if a chemical component is free to move from one part of the material to another and there are no barriers to diffusion, the chemical potential, q., for each chemical component, i, must be uniform throughout the entire material.2 So one way that a material can be out of equilibrium is if there are spatial variations in the chemical potential fii(x,y,z). However, a chemical potential of a component is the amount of reversible work needed to add an infinitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. 

The intensive variables T, P, and nt can be considered to be functions of S, V, and dj because U is a function of S, V, and ,. If U for a system can be determined experimentally as a function of S, V, and ,, then T, P, and /q can be calculated by taking the first partial derivatives of U. Equations 2.2-10 to 2.2-12 are referred to as equations of state because they give relations between state properties at equilibrium. In Section 2.4 we will see that these Ns + 2 equations of state are not independent of each other, but any Ns+ 1 of them provide a complete thermodynamic description of the system. In other words, if Ns + 1 equations of state are determined for a system, the remaining equation of state can be calculated from the Ns + 1 known equations of state. In the preceding section we concluded that the intensive state of a one-phase system can be described by specifying Ns + 1 intensive variables. Now we see that the determination of Ns + 1 equations of state can be used to calculate these Ns + 1 intensive properties. 

The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. These are therefore phase-rule variables, but they are not all independent. The phase rule gives the number of variables from this set which must be arbitrarily specified to fix all remaining phase-rule variables. 

The phase rule for nonreacting systems, presented without proof in Sec. 2.8 results from application of a rule of algebra. The number of phase-rule variable which must be arbitrarily specified in order to fix the intensive state of a syste at equilibrium, called the degrees of freedom F, is the difference between t total number of phase-rule variables and the number of independent equatio that can be written connecting these variables. 

The intensive state of a PVT system containing N chemical species and phases in equilibrium is characterized by the temperature T, the pressure P, an N - 1 mole fractionst for each phase. These are the phase-rule variables, an their number is 2 + (N - l)(ir). The masses of the phases are not phase-rul variables, because they have no influence on the intensive state of the system. 

Solution According to the phase rule (see Sec. 15.8), the system has two degrees of freedom. Specification of both the temperature and the pressure leaves no other degrees of freedom, and fixes the intensive state of the system, independent of the initial amounts of reactants. Therefore, material-balance equations do not enter into the solution of this problem, and we can make no use of equations that relate compositions to the reaction coordinate. Instead, phase equilibrium relations must... 

When JV = 2, the phase rule becomes F = 4 - it. Since there must be at least one phase (it = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be... 

At equilibrium the system consists of liquid and vapor in equilibrium, and the intensive state of the system is fixed by the specification of T and P. Therefore, one must first determine the phase compositions, independent of the ratio of reactants. These results may then be applied in the material-balance equations to find the equilibrium conversion. 

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... 

Intensive properties that specify the state of a substance are time independent in equilibrium systems and in nonequilibrium stationary states. Extensive properties specifying the state of a system with boundaries are also independent of time, and the boundaries are stationary in a particular coordinate system. Therefore, the stationary state of a substance at ary point is related to the stationary state of the system. 

The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states these variables are not all independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. 

The general VLE problem involves a multicomponent system of N constituent species for which the independent variables are T,P,N -I liquid-phase mole fractions, and N - I vapor-phase mole fractions. (Note that = 1 and yi = 1, where Xi and yi represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations ... 

For any system at equilibrium, the number of independent variables that must be arbitrarily fixed to establish its intensive state is given by the celebrated phase mle of J. Willard... 

The intensive state of a PVT system containing N chemical species and n phases in equilibrium is characterized by tlie intensive variables, temperature T, pressure P, and... 

Duhem. s theorem, is another mle, similar to the phase mle, but less celebrated. It applies to closed systems at equilibrium for which the extensive state as well as the intensive state of the system is fixed. The state of such a system is said to be completely determined, and is characterized not only by the 2 "I" (N — l) 7r intensive phase-mle variables but also by the n extensive variables represented by the masses (or mole numbers) of the phases. Thus the total number of variables is ... 

A thermodynamic system is in equilibrium when there is no change of intensive variables within the system. An equilibrium state is a state that cannot be changed without interactions with the environment. This definition includes that of mechanical equilibrium, but is a more general one. A state of thermodynamic equilibrium is a state of simultaneous chemical-, thermal-, and mechanical equilibrium. In other words, thermodynamic equihbrium is the state of the simultaneous vanishing of all fluxes ([32], p. 267). A thermodynamic system is thus... 

This relation is recognized from introductory subjects on thermodynamics. Recall that in equilibrium thermodynamics a local formulation is generally not needed, since the intensive state variables are independent of the space coordinates. This fundamental formulation of the total energy balance is known as the first law of thermodynamics for a closed system, which expresses the fundamental physical principle that the total energy of the system, Etotab is conserved (a postulate). 

Experimental observation 8 (Secs. 1.3 and 1.6). The stable equilibrium state of a system is completely characterized by values of only equilibrium propenies (and not properties that-describe the approach to equilibrium). For a single-component, single-phase sys.tem the values of only two intensive, independent state variables are needed to fi.s the thermodynamic state of the equilibrium system completely, the further specification of one extensive variable of the system fixes its size. Experimental observation 9 (Sec. 1.6j. The interrelationships between the thermodynamic state variables for a fluid in equilibrium also apply locally (i.e., at each point) for a fluid not in equilibrium, provided the internal relaxation processes are rapid with respect to the rate at which changes are imposed on the system. For fluids of interest in this book, this condition is satisfied. 

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