Intensive variables definition

The second set of equations is obtained from the first set by the Gibbs integration at constant intensive variables, as was done in obtaining Eq. III-77. It is convenient, in dealing with a surface species, to introduce some special definitions, two of which are... 

The rate is defined as an intensive variable, and the definition is independent of any partieular reaetant or produet speeies. Beeause the reaetion rate ehanges with time, we ean use the time derivative to express the instantaneous rate of reaetion sinee it is influeneed by the eomposition and temperature (i.e., the energy of the material). Thus,... 

The irreversible processes described must not occur even on open circuit. In a reversible cell, a definite equilibrium must be established and this may be defined in terms of the intensive variables in a similar way to the description of phase and chemical equilibria of electroneutral components. 

In geology it is customary to consider systems in which the intensive variables pressure (P) and temperature (T) are characteristic of the ambient and, therefore, are prefixed and constant. In these conditions, the Gibbs free energy of the system (G) is at minimum at equilibrium. The treatments presented in this chapter are based on this fundamental principle. Let us first introduce in an elementary fashion some fundamental definitions. 

The conditions for eqnilibrinm have not changed, and application of the phase rnle is conducted as in the previous section. The difference now is that composition can be counted as an intensive variable. Composition is accounted for through direct introduction into the thermodynamic quantities of enthalpy and entropy. The free energy of a mixtnre of two pure elements, A and B, is still given by the definition... 

It should be recognized that Eq. 2.27 is not in itself a conservation equation that can be "solved for anything. It may be instructive, however, to anticipate how the relationship might be used to form a conservation equation. If N represents mass, then dN/dt = 0. since a system, by definition, contains a fixed amount of mass. In the case where N represents mass, the corresponding intensive variable is rj = 1. Thus Eq. 2.27 reduces to... 

On the contrary, variables such as the temperature T, the pressure p, and the mole fraction of the component i are intensive variables since they have definite values at each point in the system. The value of an intensive variable may either be the same throughout the system or change from one point to another. 

To illustrate the distinction between extensive and intensive variables let us consider the volume F, of a single phase system containing ni moles of the component i. Since a phase is by definition homogeneous one third of the total volume V contains one third of the ni moles of the component i and, in general, a volume hV (where k is an arbitrarily chosen positive constant) contains kn moles of this component. If we write... 

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... 

The pressure, being an intensive variable, can only be a function of intensive variables so that holding V constant in the partial derivative is of no importance. From the definition of the isothermal compressibility xt s p-1(dp/dP)T it follows that (dp/dN)T,v = 1 IVp2xt)- Also from the equation dE = TdS — pdV + pdN, it follows that... 

The operational definition of y via a small change in surface area may conceptually be connected with a concomitant change in the concentration of one or several of the chemical components, AUi, in the second phase. To keep all concentrations (intensive variables) constant, the amount of An has to be replenished in that phase by adding it to the system through a leak, keeping T and V constant. The only place where the small amount of component i can go is the incremental increase in interfacial area, AA. This operation therefore defines the adsorption of component i on the interface between the two phases 1 and 2. The adsorbed amount per unit area is called the surface excess, equal to... 

For convenience, thermodynamic systems are usually assumed closed, isolated from the surroundings. The laws that govern such systems are written in terms of two types of variables intensive (or intrinsic) that do not depend on the mass and extensive that do. By definition, extensive variables are additive, that is, their value for the whole system is the sum of their values for the individual parts. For example, volume, entropy, and total energy of a system are extensive variables, but the specific volume (or its reciprocity - the density), molar volume, or molar free energy of mixing are intensive. It is advisable to use, whenever possible, intensive variables. 

Metal-solution interfaces are of obvious importance to corrosion, but they are particularly difficult to model. By definition, the interface comprises that part of the system in which the intensive variables of the two adjoining phases differ from their respective bulk values, and even in concentrated solutions this implies a thickness of the order of 15-20 A. This is too large to be modeled solely by density functional theory (DFT), which surface scientists often use as a panacea for the metal-gas interface. In addition, the two adjoining phases are of very different nature metals are usually solid at ambient temperatures, and their properties do not differ too much from those at 0 K, so that DFT, or semiempirical force fields like the embedded atom method, are good methods for their investigation. By contrast, the molecules in solutions are highly mobile, and thermal averaging is indispensable. Therefore, the two parts of the interface usually require different models, and an important part of the art consists in their combination. 

If a multiphase multicomponent system is to be at equilibrium (no change with time of the intensive variables) obviously temperature and pressure must be the same for all phases and also the chemical compositions (mole fractions of each constituent). In any given phase there are (C—1) independent mole fractions (their sum is unity by definition), so there are P.(C—1) composition variables involved and thus [P.(C—1) -1-2] intensive variables in total. But if chemical equUibrium in all phases simultaneously is to hold, the chemical potential of each constituent (a function of the composition) must be the same in each phase thus there are C.(P—1) independent constraints on the composition variables arising from the equilibrium condition (the chemical potential in one of the phases is used as the reference standard for the other phases). Thus F = [P.(C—1) -1-2] — [C.(P—1)] =C—P-F2. This is the famous Gibbs Phase Rule. 

It is understood that under the small strain state the variable of u is e and d Ua is composed of the increments of the extensive variable e and the intensive variable (see Appendix D for the definition of the extensive and intensive variables). 

Where entropy S and volume V are the extensive variables and temperature T and pressure p are the conjugated intensive variables (the signs indicated follow from the definition of the positive direction of the energy fiow, namely, one directed into the system). Thus, the thermodynamic potential of this system is E=E(S, V) if this function is known, the system can be regarded as completely described. But the variables S and V cannot be controlled in a simple, direct manner. Thus, it would be preferable to select the pressure p and the temperature T as variables because they can easily be set experimentally. Which quantity is then the proper thermodynamic potential ... 

It is noteworthy that a system, even when not in equilibrium, always has definite extensive variables as for the intensive variables, they are - as a rule - definite in a stable or metastable state only. Hence, the preference is for extensive quantities as independent variables. 

This definition suffices as the other concentration, 5Ca= Na/(Na +Nb), can be obtained from the equation Xa + Xb = I. Thus, only one additional variable, the concentration variable Xb, is required for a two-component system. In the presence of Ko components, Ko — 1 additional variables are necessary for describing the system. The concentrations so defined are intensive variables. As the number of components increases, the description of the system becomes increasingly more complicated, and its representation is quite difficult. 

Consider a system in an equilibrium state. In this state, the system has one or more phases each phase contains one or more species and intensive properties such as T, p, and the mole fraction of a species in a phase have definite values. Starting with the system in this state, we can make changes that place the system in a new equilibrium state having the same kinds of phases and the same species, but different values of some of tbe intensive properties. The number of different independent intensive variables that we may change in this way is the number of degrees of freedom or variance, F, of the system. 

The number of degrees of freedom is also the number of independent intensive variables needed to specify the equilibrium state in all necessary completeness, aside from the amount of each phase. In other words, when we specify values of F different independent intensive variables, then the values of all other intensive variables of the equilibrium state have definite values determined by tbe physical natme of the system. 

Since there are three degrees of freedom, we could, for instance, specify arbittary values of r, p, and TN2 then the values of other intensive variables such as the mole fractions TH20 and xnj would have definite values. 

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