Intensive variables independent

This is Gibbs s phase rule. It specifies the number of independent intensive variables that can and must be fixed in order to estabbsh the intensive equibbrium state of a system and to render an equibbrium problem solvable. 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

A, A,B,C,... are phenomenological coefficients that are functions of the independent intensive variables P,T,... In order to meet the equilibrium condition at rj = 0, A must vanish. Furthermore, for stable phases, A (7 ,/ )>0 and C(T,P) is also >0, otherwise G would become excessively negative for larger values of tj. (G-G°) vs. 11 is depicted in Figure 12-5a. If B2 = 4 AC, the minima for (G-G°) at t] = 21/UC and -4 AC have the same value G. This means that here two phases would coexist in equilibrium, which is characteristic of a first-order phase transition. 

The Gibbs-Duhem equation is the basis for the phase rule of Gibbs. According to the phase rule, the number of degrees of freedom F (independent intensive variables) for a system involving only PV work, but no chemical reactions, is given by... 

Since F is the symbol for the number of independent intensive variables for a system, it is also useful to have a symbol for the number of natural variables for a system. To describe the extensive state of a system, we have to specify F intensive variables and in addition an extensive variable for each phase. This description of the extensive state therefore requires D variables, where D = F + p. Note that D is the number of natural variables in the fundamental equation for a system. For a one-phase system involving only PV work, D = Ns + 2, as discussed after equation 2.2-12. The number F of independent intensive variables and the number D of natural variables for a system are unique, but there are usually multiple choices of these variables. The choice of independent intensive variables F and natural variables D is arbitrary, but the natural variables must include as many extensive variables as there are phases. For example, for the one-phase system described by equation 2.2-8, the F = Ns + 1 intensive variables can be chosen to be T, P, x, x2,xN. and the D = Ns + 2 natural variables can be chosen to be T, P, ni, n2,..., or T, P, xx, x2,..., xN and n (total amount in the system). 

This last equation is the Gibbs-Duhem equation for the system, and it shows that only two of the three intensive properties (T, P, and fi) are independent for a system containing one substance. Because of the Gibbs-Duhem equation, we can say that the chemical potential of a pure substance substance is a function of temperature and pressure. The number F of independent intensive variables is T=l — 1+2 = 2, and so D = T + p = 2 + l = 3. Each of these fundamental equations yields D(D — l)/2 = 3 Maxwell equations, and there are 24 Maxwell equations for the system. The integrated forms of the eight fundamental equations for this system are ... 

Since the chemical potential /i(H + ) depends on both the temperature and the concentration of hydrogen ions, it is not a very convenient variable when the temperature is changed. The hydrogen ion concentration can be made an independent intensive variable in the fundamental equation for G by use of the expression for the differential of the chemical potential of H + ... 

In discussing one-phase systems in terms of species, the number D of natural variables was found to be Ns + 2 (where the intensive variables are T and P) and the number F of independent intensive variables was found to be Ns + 1 (Section 3.4). When the pH is specified and the acid dissociations are at equilibrium, a system is described in terms of AT reactants (sums of species), and the number D of natural variables is N + 3 (where the intensive variables are T, P, and pH), as indicated by equation 4.1-18. The number N of reactants may be significantly less than the number Ns of species, so that fewer variables are required to describe the state of the system. When the pH is used as an independent variable, the Gibbs-Duhem equation for the system is... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

The two independent variables subject to specification may in general be either intensive or extensive. However, the number of independent intensive variables is given by the phase rule. Thus when F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive. 

To make the connection between the observed electrical, optical, etc., behavior and the type and concentration of point defects in a solid compound, one must first have some qualitative ideas as to the expected properties of a solid arising from the presence of a particular type of defect. A simple example is discussed. Further details are given by Kroger and Vink (20, 22). Secondly, some theory must be available which relates the concentration of the various defects to the independent intensive variables of the solid phase and the parameters which characterize the particular compound under study. This theory and some of its applications are discussed below. 

The calculations in Chapters 3 to 5 have been based on the use of Legendre transforms to introduce pH and pMg as independent intensive variables. But now we need to discuss the reverse process - that is the transformation of Af G ° values calculated from measured apparent equilibrium constants in the literature to Af G° values of species and the transformation of Af° values calculated from calorimetric measurements in the literature to AfH° of species. This is accomplished by use of the inverse Legendre transform defined by (7) ... 

The number of independent intensive variables, the variance, or the number of degrees of freedom w, is thus given by... 

It should be further mentioned that the choice of the ensemble also depends on whether or not the solvent is explicitly modeled. In particular, care has to be exerted concerning the number of intensive variables of the ensemble. This can be understood considering the Gibbs phase rule for interfaces [93] f = 2 + c - p, with / being the number of independent intensive variables needed to describe the interface, p the number of different phases in the interface, and c denoting the number of different components. A lipid bilayer comprising one sort of lipid embedded into an implicit solvent corresponds to a one-component system c = 1, in one-phase state so that p = 1 hence / = 2. On the other hand, a model with explicit solvent yields c = 2, p = 1, and / = 3. Thus, implicit solvent models can be simulated within nXT ensemble while for explicit-solvent models an additional intensive quantity has to be controlled, e.g., the nPzzXT ensemble is appropriate. 

The number of degrees of freedom, or variance is the number of independent intensive variables—temperature, pressure, and concentrations—that must be fixed to define the equilibrium state of the system. If fewer than S variables are fixed, an infinite number of states fits the assumptions if too many are arbitrarily chosen, the system will be overspecified. When there are only two phases, as is usually the case, in systems of two components, therefore,... 

We use now the inductive method. Assume that we have P phases. Each component is present in each phase. Then the number of total independent intensive variables is y(P). If we add one phase, then the independent intensive variables increase by a -b / - 1. The term -1 arises because we have already merged one equation, i.e., the Gibbs - Duhem equation. 

On the other hand, if we add one phase, then we must add for each independent intensive variable one equation to couple the new phase with respect to equilibrium. So the number of equations E increases by... 

Graphic Representation of the Generalized Transition Zone for Contraction Due to a Generalized Independent (Intensive) Variable... 

Generally, in x-ray spectrometry, the relationship between the independent (intensity) variable y and the dependent (composition) variable x cannot be expressed by a simple linear equation such as y = c + mx when x ranges over wide values of composition. 

From this equation we can see the number of independent intensive variables in any homogeneous phase. There are c terms containing p, i.e., c independent compositional intensive variables, plus two other intensive terms, T and P, for a total of c -l- 2 intensive variables. In a single homogeneous phase, these c -I- 2 variables are linked by one equation (4.68), so only c -f- 2 -1 of them are independent. If there are p phases, there are still only c-l-2 intensive variables, because they all have the same value in every phase (at equilibrium), but now there is one equation (4.68) for each phase. Each additional equation reduces the number of independent variables by one, so there are now c + 2-p independent intensive variables. These independent intensive variables are called degrees of freedom, /, so... 

For example, if the viscosity of a sample of water is chosen as 0.506 X 10 N s m- and its refractive index as 1.328 9, then its density is 0.988 1 g cm 8, its hotness is 50 C, etc. In the next section, instead of choosing viscosity and refractive index, we shall take as our reference variables the pressure and density, which are a more convenient choice. On this basis we shall discuss what is meant by hotness or temperature (which it is part of the business of thermodynamics to define), and thereafter we shall take pressure and temperature as the independent intensive variables, as is always done in practice. 

The degrees of freedom, F, is defined as the difference between the number of independent intensive variables of the system [which is p(c +1)] and the number of equations between these variables [which is (c -f 2)(p -1)]. 

The above relationship can be used to describe the state of a particular phase of a composite system it provides the number of independent intensive variables for specifying the system. Since chemical potential of a phase is a function of temperature, pressure, and mole fraction of components 1 to (c— 1), then one may use (c - 1.) mole fractions instead of c chemical potential for every phase. 

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