Equilibria natural variables

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

This fundamental equation for the entropy shows that S has the natural variables U, V, and n . The corresponding criterion of equilibrium is (dS) 0 at constant U, V, and n . Thus the entropy increases when a spontaneous change occurs at constant U, V, and ,. At equilibrium the entropy is at a maximum. When U, V, and , are constant, we can refer to the system as isolated. Equation 2.2-13 shows that partial derivatives of S yield 1/T, P/T, and pJT, which is the same information that is provided by partial derivatives of U, and so nothing is gained by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide any new information, we will not discuss it further. 

The preceding section was based on the fundamental equation for G in terms of the extent of reaction, but in order to identify the D natural variables for a one-reaction system at equilibrium, we need to apply the condition for equilibrium ZvjjUi = 0 (equation 3.1-6) that is due to the reaction. That is done by using each independent equilibrium condition to eliminate one chemical potential from equation 2.5-5. This is more easily seen for a simple reaction ... 

This form of the fundamental equation for G applies to a system at chemical equilibrium. Note that the number D of natural variables of G is now C + 2, rather than Ns + 2 as it was for a nonreaction system (see Section 2.5). There are... 

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... 

The preceding paragraph applies to a system in which there are no biochemical reactions. Now we consider systems with reactions that are at equilibrium (Alberty, 1992d). For a chemical reaction system, we saw (Section 3.4) that D = C + 2 and F = C + 1 for a one-phase system. For a biochemical reaction system at equilibrium, we need the fundamental equation written in terms of apparent components to show how many natural variables there are. When the reaction conditions 2 vj//f = 0 for the biochemical reactions in the system are used to eliminate one for each independent reaction from equation 4.1-18, the following fundamental equation for G in terms of apparent components is obtained ... 

The number D of natural variables for a system and the number F of intensive degrees of freedom for a one-phase system at equilibrium were discussed in Section 4.6, but now we can discuss these numbers in a more general way. Table 5.1 gives these numbers for three descriptions of a one-phase reaction... 

Table 5.1 Numbers of Natural Variables and Numbers of Intensive Degrees of Freedom for One-Phase Reaction Systems at Equilibrium...

Since concentrations of ATP, ADP, NADox, and NADred may be in steady states, it is of interest to calculate equilibrium compositions that correspond with these steady state concentrations. These calculations are referred to as level 3 equilibrium calculations because they are based on the introduction of [ATP], [ADP], and the like, as natural variables by use of a Legendre transform. 

This shows that the natural variables for G for this system before phase equilibrium is established are T, P, nAx, and nAp. When A is transferred from one phase to the other, dnAa = — dnAp. Substituting this conservation relation into equation 8.1-1 yields... 

This form of the fundamental equation, which applies at equilibrium, indicates that the natural variables for this system are T, nAx, and nAfi. Alternatively, P, nAx, and nAp could be chosen. Specification of the natural variables gives a complete description of the extensive state of the system at equilibrium, and so the criterion of spontaneous change and equilibrium is dG < 0 at constant 7( nAz, and nA/l or... 

The equilibrium relations of the preceding section were derived on the assumption that the charge transferred Q can be held constant, but that is not really practical from an experimental point of view. It is better to consider the potential difference between the phases to be a natural variable. That is accomplished by use of the Legendre transform (Alberty, 1995c Alberty, Barthel, Cohen, Ewing, Goldberg, and Wilhelm, 2001)... 

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. 

Gibbs considered the statistical mechanics of a system containing one type of molecule in contact with a large reservoir of the same type of molecules through a permeable membrane. If the system has a specified volume and temperature and is in equilibrium with the resevoir, the chemical potential of the species in the system is determined by the chemical potential of the species in the reservoir. The natural variables of this system are T, V, and //. We saw in equation 2.6-12 that the thermodynamic potential with these natural variables is U[T, //] using Callen s nomenclature. The integration of the fundamental equation for yields... 

Because of the simplicity of these relationships, we sometimes say that the natural variables of U are S and V, of // arc S and P, of A are V and T, of G are P and T, and of S are U and V. It is noteworthy that the natural variables of U are both extensive variables and those of G are both intensive variables the natural variables of H and A are mixed—one extensive and one intensive variable for each. Because Eqs. (20)-(24) only hold for systems at material equilibrium, they will become our criteria for material equilibrium under each set of conditions. For example, at constant T and P, for a system to be at material equilibrium for a process, dG must equal zero for the (infinitesimal) process. 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

The transformed Gibbs energy provides the criterion for spontaneous change and equilibrium in systems of enzyme-catalyzed reactions when the independent variables for the system are T, P, pH, and Wc - Notice that making this Legendre transform has introduced ) as a natural variable, but it has not changed the number of natural variables because there is now one less component that is conserved, the hydrogen atom component. 

It needs to be emphasized at this point that one could, of course, express each thermodynamic potential in terms of different. sets of (nonnatural) variables. The immediate consequence is that thermodynamic potentials would not necessarily attain a minimum value if the system is in a state of thermodynamic equilibrium. This point is important to realize because it implies that the set of natural variables of a given thermodjmamic potential is distinguished and unique eunong other conceivable. sets of variables. 

Helmholtz energy is displayed as the natural energy function of its natural variables T and V. The geometry of the equilibrium surface is expressed by the coefficients of the differentials. 

In principle, such an expert system could also consider the extent to which a wide variety of minerals would influence ground-water chemistry at various temperatures and reaction times, as well as the natural variability and experimental uncertainties associated with the thermochemical data used to model geochemical equilibrium. To develop a consensus in the geochemical community about the categories of information required to evaluate mineral reactivity and the properties appropriate for each of many minerals, would be a time consuming task. Thus, in the near future at least, such enhancements will probably be limited to consideration of tightly defined problems of simple mineralogy. 

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