Duhems theorem

For the system NaCl-B20, we have either the two tfaditional components NaCl and BjO, which allow us to describe the bulk composition of aU phases in this system, or we have the four basis species Na, Cl , B+, and BjO, which allow us to describe not only the compositions of the phases but also the concentration of all dissolved species in the system. Traditional components and basis species are simply different choices of components, which have different purposes and different descriptive powers. We need more basis species because they are called upon to provide more information. 

We discussed the system NaCl-BjO in 11.2.1 in terms of the traditional components, NaCl and B2O. If, however, we use basis species as components, we will have more degrees of freedom to deal with. For example, using components NaCl and BjO, we have no control over the Na/Cl ratio, but using basis species Na , Cl , B+, and BjO, we do - we can specify Na+ and Cl independently - an extra degree of freedom. The principles involved have not changed, so we can use a modified phase rule, but in dealing with aqueous 

Fortunately, this is fairly intuitive. It just says that to define an aqueous solution p = 1) at a given T and P, we have to specify the concentration of each solute element. That is, since H2O is always one of the basis species, then there are ( - 1) degrees of freedom, which is evidently the number of solute basis species (Na+, Cl, and H+). Each additional phase present fixes the value of one basis species, and hence reduces by one the number of basis species that must be specified.  

Each of the phase rules above is used to define the equilibrium state, which means that they each relate the number of properties (understood to be intensive variables) of the system to the number of degrees of freedom. This defines the equilibrium state, but it does not define how much of the equilibrium state we have. The equilibrium state of 1 kg of water saturated with halite is the same whether we have 1 g or 1 kg of halite. But modeling programs commonly want to do more than to define the equilibrium state. They want to dissolve or precipitate phases during processes controlled by the modeler, and to keep track of the masses involved, so as to know when phases should appear or disappear. To do this, the mass of each phase is required, not just its presence or absence. Therefore, an additional piece of information is required for each phase present, or p quantities. Almost invariably, the mass of HjO is chosen 

This assumes T and P have been chosen, so they are not included as degrees of freedom. To include these degrees of freedom, we would write (11.5) as 

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Derive Eq. (2.107) using this relationship and the Gibbs-Duhem theorem,... 

The difference between variables and equations, equal to 2, suggests that specification of two variables suffices for the complete determination of the equilibrium state of a multiphase system, provided that the initial amounts of its components are known. This is referred to as the Duhem Theorem. These two variables can be intensive or extensive. Keep in mind, however, that the number of independent intensive variables is determined by the phase rule (see next Example). 

We then move to the more complex cases reactions taking place in heterogeneous systems and multiple reactions. To this purpose we consider first the Phase Rule and the Duhem theorem - as they apply to reacting systems - and discuss the methodology for identifying the number of independent reactions for the formation of a given equilibrium mixture. 

The Phase Rule and the Duhem Theorem for Reacting Systems... 

In developing the Duhem theorem for non-reacting systems in Section 12.7, we subtracted the number of equations from the number of vari-... 

The Duhem theorem therefore can be stated in the following general form An equilibrium system, resulting from specified initial amounts of its components, is completely determined if two variables that vary independently at equilibrium are fixed, independently of the number of reactions and phases involved. This theorem is very important in reacting systems, for we are often interested in determining the equilibrium composition of a system of specified initial amounts of reactants at constant temperature and pressure. 

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. 

Dolezalek s theory of mixtures, 402 rule, 403 Double-layer, 454, 470 Duhem s theorem, 219 Margule s equation, 395 Duhring s rule, 180... 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

This statement is similar to Duhem s theorem, which states that values must be assigned to only two independent variables in order to define the state of a closed system for which the original number of moles of each component is known. 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

Duhem s theorem states that, for any closed system formed initially given masses of particular chemical species, the equilibrium state is compl determined (extensive as well as intensive properties) by specification of any independent variables. This theorem was developed in Sec. 12.2 for nonrea systems. It was shown there that the difference between the number of indepet] variables that completely determine the state of the system and the number independent equations that can be written connecting these variables is... 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

Duhem s theorem is another rule, similar to the phase rule, but less celebratec It applies to closed systems for which the extensive state as well as the intensiv state of the system is fixed. The state of such a system is said to be completel determined and is characterized not only by the 2 + (iV—l)ir intensive phase rule variables but also by the it extensive variables represented by the masse (or mole numbers) of the phases. Thus the total number of variables is... 

On the basis of this result, Duhem s theorem is stated as follows ... 

By relating the fluctuations to the reaction coordinate v, de = SNh Eqs. (12.10) and (12.12) imply that if a system is stable to fluctuations in diffusion, it is also stable to fluctuations in chemical reactions, which is called the Duhem-Jougeut theorem (Kondepudi and Prigogine, 1999). However, a nonequilibrium steady state involving chemical reactions may be unstable even if the system is stable with respect to diffusion. 

Galvanostatic transients, 66, 357, 359, 394 Gas-diflusion electrodes, 484 Gauss theorem. 192, 339 Gibbs adsorption isotherm, 228 Gibbs-Duhem equation, 235... 

The complete investigation of this important theorem (usually attributed to Lejeune Dirichlet) is difficult see Duhem, Lemons sur l lectricit6 et le Magn6tisme, Paris, 1891, 1, 159 Maxwell, Treatise on Electricity and Magnetism, Oxford, 1892, 1, 136. 

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