Duhem theorem

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. 

Spear F. S. (1988) The Gibbs method and Duhem s theorem the quantitative relationships among P, T, chemical potential, phase composition and reaction progress in igneous and metamorphic systems. Contrib. Mineral. Petrol 99, 249-256. 

PHASE RULE AND DUHEM S THEOREM FOR REACTING SYSTEMS... 

Phase Rule and Duhem s Theorem for Reacting Systems... 

Most chemical-reaction equilibrium problems are so posed that it is Duhem s theorem that makes them determinate. The usual problem is to find the composition of a system that reaches equilibrium from an initial state of fixed amounts of of reacting species when the fu o variables T and P are specified. 

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