Duhem’s 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... 

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. 

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 ... 

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. 

This means that in the case of a system which forms an azeotrope, Duhem s theorem ceases to be valid if the temperature and pressure are chosen corresponding to the point at which the compositions of the two phases are equal. If instead of T and p we choose T and F as the variables then these systems do not present any anomaly. 

Let us now consider a closed system which is at least bivariant. Then, apart from the case of indifferent states, we can, from Duhem s theorem (c/. chap. XIII, 6 and 7), describe all equilibrium states of the system in terms of two variables, T and p. We have... 

We now examine the application of the phase rule and Duhem s theorem to an osmotic system. The phase rule was deduced in chap. XIII by assuming that all phases of the system were subject to the same applied pressure p. The introduction of a semi-permeable membrane increases the variance of the system by one, and renders possible the existence of two different pressures p and p". Instead of the 2 + C(/> variables T, p,. .. we have now 3-fc< variables T, p p", x. .. x. ... 

We found in chapter XIII that the equilibrium state of a closed system, corresponding to given initial conditions, is completely determined by two variables (Duhem s theorem). For polyvariant systems... 

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