Phase rule 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. 

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... 

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

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

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. ... 

The term Extensive Phase Rule is our own terminology, and may prove confusing to geochemists more used to seeing it referred to as Duhem s Theorem. As expressed by Prigogine and Defay (1965), p. 188, Duhem s Theorem says... 

Duhem s theorem is applied to closed systems at equilibrium, when both intensive and extensive parameters are known, i.e., for systems of totally defined state. Both extensive and intensive parameters may be independent. However, their interrelation is defined by Gibbs phase rule. At C = 0 both parameters must be extensive, and at C = 1 at least one of them must be extensive. 

We start in 9.1 by giving prescriptions for determining the number of properties needed to identify the thermodynamic state in multicomponent mixtures. Those prescriptions include Duhem s theorem and the Gibbs phase rule as special cases. The required number of properties determines the dimensionality of the state diagram needed to represent phase behavior. Then in 9.2 we summarize some features of pure-component diagrams that have not been discussed in earlier chapters. 

In addition to the phase rule discovered by Gibbs, there is another general observation which Pierre Duhem made in his treatise Traite elementaire de Mecanique Chimique, which is referred to as Duhem s theorem. It states ... 

Use of the equations that must be satisfied by a system in equilibrium combined with the number of system variables, leads to the phase rule and Duhem s theorem that provide for the complete specification of the state of the system. 

Consider a vapor-liquid equilibrium mixture containing three components. According to the phase rule P = 3 + 2- 2 = 3, but according to Duhem s theorem, specification of two of these variables suffices to completely describe this system. Is there a paradox here ... 

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