Phase Rule and Duhems Theorem for Reacting Systems

The phase mle (applieable to intensive properties) as discussed in Secs. 2.7 and 10.2 for nonreacting systems of tt phases and N ehemieal species is  

It must be modified for application to systems in whieh chemical reactions occur. The phase-rule variables are unchanged temperature, pressure, and N — 1 mole fractions in each phase. The total lumiber of tliese variables is 2 (N — 1)(7t). The same phase-equilibrium equations apply as before, and tliey lumiber (ti — 1)(A ). However, Eq. (13.8) provides for each independent reaction an additional relation tliat must be satisfied at equilibrium. Since the /r,- s are functions of temperature, pressure, and the phase compositions, Eq. (13.8) represents a relation comiect-ing phase-mle variables. If tliere are r independent chemical reactions at equilibrium within the system, then there is a total of (rr — 1)(A ) H-r independent equations relating tlie phase-rule variables. Taking the difference between the number of variables and the nmnber of equations gives  

The only remaining problem for application is to detennine the nmnber of independent chemical reactions. This can be done systematically as follows  

The set of r equations resulting from this reduction proeedure is a complete set of independent reactions for the N speeies eonsidered present in tlie system. However, more tlian one sueh set is possible, depending on how the reduction procedure is earned out, but all sets number r and are equivalent. The reduetion procedure also ensures tlie following relation  

The phase-equilibrium and ehemieal-reaetion-equilibrium equations are the only ones considered in the foregoing treatment as interrelating the phase-rule variables. However, in certain situations special constraints may be plaeed on tlie system that allow additional equations to be written over and above tliose eonsidered in the development of Eq. (13.36). If die number of equations resulting from speeial eonstraints is s, theiiEq. (13.36) must be modified to take accoimt of these 5 additional equations. The still more general fomi of the phase rule that results is  

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