The Uniqueness of Equilibrium under Adiabatic Conditions

In 4 of P it was shown that the equilibrium conditions for a set of r independent reactions, namely. 

Suppose now that the reaction takes place adiabatically at constant pressure from an initial composition cf and temperature To. Then the concentrations will be governed by 

In theorem 12 of P it was shown that the first term is that of a positive definite Gramian matrix. The second is clearly the general term of a positive semi-definite matrix we cannot assert that it is definite since one or more of the A Hi might be zero. But the sum of a positive definite and a positive semi-definite matrix is definite and so the Jacobian nowhere vanishes. 

It is evident from the method of proof that a range of semi-adiabatic cases in which (dTldt r) = 6 -h.H)rICp will also yield unique equilibria it is only necessary that B be independent of r. 

A theorem of this kind is needed to establish the existence and properties of the solution of the equations for a tubular reactor, as in the work of Amundson Luss [8]. 

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