Advances in Chemical Physics, Volume 43 by Ilya Prigogine, Stuart A. Rice

By Ilya Prigogine, Stuart A. Rice

The Advances in Chemical Physics sequence presents the chemical physics and actual chemistry fields with a discussion board for serious, authoritative reviews of advances in each region of the self-discipline. jam-packed with state-of-the-art study pronounced in a cohesive demeanour no longer came across somewhere else within the literature, every one quantity of the Advances in Chemical Physics sequence serves because the excellent complement to any complicated graduate category dedicated to the learn of chemical physics.

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First let us compare the number of parameters in the general open system, and in a similar closed system in thermodynamic equilibrium. 37) where one phase is assumed ( P = I), and the number of components is C = n - d. As parameters for this system, choose the reciprocal temperature /3 = 1/ RT, and the dimensionless product p*, j E [ 1, n - d], where is the chemical potential of speciesj . These parameters are the fundamental variables of the entropy representation of thermodynamics (Callen) and are the most profound parameters because the partition functions of the statistical mechanical ensembles all take the same general form only when expressed in these parameters.

This process may be continued until we obtain a cone that does not intersect any coordinate hyperplane except at the origin. At the kth stage the cone eUkhas dimension r - d - k + 1. If the dimension of is greater than one, S,, must intersect some coordinate axes and the process can always be continued. However when k = r - d, eOkis a one-dimensional cone or hufffine, and after dropping all components up STABILITY OF COMPLEX REACTION NETWORKS 23 containing the half-line, S,-d will not intersect the coordinate hyperplanes of any remaining component u,?

55) must have a double root. A polynomial has this property if and only if it can be put into the form CU,(Xi - l)(Xi - 1) where ug is a polynomial in j and x. The relationships among the components of j that must be satisfied to give the polynomial this form are the equations of the fold. Since convex parameters are nonnegative,' it is easy to recognize whether such conditions can be satisfied for physical steady states. Example Il9. 7). We inspect the coefficients of x - I,y - 1, z - 1 to see if there is a polynomial where all coefficients can vanish for some Ej E II,.

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