Chemical Equilibrium Problems With Unbounded Constraint Sets

An investigation of the use of mathematical models to explore the chemical aspects of physiological systems; this deals with the theoretical and computational aspects of understanding the chemistry of human physiological function. The question of existence of solutions to problems having unbounded constraint sets is investigated by relating their existence (or nonexistence) to a property of a solution to an auxiliary chemical equilibrium problem with a bounded constraint set. An example system is selected consisting of gases in contact with an aqueous buffer solution at a uniform total hydrostatic pressure and temperature. The numerical problem of determining the amount of CO2 to be added to achieve a specified partial pressure of CO2 in the gas phase, and its effects on the composition of the total system, is solved by using a procedure suggested by the concept of unbounded constraint sets. Findings may apply to design of artificial life-support systems needed in extraterrestrial environments related to Air Force missions. (Author).

A single- or multi-phase chemical equilibrium problem may be though of as the problem of minimizing a particular nonlinear function (the free energy) of composition subject to the conditions that the composition vector be nonnegative and satisfy a system of linear equations (the mass-balance laws). It was pointed out in a previous paper (AD-706 018) that the free energy is convex and homogeneous of degree one, but that as a variable approaches zero, the free energy may behave badly. In this paper, the phase 'chemical equilibrium problem' refers only to a problem with a particular mathematical form. Problems of this form arise in many situations that are not classically denoted chemical equilibrium problems. (Author).

The aim of the book is to cover the three fundamental aspects of research in equilibrium problems: the statement problem and its formulation using mainly variational methods, its theoretical solution by means of classical and new variational tools, the calculus of solutions and applications in concrete cases. The book shows how many equilibrium problems follow a general law (the so-called user equilibrium condition). Such law allows us to express the problem in terms of variational inequalities. Variational inequalities provide a powerful methodology, by which existence and calculation of the solution can be obtained.

In physical chemistry, the 'chemical equilibrium problem' is the problem of determining the distribution of chemical species that minimizes the free energy of a system while conserving the mass of each of the chemical elements. The reactions occurring within the chemical system may be quite complex. However, in a great number of cases, the mathematical statement of the problem can be simplified to a particular mathematical form (AD-605 316 and Dantzig and DeHaven, J. Chem. Phys. 36:2620-2627 (1962)) involving the minimization of a nonlinear objective function over a set of linear constraints. This Memorandum presents the numerical solution of the chemical equilibrium problem by describing methods for starting the solution when an initial estimate is not available, and for improving an initial estimate to make it feasible. It presents a firstorder method and a second-order method for solving the chemical equilibrium problem in the context of the linearlogarithmic programming problem (AD-407 547) and provides convergence criteria for the majority of problems of this type that are likely to be attempted. (Author).