Handling multicomponent systems in $\mathbb{R}^n$. I: Theoretical results

Abstract

The phases of multicomponent systems (mixtures, states, etc.) containing the compounds $A_1 ,...,A_n$ are $x_1 A_1 + ... + x_n A_n$, where $0 \leqslant x_i \leqslant 1$ and $x_i + ... + x_n = 1$. For $n \geqslant 4$ (quaternary or higher dimensional systems), the displaying methods and visual investigations in the $n$‐dimensional Euclidean space $\mathbb{R}^n$ are tangentially or not at all described in the literature. In this paper we first develop the theoretical (both mathematical and computational) background in any dimension in $\mathbb{R}^n$. We focus not only on the important points, lines, surfaces of these systems, and computing method of the states of some processes in such systems, but also on the approximating methods of the above mentioned lines and surfaces, and, finally, on the question “which is the region where a state (a point) falls into”. Using the above results a computer program for PC's was created for evaluating and displaying the approximated surfaces. This program is described in I. Szalkai, SALT3DIM.exe – A program for handling 4 component mixtures, Preprint No. 047, University of Veszprém (1996), and the computing results are planned to be published in a forthcoming paper (I. Szalkai, Handling multicomponent systems in $\mathbb{R}^n$. II: Computational results, J. Chem. Inf. Comput. Sci., submitted).