Determining the Number of Independent Components by Brinkley's Method Muyu Zhao, Zichen Wang, and Liangzhl Xiao Jilin University, Changchun 130023, China Gibbs phase rule describes a system with the variables temperature T and pressure P at equilibrium by
where f is the number of degrees of freedom; C is the number of independent components; and 4 is the number of phases present in the equilibrium system. According to Jouguet's method, which is extensively used in physical chemistry, C may be calculated.
where N is the number of chemical species (or components); r is the number of independent chemical reactions in the system; and Z is the number of constrained conditions other than those given by z x , = l and zr,=l
wherex; and x, are the mole fractions of the ith component in the system in the jth phase and where i = 1,2, ...N a n d j = 1,2, ... 4. As universal as Jouguet's method is, there are still many problems applying it, especially for the systems in which only the initial compositions are known. In the equilibrium state, i t is difficult to determine both N and r. Due to interactions among chemical species (e.g., chemical reaction, ionization, and association) there might be many species present in the equilibrium system, and they usually are not the same as those present in the initial state. When this is the case, it is very difficult to determine N. Moreover, as N increases, the number of chemical reactions among all the components will increase dramatically. Then i t becomes very difficult to wnsider all the probable chemical reactions in the system and to select the independent ones. For example, in water AICl3 dissolves, ionizes, hydrolyzes, and partially deposits AI(OH)3. Such a system is actually very complicated. It is inconvenient to determine C in a complicated system using Jouguet's method. We will introduce a convenient method to deal with the number of independent components in the complex system. Basic Principles of Brinkley's Method Early in 1947, a method was suggested by Brinkley ( I ) , but it received little attention because Brinkley's original formula was flawed. The method has since been improved, and is now widely used (2)with the revised Brinkley's formula
where M is the number of elements in the system; Z is the same as in Jouguet's method described above; and r" is the number of independent reactions that do not occur due to restrictions imposed by kinetic conditions. For example, a t
room temperature and with no catalyst, there will be no reaction betweenHz and Oz,in the system composed of Hp, OZ,and HzO. The following reaction does not occur.
Thus, r" is equal to one. The principle on which Brinkley's method relies is simple. The chemical components of a system are composed of the atoms of various elements that are present in the different comoonents in various ratios. These ratios are described hy chemical stoichiometric coefficients. Brinklev's method attemots to determine the relationships that exist among the stoichiometric coefficients of the wmponents. Then the number of independent components can be determined. In a closed system, M elements compose N components. The chemical formula of the ith componentAi (i = 1, 2,...M may be expressed by where E. is the symbol of eth element; e = 1,2,...M; and a, is the number of atoms of E. in the chemical formula A;. Take a matrix to represent the chemical formulas of all the components.
If the rank of the matrix a, is C, the number of independent components is C. Brinkley pointed out the following. 'IfN>M,thenC
