Stoichiometric Formulation

In these equations G(n) is the Gibbs energy function for the mixture and n G 77. is a column vector (sometimes called the species-abundance vector) containing the unknown variables representing the number of moles of each of the q species (molecules) in the mixture. A is the predefined formula 

In this formulation the solution task consist in identifying the species composition vector Ueq that minimizes G, for fixed T and p. The governing equations to be solved in the equilibrium calculations can be derived from the total differential of the state function. Assuming G = G T,p,n), the total differential of the Gibbs function can be expressed like  

However, the number of unknowns and thus the mathematical problem to be solved is still quite large even for systems in which only a few reactions occur. To reduce the number of unknown variables, it is convenient to introduce the extent of reaction quantity. For a specific stoichiometric equation defining a particular reaction, the extent of reaction is defined by  

By definition the stoichiometric coefficient is positive for a product and negative for a reactant. 

For a system with r reactions a matrix notation may be more appropriate, hence the r stoichiometric equations and the species-abundance vector n are related through the extent of reaction by  

In these equations G(n) is the Gibbs energy function for the mixture and n e TZ is a column vector (sometimes called the species-abundance vector) containing the unknown variables representing the number of moles of each of the q species (molecules) in the mixture. A e is the predefined formula matrix. This coefficient matrix defines how the m elements in the mixture are distributed within the q species (molecules) that can exist in the system, m is the total number of elements in the mixture, b e 7 is a column vector (sometimes called the element-abundance vector) containing the known values for the total amount of the different elements in the mixture. 

However, the number of unknowns and thus the mathematical problem to be solved is still quite large even for systems in which only a few reactions occur. To reduce 

---

We regard here a catalytic reaction with the following stoichiometric formulation ... 

Series of nonideal networks obtained using off-stoichiometric epoxyamine formulations have also been employed for yielding studies. As the highest Tg corresponds to the stoichiometric formulation, it is expected that <7y, measured at a constant temperature, should also be a maximum for the same formulation. This was the trend reported in some of these studies (Morgan et al., 1984 Vallo et al., 1993). 

H. GERISCHER I do not think that a stoichiometric formulation should be used for the consumption of holes and electrons in the dissolution of germanium. We found no... 

As shown in Figure 15.7 and 15.8, simple correlations in AN, AP, and AO2 are typically observed in lakes and in the ocean. The stoichiometric formulation of equation 3 reflects in a simple way Liebig s law of the minimum. It follows from Figure 15.7a,b that seawater becomes exhausted simultaneously in dissolved phosphorus and nitrogen as a result of photosynthetic assimilation. We infer that nitrogen and phosphorus together determine the extent of organic... 

The stoichiometric formulation, in which the constraints are incorporated through stoiciometric reaction equations. These procedures are sometimes referred to as the classical equilibrium constant methods because the equilibrium constant and the extent of reaction are the primary quantities. 

The non-stoichiometric formulation, in which the stoichiometric equations are not used, instead the material balance constraints are treated by means of Lagrange multipliers. In these direct free energy minimization methods the problem is usually expressed as minimizing G, for fixed T and p, subject to the material balance constraint. 

This model represents the most frequently used description of chemical reaction equilibrium and should be familiar to most chemical engineering students. However, for multicomponent mixtures in which multiple reactions may take place, this type of non-linear problems may be cumbersome to solve numerically. One important obstacle is that the non-linear equilibrium constant definitions may give rise to multiple solutions, hence we have to identify which of them are the physical solutions. The stoichiometric formulation might thus be inconvenient for mixtures containing just a few species for which only a few reactions are taking place. 

An important feature of the non-stoichiometric formulation is that no information about the reaction stoichiometry is required. However, the species that the mixture is composed of must be specified. 

Stepwise Polymerizations For stepwise polymerizations of stoichiometric formulations of comonomers with / and g active sites per molecule exhibiting an ideal behavior (equal reactivity of functional groups, absence of both substimtion effects and intramolecular cycles), the gel conversion (Xg i) is given by ... 

CHEAT ShEET TABLE THREE STOICHIOMETRIC FORMULATIONS TO PRODUCE THE COMPOUNDS INDICATED OR MELTS OF THOSE STOICHIOMETRIES TO AN APPROXIMATION (CHARCOAL CALCULATED AS PURE CARBON)... 

---