Elements balance, molecular matrix

Given a set of components of known atomic composition, establish which of them are key to determine, together with the element balances, the amounts of all others. Furthermore, find a way of generating all possible reactions involving these components. This will be addressed by the augmented molecular matrix, introduced in Section 2.2. 

In practice, it is often crucial to calculate the balance for each of the elements present in the components of a mixture. Now the questions arise how many of these balances are independent and which amounts can be deduced from other ones given the element balance values. To solve this problem in a systematic way, a certain ordering of the components must be chosen typically, the best known or most easily measured components should be listed first. Then the molecular matrix can be augmented by adding a unit matrix. 

If we wish to verify, for instance, that hydrogen atoms are conserved by the first reaction, Eq. (2.47), we must count them in every term on either side, and then subtract these values there are four hydrogen atoms on both sides and subtraction yields zero. An equivalent but mathematically more elegant method is to use the entries of the stoichiometric matrix and the molecular matrix as follows we read —2 for the coefficient of H2 in the first row of the stoichiometric matrix and multiply by 2, the number of H atoms in H2 as indicated by the molecular matrix, and so on. The result is then (—2)(2)+(—1)(0)+(2)(2)+(0)(2)=0, which by definition is precisely the entry (1,1) of the product of the stoichiometric and molecular matrices SM. Similarly, the entry (//) of that product expresses the difference in atom counts of element j in reaction /. Consequently, we can express the balance of all reactions for all elements compactly as the zero matrix with all entries zero ... 

When considering the molecular matrix, the element balances (C, Hj, etc.) provide us with absolute values. In the current setting of reactions, we can state facts only about changes in the number of molecules of the components, not about their total amounts. 

The principle of mass conservation, on a macroscopic basis, using equations representing the conservation of total mass, the conservation of each chemical element and the mass balance for each of the molecular species was applied. Chemical reactions were taken into account in these equations. The stoichiometric coefficients of individual reactants were evaluated using Microsoft Excel software by expressing the equations in matrix notation. 

Despite the vast literature that deals with the observations of the previous paragraphs not even a speculative account to explain the paradoxical asymmetries could be found. Experience points to the conclusion that at a large enough scale the asymmetries in time, material and molecular activity must balance out, at least to within an approximate symmetry of high degree. If PCT invariance holds for the total Hamiltonian H, the matrix element of H between any two states A) and B) is related to that between their CPT-conjugate states A) and B)... 

An irreversible, first order reaction takes place in the porous matrix. The mass-transport is represented by a molecular diffusion. A steady-state mass balance over a differential volume element yields ... 

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