Mass action constraints

Measurements carried out in solutions of different HCO /CO ratios (at constant total carbonate) indicate a decrease of n from 4 to 1 as the proportion of bicarbonate ions decreased and that of carbonate ions increased. From this observation, Veyland concludes that the presence of HCOj favours the formation of Zr(CO,)4". In our opinion, this interpretation is not entirely correct, as it conflicts with law of mass action constraints. If the only complex involved were Zr(C03)4, n would necessarily increase with increasing concentration however, exactly the opposite is ob-... 

The subscript i labels the six concentrations and energies defined previously. It is straightforward to show that by minimising G with respect to the c, subject to the three constraints, we recover the three equations for the q predicted by the law of mass action. 

SCMs have many similarities with the way problems for soluble complex equilibria are formulated. For example, the reactions that describe the formation of surface species are described by mass action equations coupled with mass balance constraint. However, the major differences arise from the fact that the complexant ligands are electrically charged and this requires the introduction of an electrostatic component into the mass action equations. In addition, an electrical balance constraint equation is needed to solve the equations. 

The free—energy function (Equation i) is related to the mass action laws, and the constraints (Equation ii) are simply the mass balance laws. See Ref. [5],... 

Formal models, with two to four variables, the only ones that might be tractable, may have a stange chemical look. Yet, a mathematician would probably not even bother to keep any chemical structure to his models, when looking,for instance, for a particular type of bifurcation. In the most reduced form they are known to be described by normal forms" , which no longer bear any chemical appearance. Nevertheless, although chemistry, through mass action kinetics, provides all the ingredients necessary to exhibit all those exotic behaviours, it may also impose a number of constraints, that make the reaction or reaction-diffusion equations a... 

For reactions (2) and (3), we establish the following system of four equations, which include the mass action constants together with the mass balance and charge balance constraints ... 

Mass action relationships for aluminosilicate-carbonate-water reactions will be the most important constraint in buffering pH if their reaction rates are fast enough to accommodate changing environmental conditions (kinetic constraint), and there are sufficient amounts of these minerals in the rock (mass balance constraint). To attempt to address the kinetic constraint first, we can consider how fast a mineral typical of low temperature alteration in both sediments and fractured hard rocks (albite) behaves. Albite (Na-plagioclase) has a congruent dissolution... 

A system is at chemical equilibrium when it has adjusted its chemical compositions so that the overall Gibbs energy of the system is the minimum possible, subject to the external constraints (T, P, initial chemical composition, etc.). The shorthand way of showing this is the law of mass action or chemical equilibrium constant. The relation between the two is explored in Chapter 12. 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

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