Equations of State in Chemical Reacting Systems

SELVA PEREDA, ESTEBAN BRIGNOLE AND SUSANA BOTTINI 

Planta Piloto de Ingenieria Quimica (PLAPIQUI) - CONICET, Universidad Nacional del Sur, Camino La Carrindanga Km 7 - C.C 717, Argentina 

Phase and chemical equilibrium calculations are essential for the design of processes involving chemical transformations. Even in the case of reactions that cannot reach chemical equilibrium, the solution of this problem gives information on the expected behaviour of the system and the potential thermodynamic limitations. There are several problems in which the simultaneous calculation of chemical and phase behaviour is mandatory. This is the case, for example, of reactive distillations where phase separation is used to shift chemical equilibrium. Also, the calculation of gas and solid solubility in liquids of high dielectric constants requires at times the resolution of chemical equilibrium between the different species that are formed in the liquid phase. Several algorithms have been proposed in the literature to solve the complex non-linear problem however, proper thermodynamic model selection has not received much attention. 

In recent times, the use of supercritical solvents has emerged as an important technique to improve rates and selectivities in diffusion-controlled reactions. Phase behaviour near the critical point of mixtures is very sensitive to process operating conditions and mixture compositions. The selection and design of the appropriate phase conditions to exploit process potential require 

Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Peters International Union of Pure and Applied Chemistry 2010 Published by the Royal Society of Chemistry, www.rse.org 

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Equations of State in Chemical Reacting Systems with... 

It should be emphasized that the choice of standard states implied by equation 2.2.9 is not that which is conventionally used in the analysis of chemically reacting systems. Furthermore,... 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

Direct solution of the master equation is impractical because of the huge number of equations needed to describe all possible states (combinations) even of relatively small-size systems. As one example, for a three-step linear pathway among 100 molecules, 104 such equations are needed. As another example, in biological simulation for the tumor suppressor p53, 211 states are estimated for the monomer and 244 for the tetramer (Rao et al., 2002). Instead of following all individual states, the MC method is used to follow the evolution of the system. For chemically reacting systems in a well-mixed environment, the foundations of stochastic simulation were laid down by Gillespie (1976, 1977). More... 

Saim and Subramanian addressed the effect of supercritical CO2 as solvent media in the homogeneous chemical equilibria of different reacting systems, including the isomerisation of hexane and 1-hexene and the oxidation of 2-methylpropane. The Peng-Robinson equation of state was used to calculate the critical loci of CO2 with each hydrocarbon the information was used to avoid the two-phase region in the computation of conversion for each system studied. 

Phoenix and Heidemann on the other hand, adapted Michelsen s approach " for chemical equilibrium calculation in multiphase reacting systems, applying iterative corrections to ideality with the Soave-Redlich-Kwong equation of state. Phoenix and Heidemann applied this procedure with the Peng-Robinson equation of state to study the phase behaviour of natural gases containing elemental sulfur, which is known to exist as a number of species up to... 

As a direct consequence of the particular role of Dynamics, as such,in the study of non-equilibrium behaviour of chemical systems, two classes of models are to be considered, depending on which aspect one is insisting on. Formal models, of mathematical or chemical-like nature, are designed to exhibit specific dynamical behaviours, without too much concern about chemical significance. Their aim is to provide examples of evolution equations of chemical reacting systems, as described by mass action kinetics, that are able to produce those exotic behaviours, such as bistability or multistability, between various types of attractors, like steady states, oscillations or deterministic chaos. A typical historical model of that kind is the "Brusselator ... 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

The law of conservation of matter states that in a closed system when a chemical change occurs, there is no change in mass. This is because atoms are conserved in a chemical change so atoms must be balanced in a chemical equation. In a balanced equation, coefficients tell the number of reactant and product substances that react and are produced. Subscripts tell the number of atoms of each kind in these substances. When a coefficient is multiplied by a subscript in a substance formula, the number of atoms is determined. Since a mole is an amount of a substance, the coefficients in a chemical equation can stand for the number of moles that react and are produced. 

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