Independent chemical reaction

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

In section 11.3 vie showed that the difficult problem of solving the flux relations can be circumvented rather simply when the stoichiometric relations are satisfied by the flux vectors, but the treatment given there was limited to the case of a single Independent chemical reaction, when the stoichiometric relations permit all the flux vectors to be expressed in terms of any one of them. The question then arises whether any comparable simplification is possible v en the reactants participate in more than one independent reaction. 

Let us suppose that there are several Independent chemical reactions, which are numbered, and that ft C ,T) denotes the rate of the re-... 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

Equation (4-274) for each independent chemical reaction, giving r equations. 

The number of independent chemical reactions / can be determined as follows ... 

Example 7.17 illustrates the utility of the reaction coordinate method for solving equilibrium problems. There are no more equations than there are independent chemical reactions. However, in practical problems such as atmospheric chemistry and combustion, the number of reactions is very large. A relatively complete description of high-temperature equilibria between oxygen and... 

To derive the governing equations we need to identify each independent chemical reaction that can occur in the system. It is possible to write many more reactions than are independent in a geochemical system. The remaining or dependent reactions, however, are linear combinations of the independent reactions and need not be considered. 

If we had chosen to describe composition in terms of elements, we would need to carry the elemental compositions of all species, minerals, and gases, as well as the coefficients of the independent chemical reactions. Our choice of components, however, allows us to store only one array of reaction coefficients, thereby reducing memory use on the computer and simplifying the forms of the governing equations and their solution. In fact, it is possible to build a complete chemical model (excluding isotope fractionation) without acknowledging the existence of elements in the first place ... 

The study described above for the water-gas shift reaction employed computational methods that could be used for other synthesis gas operations. The critical point calculation procedure of Heidemann and Khalil (14) proved to be adaptable to the mixtures involved. In the case of one reaction, it was possible to find conditions under which a critical mixture was at chemical reaction equilibrium by using a one dimensional Newton-Raphson procedures along the critical line defined by varying reaction extents. In the case of more than one independent chemical reaction, a Newton-Raphson procedure in the several reaction extents would be a candidate as an approach to satisfying the several equilibrium constant equations, (25). 

For a system undergoing R independent chemical reactions among N chemical species, R equilibrium expressions are to be added to the relationships among the intensive variables. From Equation (13.1), the total number of intensive variables in terms of N becomes... 

Gibbs phase rule phys chem A relationship used to determine the number of state variables F, usually chosen from among temperature, pressure, and species composition in each phase, which must be specified to fix the thermodynamic state of a system in equilibrium F = C - P - M+2, where C is the number of chemical species presented at equilibrium, P is the number of phases, and M is the number of independent chemical reactions. Also known as Gibbs rule phase rule. gibz faz, rijl I... 

In carrying out the procedure for determining mechanisms that is presented here, one obtains a set of independent chemical reactions among the terminal species in addition to the set of reaction mechanisms. This set of reactions furnishes a fundamental basis for determination of the components to be employed in Gibbs phase rule, which forms the foundation of thermodynamic equilibrium theory. This is possible because the specification of possible elementary steps to be employed in a system presents a unique a priori resolution of the number of components in the Gibbs sense. 

A similar proof goes through if there are R independent chemical reactions taking place. If the suffix i denotes a particular reaction 2al7Ay = 0,... 

For sake of simplicity, only homogeneous, independent chemical reactions are considered in scheme (3.VI)... 

The mole numbers in Equation (5.63) could represent the species present in the system, rather than the components. However, in such a case they are not all independent. For each independent chemical reaction taking place within the system, a relation given by Equation (5.46) must be satisfied. There would thus be (S — R) independent mole numbers for the system if S is the number of species and R is the number of independent chemical reactions. In fact, this relation may be used to define the number of components in the system [6]. If the species are ions, an equation expressing the electroneutrality of the system is another condition equation relating the mole numbers of the species. The total number of components in this case is C = S — R — 1. 

If one or more chemical reactions are at equilibrium within the system, we can still set up the set of Gibbs-Duhem equations in terms of the components. On the other hand, we can write them in terms of the species present in each phase. In this case the mole numbers of the species are not all independent, but are subject to the condition of mass balance and to the condition that , vtpt must be equal to zero for each independent chemical reaction. When these conditions are substituted into the Gibbs-Duhem equations in terms of species, the resultant equations are the Gibbs-Duhem equations in terms of components. Again, from a study of such sets of equations we can easily determine the number of degrees of freedom and can determine the mathematical relationships between these degrees of freedom. 

R number of independent chemical reactions s solid state s displacement S entropy... 

The corresponding conservation equations are less familiar, but they contain the same information as a set of independent chemical reactions. The conservation equations for a system containing Ns species are given by... 

When equcalcrx[nt,lnkr,no] is applied to a system of R independent chemical reactions, it requires a R x N transposed stoichiometric number matrix nt, a vector of natural logarithms of the equilibrium constants of independent reactions, and a vector no of the initial concentrations. It can be used at a specified pH by using a R x N transposed stoichiometric number matrix nt, a vector lnkr of natural logarithms of the apparent equilibrium constants of independent biochemical reactions, and a vector no of the initial concentrations. 

Each independent chemical reaction at equilibrium gives rise to a restrictive condition. 

Chemical reactions reduce the number of degrees of freedom of a system. For each independent chemical reaction that can occur in a system, AnnG = 0 is required. If r is the number of independent reactions at equilibrium in the system, this reduces the degrees of freedom by r. 

The governing equations for each of the three ideal reactors are material balances for reactants and products of the reaction. In general, one material balance equation must be written for each independent reaction taking place in the reactor. For a group of Nsp species (i.e., reactants and products) in the reactor consisting of iVei elements, the number of independent chemical reactions typically is equal to (/Vsp - Nei). 

CO, CO2 and 0 In the reaction mixture, hence, we need two Independent chemical reactions to describe the system completely. If in addition carbon is deposited, then we need an additional reaction. 

When the number of independent chemical reactions equals C - p, where p is the rank of the atom matrix (mjk), Gibbs free energy is minimized subject to atom balance constraints ... 

Sanderson and Chien (18) solve Equations (7), (8), and (13) to determine compositions of vapor and liquid phases in chemical and phase equilibrium given temperature and pressure. A set of independent chemical reactions is selected with guesses for extent of reaction. Solution of Equation (13) leads to compositions in phase equilibrium, but applies only for a vapor and liquid in equilibrium. Residuals of Equations (7) and (8) are computed and extents of reaction, g, and moles of species j, n, are adjusted using Marquardt s method (15). 

Finding the independent chemical reactions provides consistency and proper specification for both material balance and chemical kinetics. By definition, a set of... 

The final visualization of the reduced B matrix allows finding the basic set of independent chemical equations. Note that C = rank (B) gives the number of component species that may form all the other noncomponent species by a minimum of independent chemical reactions. The procedure can be applied by hand calculations for simple cases, or by using computer algebra tools for a larger number of species. More details can be found in the book of Missen et al. [7], or at www.chemical-stoichiometry.net. 

At this point, we should mention the difference between independent chemical equations and independent chemical reactions. The former are of mathematical significance, being helpful to carry out consistent material balance. The latter are useful for describing the chemical steps implied in a chemical-reaction network. They may be identical with the independent stoichiometric equations, or derived by linear combination. This approach is useful in formulating consistent kinetic models. 

Identify the number of independent chemical reactions associated with each reaction step in the manufacture of the desired product. Include intermediates that can be separated and recycled. At each step indicates the phase(s), as well as the range of feasible temperatures and pressures. 

Solution The data regarding the species distribution are converted in stoichiometric equations. Table 2.5 gives the independent chemical reactions. Note the presence of a reaction for describing the formation of heavies, lumped as dinitrile succinate. The material balance around the chemical reactor can be easily calculated by using a spreadsheet Table 2.6 presents the results for lkmol propylene, 1.2kmol NH3 and 9.5kmol air. Propylene conversion is 0.983 and the selectivity in acrylonitrile 79.6%. 

It must be modified for application to systems in which chemical reactions occur. The phase-rule variables are the same in either case, namely, temperature, pressure, and N - 1 mole fractions in each phase. The tothl number of these variables is 2 + (N - 1)(7r). The same phase-equilibrium equations apply as before, and they number (it - 1)(N). However, Eq. (15.8) provides for each independent reaction an additional relation that must be satisfied at equilibrium. Since the Hi s are functions of temperature, pressure, and the phase compositions, Eq. (15.8) represents a relation connecting the phase-rule variables. If there are r independent chemical reactions at equilibrium within the system, then there is a total of (it - 1)(N) + r independent equations relating the phase-rule variables. Taking the difference between the number of variables and the number of equations, we obtain... 

Find the independent chemical reactions. Write equations for the formation of each compound present ... 

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