Species mole balances

Finally, we write mole balances on each species. Mole balances ... 

In chap 1 it was shown that when the total mole balance is used instead, the reaction term does not always vanish on the RHS because the number of moles may change in a chemical process. This total molar balance can be obtained starting out from the species mass balance (6.4), after dividing by the molecular mass for each species to obtain a species mole balance and finally sum these equations for all species in the mixture. 

For a single reaction, a generic cross sectional average species mole balance for component A in the bubble phase gas can thus be written as ... 

The corresponding cross sectional average species mole balance for component... 

Similarly, for species B, the species mole balance yields ... 

We begin with overall and species mole balances on the entire column using the notation in Fig. 13.3-1 ... 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

These equations apply to the total mass or mass density of the system, while we use moles when describing chemical reaction. Therefore, whenever we need to solve these equations simultaneously, we must transform our species mass balances into weight fraction when including momentum and total mass-balance equations. 

Mole balance expressions were developed for a general series reaction by Agarwalla and Lund [16], and the same procedures were used here to develop the species balance equations shown in Table I. Boundary conditions and parameter definitions are presented in Tables II and III. Note that the boundary conditions are given only for co-current flow of reactants and inert, which is the only configuration studied. Previous work [16], has shown that counter-current operation is less effective than co-current operation. 

For infinitely dilute solutions activity coefficients approach unity so the activity and the concentration of an ion will be equal. For calculations involving more concentrated solutions corrections must be made using activity coefficients, especially when relating the calculated concentration of species to an imposed mass (mole) balance constraint. The activity coefficients can be calculated from a number of ion activity theories and the relevant equations for some of the commonly used ones are shown below. 

The characteristic features of parameter estimation in a molecular model of adsorption are illustrated in Table 9.9, taking the simple example of the constant-capacitance model as applied to the acid-base reactions on a hydroxylated mineral surface. (It is instructive to work out the correspondence between equation (9.2) and the two reactions in Table 9.9.) Given the assumption of an average surface hydroxyl, there are just two chemical reactions involved (the background electrolyte is not considered). The constraint equations prescribe mass and charge balance (in terms of mole fractions, x) and two complex stability constants. Parameter estimation then requires the determination of the two equilibrium constants and the capacitance density simultaneously from experimental data on the species mole fractions as functions of pH. 

SPECIES cone. cone. Act. coef. Activity Moles balance... 

Kinetic models utilize a set of algebraic or differential equations based on the mole balances of the main species involved in the process (ozone in water and gas phases, compounds that react with ozone, presence of promoters, inhibitors of free radical reactions, etc). Solution of these equations provides theoretical concentration profiles with time of each species. Theoretical results can be compared with experimental results when these data are available. In some cases, kinetic modeling allows the determination of rate constants by trial and error procedures that find the best values to fit the... 

Macroscopic chemical techniques can be used to characterize overall reactions like those in Eqs. 1.3, 1.8, and 1.9. Given the complexity of reaction mechanisms, however, measurements of the composition of the aqueous system in which an overall reaction occurs over the course of time may not always yield data that conform to the expected stoichiometry. For example, if the reaction of carbon dioxide and water to produce protons and bicarbonate ions is initiated at high pH (very low proton concentration), the disappearance of 1 mol C02 need not be accompanied by the disappearance of 1 mol H20 (because of Eq. 1.5) or by the appearance of 1 mol H+ (because of Eq. 1.1).2,7 The unaccounted-for presence of intermediate species (like H2COj in Eq. 1.1) can lead typically to a delay in the formation of one or more final product species relative to the others, such that the expected stoichiometry in an overall reaction is violated when the reaction progress is monitored. This transient feature of mole balance in overall reactions has important ramifications when the kinetics of soil chemical processes are investigated (Section 1.3). 

Of course, redox reactions do not occur in isolation but are coupled through complexation reactions to other species. For example, Eq. 2.30 could include terms for nitrate and amine complexes, in addition to those for free nitrate, nitrite, and ammonium ions, if a typical soil solution were under consideration. The calculation of nitrogen speciation then would proceed just as described above. Indeed, redox reactions introduce no new mathematical elements into a speciation computation, any more than would the consideration of, for example, C02 reactions. The only new item brought in is an additional variable, the pE value, which, like the partial pressure of C02(g), must be specified in order to solve mole balance equations for distribution coefficients. 

The way in which conditional stability constants are used to calculate the distribution of chemical species can be illustrated by consideration of the forms of dissolved Cu(II) in a dilute, acidic soil solution. Suppose that the pH of a soil solution is 6.0 and that the total concentration of Cu is 0.1 mmol m 3. The concentrations of the complex-forming ligands sulfate and fulvic acid have the values 50 and 10 mmol m 3, respectively. The important complexes between these ligands and Cu are CuS04 and CuL where L refers to fulvic acid ligands (see Section 2.3). These illustrative complexes are not the only ones formed among Cu, S04, or L, nor are the three ligands the only ones that form Cu complexes in soil solution.29 Under the conditions assumed, the equation of mole balance for Cu is (cf. Eqs. 2.11 and 2.30)... 

Mole-balance equations are established for each metal and each ligand in terms of the concentrations of all species assumed to exist in solution. 

The mole-balance equations are expressed in terms of conditional stability constants and the concentrations of the species chosen as components31 (e.g., free metals and ligands) to provide a set of coupled, nonlinear algebraic equations in the component concentrations. 

The formal similarity between adsorption and complexation reactions can be exploited to incorporate adsorbed species into the equilibrium speciation calculations described in Sections 2.4 and 3.1. To do this, a choice of adsorbent species components (SR r in Eq. 4.3) must be made and equilibrium constants for reactions with aqueous ions must be available. A model for computing adsorbed species activity coefficients must also be selected.8 Once these choices are made and the thermodynamic data are compiled, a speciation calculation proceeds by adding adsorbent species and adsorbed species (SR Mp(OH)yHxLq in Eq. 4.3) to the mole-balance equations for metals and ligands, and then following the steps described in Section 2.4 for aqueous species. For compatibility of the units of concentration, njw) in Eq. 4.2 is converted to an aqueous-phase concentration through division by the volume of aqueous solution. 

Reaction equilibrium, which should not be used to compensate charge balances, has to be marked with no check . If the stoichiometry of a species has to be defined explicitly, like the polysulphide species (see Table 23, S22" contains 2 S atoms, but only one will be used for the combination of HS"), the declarations have to be made under mole balance . 

Now, partial pressure of species A must be related to total-system pressure. This may be done easily by a general mole balance on the system, resulting in the following relationships ... 

In batch photocatalytic reactors working in liquid-solid regime, the depletion of a species is the combined result of photoadsorption and photoconversion processes. To describe this depletion, a mole balance applied to the species at whatever time (de Lasa et al., 2005) can be represented as... 

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