Balance Equation for Concentration

Using the precondition of local geochemical equilibrium, the mass balance equation for concentration TDS C can be written as ... 

In earlier sections of this chapter, we showed how to write and balance equations for precipitation reactions (Section 4.2) and acid-base reactions (Section 4.3). In this section we will concentrate on balancing redox equations, given the identity of reactants and products. To do that, it is convenient to introduce a new concept, oxidation number. 

Sometimes the rate expression obtained by the process just described involves a reactive intermediate, that is, a species produced in one step of the mechanism and consumed in a later step. Ordinarily, concentrations of such species are too small to be determined experimentally. Hence they must be eliminated from the rate expression if it is to be compared with experiment. The final rate expression usually includes only those species that appear in the balanced equation for the reaction. Sometimes, the concentration of a catalyst is included, but never that of a reactive intermediate. 

Reality Check Notice that the concentration of 02, a product in the reaction, appears in the denominator. The rate is inversely proportional to the concentration of molecular oxygen, a feature that you would never have predicted from the balanced equation for the reaction. As the concentration of 02 builds up, the rate slows down. 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Although a catalyst does not appear in the balanced equation for a reaction, the concentration of a homogeneous catalyst does appear in the rate law. For example, the reaction between the triiodide ion and the azide ion is very slow unless a catalyst such as carbon disulfide is present ... 

As with most modeling efforts, the mathematical formulation is the easy part. Picking the right values from the literature or experiments is more work. An immediate task is to decide how to characterize the substrate and product concentrations. The balance equations for substrate and product apply to the carbon content. The glucose molecule contains 40% carbon by weight so S will be 0.4 times the glucose concentration, and 5q = 0.04. Similarly,... 

From the above reaction mechanism it is easy to write balance equations for the initiator and monomer concentrations. 

Each equilibrium expression described so far contains a ratio of concentrations of products and reactants. Moreover, each concentration is raised to a power equal to its stoichiometric coefficient in the balanced equation for the overall reaction. Concentration ratios always have products in the numerator and reactants in the... 

Concentrated aqueous sulfuric acid, H2 SO4, is a strong oxidizing agent that can react with elemental carbon H2 SO4 + C CO2 + SO2 (unbalanced) What is the balanced equation for this process ... 

Table 5.4-3 summarizes the design equations and analytical relations between concentration, C/(, and batch time, t, or residence time, t, for a homogeneous reaction A —> products with simple reaction kinetics (Van Santen etal., 1999). Balance equations for multicomponent homogeneous systems for any reaction network and for gas-liquid and gas-liquid-solid systems are presented in Tables 5.4-7 and 5.4.8 at the end of Section 5.4.3. 

Cf, Cm, and c ui are reactant concentrations in the feed, and at the inlet and outlet of the catalyst bed, respectively and Vr is the reactor volume. The mass balance equation for mixing feed with recycle is as follows ... 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

By writing the mass balance equation for sludge in the entire system, as shown in Figure 27.11, and assuming X, is in negligible amounts (X0 is the sludge concentration in the primary effluent), one can obtain... 

Both the Chen and Gross [48] and the Gallo et al. [49] methods have been applied to eliminating compartments. Both derivation methods are based on the specific mass balance equations for the given model structure. Monte Carlo investigations have demonstrated that both methods provide reasonably accurate and precise estimates of partition coefficients from concentration-time data sets containing error, data one is likely to encounter from in vivo studies. 

Most chemical reactions do not progress completely from reactants to products. Instead, the net reaction stops in the forward direction when equilibrium is established. Analysis of the contents of the reaction vessel would show a constant concentration of monomers and polymer once equilibrium is reached. This situation is actually a dynamic equilibrium, where the monomers are forming polymers at the same rate as the polymers depolymerize to monomer. Therefore, at equilibrium, the net concentrations of any one species remains constant. The amount of monomer converted into polymer will be defined by the equilibrium constant, K. This constant is the ratio of the concentration of the products to the reactants, with each concentration raised to the stoichiometric coefficients in the balanced equation. For Eq. 3.5 ... 

Here the square brackets indicate the concentration of the chemical species within the bracket. That is, [A] means the concentration of A, and so forth. [A]" means the concentration of A raised to the a power, where a is the value of the coefficient of A in the balanced equation for the chemical equilibrium. The value of the ratio of concentration terms is symbolized by the letter K, called the equilibrium constant. For example, for the reaction of nitrogen and hydrogen referred to in Sec. 19.3,... 

For reaction in a constant-volume BR, with only A present initially, the concentrations of A, B and C as functions of time t are governed by the following material-balance equations for A, B and C, respectively, incorporating the two independent rate... 

Approximating the concentrations entering and leaving each section by an arithmetical mean of the neighbouring concentrations, as shown in Section 4.3.5, the component balance equations for stage n become... 

The design engineer can use the dispersion coefficients determined in this way for the calculation of the real course of concentrations, c, of any component in the dispersed d) and continuous (c) phases along the countercurrent column. If the mass transfer between the two phases, the actual task of an extractor, is included in the balance, the balance equations for an element of height dh of the extractor for stationary conditions is ... 

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