Rate of reaction for each species

Next, we (toermine the net rate of reaction for each species by using the appropiiate stoichiometric coefficients and then summing the rates of the individual reaction.s. ... 

Figure 4-11). After numbering each reaction, we write a mole balance on eac species similar to those in Figure 4-11. The major difference between the iw algorithms is in rate law step. Here we have four steps ( , , . and ) t find the net rate of reaction for each species in terms of the concentration c the reacting species in order to combine them with their respective mole ba ances. The remaining steps are analogous to those in Figure 4-8.

Figure 8.14 shows the proposed mechanisms of reaction for each species. The overall rate of the S(IV) oxidation can then be represented by... 

We now substitute the rate laws for each species in each reaction to obtain t net rate of reaction for that species. Again, considering only Reactions 1 anc... 

There are a number of instances when it is much more convenient to work in terms of the number of moles (iV, N-g) or molar flow rates (Fj, Fg, etc.) rather than conversion. Membrane reactors and multiple reactions taking place in the gas phase are two such cases where molar flow rates rather than conversion are preferred. In Section 3.4 we de.scribed how we can express concentrations in terms of the molar flow rates of the reacting species rather than conversion, We will develop our algorithm using concentrations (liquids) and molar flow rates (gas) as our dependent variables. The main difference is that when conversion is used as our variable to relate one species concentration to that of another species concentration, we needed to write a mole balance on only one species, our basis of calculation. When molar flow rates and concentrations are used as our variables, we must write a mole balance on each species and then relate the mole balances to one another through the relative rates of reaction for... 

Having written the mole balances, the key point for multiple reactions is to write the net rate of formation of each species (e.g.. A, B). That is, we have to sum up the rates of formation for each reaction in order to obtain the net... 

We saw in section 1.1.1, how atoms with identical atom number but with different amount of neutrons are called isotopes. Likewise did we see that the combined number of protons and neutrons are called nucleons and that radioactive species decay under emission of different types of radiation. The rate of such decay is in principle similar to the rate of reaction for the transition of reactants to products in a chemical reaction. We imagine that for a specific time r = 0 we have an amount of specie with No radioactive nuclei. It has been found that all nuclei have a specific probability of decaying within the next second. If this probability is e.g. 1/100 pr. second this means that on an average 1% of all nuclei decay each second. The number of radioactive nuclei is thereby a decreasing function with time and may formally be written as N(t). The rate for the average number of decays pr. time is thereby defined analogously to equation (3-1) as ... 

When the reaction rate depends on the concentrations of several species, or when more than one reaction is involved, analytical solutions of the pore diffusion equations are impossible or too complicated to be useful. The equations for simultaneous diffusion and reaction of several species can be solved numerically if concentrations at the center are specified, but then many cases must be solved to match given external concentrations. For such cases, a simplified method can be used instead to show the approximate effect of gradients for each species. 

In short, each reaction family could be described with a maximiun of three parameters (A, Eo, a). Procurement of a rate constant from these parameters required only an estimate of the enthalpy change of reaction for each elementary step. In principle, this enthalpy change of reaction amoimted to the simple calculation of the difference between the heats of formation of the products and reactants. However, since many model species, particularly the ionic intermediates and olefins, were without experimental values, a computational chemistry package, MOPAC, ° was used to estimate the heat of formations on the fly . Ihe organization of the rate constants into quantitative structure-reactivity correlations (QSRC) reduced the number of model parameters greatly Ifom O(IO ) to 0(10). 

Division by the stoichiometric coefficients takes care of the stoichiometric relations between the reactants and products. There is no need to specify the species when reporting the unique average reaction rate, because the value of the rate is the same for each species. However, the unique average rate does depend on the coefficients used in the balanced equation, and so the chemical equation should be specified when reporting the unique rate. 

Solving for rates of production of chemical species requires as input an elementary reaction mechanism, rate constants for each elementary reaction (usually in Arrhenius form), and information about the thermochemistry (Aff/, 5, and Cp as a function of temperature) for each chemical species in the mechanism. 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

A similar procedure allows us to assign the additional peaks produced on ArF photolysis as being due to Cr(C0>3 and Cr(C0>2 [9]. The frequencies of the gas phase absorptions for these coordinatively unsaturated fragments is presented in table II. Data for the rates of reaction of each of these species with CO is summarized in table I. 

The rate constants (/c[and k]) and the stoichiometric coefficients (t and 1/ ) are all assumed to be known. Likewise, the reaction rate functions Rt for each reaction step, the equation of state for the density p, the specific enthalpies for the chemical species Hk, as well as the expression for the specific heat of the fluid cp must be provided. In most commercial CFD codes, user interfaces are available to simplify the input of these data. For example, for a combusting system with gas-phase chemistry, chemical databases such as Chemkin-II greatly simplify the process of supplying the detailed chemistry to a CFD code. 

The slopes of the lines in the plot give the reaction coefficients for each species and mineral in the overall reaction. Species with negative slopes appear to the left of the reaction (with their coefficients set positive), and those with positive slopes are placed to the right. The reactant plotted on the horizontal axis appears to the left of the reaction with a coefficient of one. If there are additional reactants, these also appear on the reaction s left with coefficients equal to the ratios of their reaction rates nr to that of the first reactant. 

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 simplest theories of reactions on surfaces also predict surface rate laws in which the rate is proportional to the amount of each adsorbed reactant raised to the power of its stoichiometric coefficient, just like elementary gas-phase reactions. For example, the rate of reaction of adsorbed carbon monoxide and hydrogen atoms on a metal surface to produce a formyl species and an open site,... 

The elementary steps in gas-phase reactions have rate laws in which reaction order for each species is the same as the corresponding molecularity. The rate constants for these elementary reactions can be understood quantitatively on the basis of simple theories. For our purpose, reactions involving photons and charged particles can be understood in the same way. 

We illustrate the development of the model equations for a network of two parallel reactions, A -> B, and A - C, with kt and representing the rate constants for the first and second reactions, respectively. Continuity equations must be written for two of the three species. Furthermore, exchange coefficients (Kbc and Kce) must be determined for each species chosen (here, A and B). 

Inputs + Sources = Outputs + Sinks + Accumulations where each of these terms may be a quantity or a rate. Inputs and Outputs are accomplished by crossing the boundary of the reference volume. In case of mass transfer this occurs by bulk flow and diffusion. Sources and Sinks are accretions and depletions of a species without crossing the boundaries. In a mass and energy balance, sinks are the rate of reaction, rdVr, or a rate of enthalpy change, AHrpdC. Accumulation is the time derivative of the content of the species within the reference volume, for example, (<9C/3t)dVr or... 

A chemical equilibrium results when two exactly opposite reactions are occurring at the same place, at the same time and with the same rates of reaction. When a system reaches the equilibrium state the reactions do not stop. A and B are still reacting to form C and D C and D are still reacting to form A and B. But because the reactions proceed at the same rate the amounts of each chemical species are constant. This state is a dynamic equilibrium state to emphasize the fact that the reactions are still occurring—it is a dynamic, not a static state. A double arrow instead of a single arrow indicates an equilibrium state. For the reaction above it would be ... 

In this expression, k is the rate constant—a constant for each chemical reaction at a given temperature. The exponents m and n, called the orders of reaction, indicate what effect a change in concentration of that reactant species will have on the reaction rate. Say, for example, m = 1 and n = 2. That means that if the concentration of reactant A is doubled, then the rate will also double ([2]1 = 2), and if the concentration of reactant B is doubled, then the rate will increase fourfold ([2]2 = 4). We say that it is first order with respect to A and second order with respect to B. If the concentration of a reactant is doubled and that has no effect on the rate of reaction, then the reaction is zero order with respect to that reactant ([2]° = 1). Many times the overall order of reaction is calculated it is simply the sum of the individual coefficients, third order in this example. The rate equation would then be shown as ... 

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