Chemical Reaction Equilibrium Calculations

Given a balanced equation for a reversible chemical change and gas pressures for reactants and products at equilibrium, calculate tbe reaction s Kp. 

Vapor-Liquid Equilibrium Plus Chemical Reaction. To conclude this discussion of VLE calculations, I d like to mention that we have initiated a program to allow VLE plus chemical reaction in column calculations. A modification of Frank s dynamic simulation method ( ) is being used. The initial trials with complex VLE and kinetic models have been moderately successful, but further testing is required. Other simulation techniques are also being investigated. 

It is interesting to compare bromination against chlorination. Identical base case conditions as those of chlorination (except that temperature variations are limited to 500 Q were used for the chemical equilibrium calculations of Reaction (19). Major conclusions were ... 

In reality, undesirable" olefin oligomerization reactions also take place, resulting in undesirable C2H4 consumption. To illustrate this point, additional CjHg and C4H8 formations were included in the chemical equilibrium calculations. The reactions are... 

Chemical equilibrium calculations of Reaction (52) using 800-1,000 C showed that... 

The thermodynamic basis for calculating chemical reaction equilibrium was developed in Section 12.5. There we showed that if a system of componentsA2,. .., A/, reacting according to the equation ... 

The most important themiodynamic property of a substance is the standard Gibbs energy of fomiation as a fimetion of temperature as this infomiation allows equilibrium constants for chemical reactions to be calculated. The standard Gibbs energy of fomiation A G° at 298.15 K can be derived from the enthalpy of fomiation AfT° at 298.15 K and the standard entropy AS° at 298.15 K from... 

Having calculated the standai d values AyW and S" foi the participants in a chemical reaction, the obvious next step is to calculate the standard Gibbs free energy change of reaction A G and the equilibrium constant from... 

The standard Gibbs-energy change of reaction AG° is used in the calculation of equilibrium compositions. The standard heat of reaclion AH° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating AH° and AG° with T. Numerical values for AH° and AG° are computed from tabulated formation data, and AC° is determined from empirical expressions for the T dependence of the C° (see, e.g., Eq. [4-142]). 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Some chemical reactions are reversible and, no matter how fast a reaction takes place, it cannot proceed beyond the point of chemical equilibrium in the reaction mixture at the specified temperature and pressure. Thus, for any given conditions, the principle of chemical equilibrium expressed as the equilibrium constant, K, determines how far the reaction can proceed if adequate time is allowed for equilibrium to be attained. Alternatively, the principle of chemical kinetics determines at what rate the reaction will proceed towards attaining the maximum. If the equilibrium constant K is very large, for all practical purposes the reaction is irreversible. In the case where a reaction is irreversible, it is unnecessary to calculate the equilibrium constant and check the position of equilibrium when high conversions are needed. 

Chemical reaction equilibrium calculations are structured around another thermodynamic term called tlie free energy. Tliis so-callcd free energy G is a property that also cannot be defined easily without sonic basic grounding in tlicmiodynamics. However, no such attempt is made here, and the interested reader is directed to tlie literature. " Note that free energy has the same units as entlialpy and internal energy and may be on a mole or total mass basis. Some key equations and information is provided below. 

The following equation is used to calculate tlie chemical reaction equilibrium constant K at a temperature T. 

Several basic principles that engineers and scientists employ in performing design calculations and predicting Uie performance of plant equipment includes Uieniiochemistiy, chemical reaction equilibrimii, chemical kinetics, Uie ideal gas law, partial pressure, pliase equilibrium, and Uie Reynolds Number. 

The molecular mechanics calculations discussed so far have been concerned with predictions of the possible equilibrium geometries of molecules in vacuo and at OK. Because of the classical treatment, there is no zero-point energy (which is a pure quantum-mechanical effect), and so the molecules are completely at rest at 0 K. There are therefore two problems that I have carefully avoided. First of all, I have not treated dynamical processes. Neither have I mentioned the effect of temperature, and for that matter, how do molecules know the temperature Secondly, very few scientists are interested in isolated molecules in the gas phase. Chemical reactions usually take place in solution and so we should ask how to tackle the solvent. We will pick up these problems in future chapters. 

Mukherjee studied the gas phase equilibria and the kinetics of the possible chemical reactions in the pack-chromising of iron by the iodide process. One conclusion was that iodine-etching of the iron preceded chromis-ing also, not unexpectedly, the initial rate of chromising was controlled by transport of chromium iodide. Neiri and Vandenbulcke calculated, for the Al-Ni-Cr-Fe system, the partial pressures of chlorides and mixed chlorides in equilibrium with various alloys and phases, and so developed for pack aluminising a model of gaseous transport, solid-state transport, and equilibria at interfaces. 

In this generalized equation, (75), we see that again the numerator is the product of the equilibrium concentrations of the substances formed, each raised to the power equal to the number of moles of that substance in the chemical equation. The denominator is again the product of the equilibrium concentrations of the reacting substances, each raised to a power equal to the number of moles of the substance in the chemical equation. The quotient of these two remains constant. The constant K is called the equilibrium constant. This generalization is one of the most useful in all of chemistry. From the equation for any chemical reaction one can immediately write an expression, in terms of the concentrations of reactants and products, that will be constant at any given temperature. If this constant is measured (by measuring all of the concentrations in a particular equilibrium solution), then it can be used in calculations for any other equilibrium solution at that same temperature. 

The acidity of a solution has pronounced effects on many chemical reactions. It is therefore important to be able to learn and control the hydrogen ion concentration. This control is obtained through application of the Equilibrium Law. Common types of calculation, based on this law, are those needed to determine KA from experimental data and those using KA to find [H+], We will illustrate both of these types, using benzoic acid, QH6COOH, as an example. 

The preceeding discussion was confined mostly to the carbon deposition curves as a function of temperature, pressure, and initial composition. Also of interest, especially for methane synthesis, is the composition and heating value of the equilibrium gas mixture. It is desirable to produce a gas with a high heating value which implies a high concentration of CH4 and low concentrations of the other species. Of particular interest are the concentrations of H2 and CO since these are generally the valuable raw materials. Also, by custom it is desirable to maintain a CO concentration of less than 0.1%. The calculated heating values are reported as is customary in the gas industry on the basis of one cubic foot at 30 in. Hg and 15.6°C (60°F) when saturated with water vapor (II). Furthermore, calculations are made and reported for a C02- and H20-free gas since these components may be removed from the mixture after the final chemical reaction. Concentrations of CH4, CO, and H2 are also reported on a C02 and H20-free basis. 

The determination of ArG° for a chemical reaction is very useful in predicting the course of the reaction. Qualitatively, we will show in Chapter 5 that with ArC°<0, the reaction is spontaneous, at least when products and reactants are in their standard state condition. Quantitatively, we will see in Chapter 9 that ArG° can be used to calculate the equilibrium constant for the reaction, from which the final equilibrium conditions can be determined. 

Equation (9.5) enables us to calculate ArG for a chemical reaction under a given set of activity conditions when we know the free energy change for the reaction under the standard state condition. Of special interest are the activities when reactants and products are at equilibrium. Under those conditions,... 

The decrease in Gibbs free energy as a signpost of spontaneous change and AG = 0 as a criterion of equilibrium are applicable to any kind of process, provided that it is occurring at constant temperature and pressure. Because chemical reactions are our principal interest in chemistry, we now concentrate on them and look for a way to calculate AG for a reaction. 

What Do We Need to Know Already The concepts of chemical equilibrium are related to those of physical equilibrium (Sections 8.1-8.3). Because chemical equilibrium depends on the thermodynamics of chemical reactions, we need to know about the Gibbs free energy of reaction (Section 7.13) and standard enthalpies of formation (Section 6.18). Ghemical equilibrium calculations require a thorough knowledge of molar concentration (Section G), reaction stoichiometry (Section L), and the gas laws (Ghapter 4). 

A knowledge of the concentrations of all reactants and products is necessary for a description of the equilibrium state. However, calculation of the concentrations can be a complex task because many compounds may be Imked by chemical reactions. Changes in a variable such as pH or oxidation potential or light intensity can cause large shifts in the concentrations of these linked species. Aggregate variables may provide a means of simplifying the description of these complex systems. Here we look at two cases that involve acid-base reactions. 

While these calculations provide information about the ultimate equilibrium conditions, redox reactions are often slow on human time scales, and sometimes even on geological time scales. Furthermore, the reactions in natural systems are complex and may be catalyzed or inhibited by the solids or trace constituents present. There is a dearth of information on the kinetics of redox reactions in such systems, but it is clear that many chemical species commonly found in environmental samples would not be present if equilibrium were attained. Furthermore, the conditions at equilibrium depend on the concentration of other species in the system, many of which are difficult or impossible to determine analytically. Morgan and Stone (1985) reviewed the kinetics of many environmentally important reactions and pointed out that determination of whether an equilibrium model is appropriate in a given situation depends on the relative time constants of the chemical reactions of interest and the physical processes governing the movement of material through the system. This point is discussed in some detail in Section 15.3.8. In the absence of detailed information with which to evaluate these time constants, chemical analysis for metals in each of their oxidation states, rather than equilibrium calculations, must be conducted to evaluate the current state of a system and the biological or geochemical importance of the metals it contains. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

Equilibrium conditions are determined by the chemical reactions that occur in a system. Consequently, it is necessary to analyze the chemistry of the system before doing any calculations. After the chemistry is known, a mathematical solution to the problem can be developed. We can modify the seven-step approach to problem solving so that it applies specifically to equilibrium problems, proceeding from the chemistry to the equilibrium constant expression to the mathematical solution. 

As the LiF example illustrates, the most direct way to determine the value of an equilibrium constant is to mix substances that can undergo a chemical reaction, wait until the system reaches equilibrium, and measure the concentrations of the species present once equilibrium is established. Although the calculation of an equilibrium constant requires knowledge of the equilibrium concentrations of all species whose concentrations appear in the equilibrium constant expression, stoichiometric analysis often can be used to deduce the concentration of one... 

J. P. Guthrie, No Barrier Theory Calculating Rates of Chemical Reactions from Equilibrium Constants and Distortion Energies, ChemPhysChem 2003,4, 809. 

However, the value of the equilibrium electrode potential is often not well defined (e.g. when the electrode reaction produces an intermediate that undergoes a subsequent chemical reaction yielding one or more final products). Often, an equilibrium potential is not established at all, so that the calculated equilibrium values must often be used. 

If 1.0 mol of W and 3.0mol of Q are placed in a 1.0-L vessel and allowed to come to equilibrium, calculate the equilibrium concentration of Z using the following steps (a) If the equilibrium concentration of Z is equal to x, how much Z was produced by the chemical reaction (b) How much R was produced by the chemical reaction (c) How much W and Q were used up by the reaction (d) How much W is left at equilibrium (e) How much Q is left at equilibrium (/) With the value of the equilibrium constant given, will x (equal to the Z concentration at equilibrium) be significant when subtracted from 1.0 (g) Approximately what concentrations of W and Q will be present at equilibrium (/t) What is the value of x (/) What is the concentration of R at equilibrium (7) Is the answer to part (/) justified ... 

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