Equilibrium compositions, single reactions

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

Problems with the determination of chemical equilibria in multiphase systems are solved in practice by assuming that the reaction takes place in any phase and all components are also equilibrated between phases. Accordingly, for a single reaction any of Eqns. (5.4-32) and (5.4-33) must be solved, while the relationships of Eqn. (5.4-31) must also be fulfilled. Since vapour-liquid equilibrium coefficients are functions of the compositions of both phases, the search for the solution is an iterative procedure. Equilibrium compositions are assumed, vapour-liquid equilibrium coefficients are then estimated, and new equilibrium compositions are evaluated. If the new equilibrium compositions are close to those assumed initially one may consider the assumed values to be the solution of the problem. Otherwise the evaluated compiositions are taken as the start for repetition of the procedure until a reasonable agreement between tissumed and evaluated comfiositions has been reached. 

This equation is extremely useful for calculating the equilibrium composition of the reaction mixture. The mole numbers of the various species at equilibrium may be related to their values at time zero using the extent of reaction. When these values are substituted into equation 2.6.9, one has a single equation in a single unknown, the equilibrium extent of reaction. This technique is utilized in Illustration 2.1. If more than one independent reaction is occurring in a given system, one needs as many equations of the form of equation 2.6.9 as there are independent reactions. These equations are then written in terms of the various extents of reaction to obtain a set of independent equations equal to the number of unknowns. Such a system is considered in Illustration 2.2. 

In thermodynamics we learned how to describe the composition of molecules in chemical equilibrium. For the generalized single reaction... 

We will find it convenient to use X rather than Ca as the composition variable for a single reaction and consider C e and as the corresponding quantities at equilibrium. For an endothermic reaction A Hg > 0 and increases with T, while for an exothermic reaction AHg < 0 and Xg decreases with T. 

Calculation of equilibrium compositions of a single biochemical reaction or a system of biochemical reactions at specified pH... 

When the equilibrium state in a reacting system depends on two or more s taneous chemical reactions, the equilibrium composition can be found by a < extension of the methods developed for single reactions. One first deter set of independent reactions as discussed in Sec. 15.8. With each indep reaction there is associated a reaction coordinate in accord with the trea of Sec. 15.1. In addition, a separate equilibrium constant is evaluated for reaction, and Eq. (15.13) becomes... 

We have been restricting our attention to the equilibrium condition for a single reaction, (17). It is also worthwhile to investigate the problem of computing equilibrium compositions for systems in which many reactions may occur simultaneously. Explicitly, the problem is the following given the pressure, the temperature, and the total number of moles of atoms in the system (irrespective of the number of moles of the chemical compounds in which these atoms may appear), find the number of moles of all chemical species (compounds and free atoms) in the system at equilibrium. 

Some reactions are essentially irreversible that is, the reaction proceeds only in a single direction (from reactants to products) and the concentration of the limiting reactant eventually approaches zero (although eventually could mean seconds tor some reactions and years for others). The equilibrium composition for such a reaction is therefore the composition corresponding to complete consumption of the limiting reactant. 

If the position of an equilibrium (that is, the composition of a chemical reaction system at equilibrium) can depend on the amounts of substances brought together (the active masses as defined by Guldberg and Waage s Law of Mass Action), an important question arises is there a single, measurable property that is unique to any chemical reaction system that can be used to predict its equilibrium composition for all possible initial amounts of the substances involved in the reaction The answer, which of course is yes , first became evident through a careful examination of a reaction studied by Berthollet ... 

We show how the initial composition of the gas in a closed vessel allows us to determine the equilibrium composition. Notice in particular that we do not know the volume of the vessel or the initial number of moles, just that the vessel initially contains an equimolar mixture of the two reactants. We proceed to describe the mole fractions of all components in terms of a single variable, the extent of reaction. Let njQ represent the unknown number of moles of each component initially contained in the vessel. Since the vessel is closed, the number of moles at some other time can be related to the reaction extent from the stoichiometry US.ing Mj = nyo + Vj ,... 

In the single stage experiments the degree of conversion of CO into CH3OH and carbon black were 1.57 and 51.0%, respectively. In the two-stage experiments 3.7-5.0% of CO was converted to CH4, 1.9-2.62% of CO into CH3OH, 20.4-32.6% of CO into carbon black, and 4.02-6.38% of CO into CO2. Calculations of the equilibrium compositions of the gaseous mixtures in the reaction... 

In the typical reaction-equilibrium problem for a single reaction, we are to determine the equilibrium composition at the end of the reaction. The problem is solved when we find a set of mole fractions x that minimize G at fixed T and P, or that minimize A at fixed T and v, etc. That is, in general we have a minimization problem of this form. 

How do we use the nonstoichiometric method to set up equations for computing the equilibrium composition at the completion of a single reaction ... 

The expressions for the equilibrium constants in (10.4.26)-(10.4.28) apply only when the same standard state has been chosen for all species. When different standard states are used for different species, K is still given by (10.3.13), with the appropriate expression for each activity taken from 10.4.1. In any event, (10.3.26)-(10.3.28) illustrate how the compositions occur in the expressions for K. On substituting (7.4.22) for those mole fractions, we obtain R algebraic equations that can be solved for the equilibrium values of (R extents of reaction. Then with those values for the j, the equilibrium mole fractions are obtained from (7.4.22). This procedure is illustrated for a single reaction in the following example. More elaborate reaction-equilibrium problems are discussed in Chapter 11. 

How do we compute the equilibrium composition from a reaction carried out in a single phase at fixed T and P ... 

There is no difficulty in calculating the chemical equilibrium of a system, in which a single chemical reaction takes place. The calculation, however, becomes increasingly difficult with the rising number of simultaneous reactions, until application of the same procedure to systems with more than three reactions proceeding simultaneously is practically impossible. Therefore, techniques have had to be worked out for more complicated chemically reacting systems, based on principles somewhat different from those of simple equilibrium calculation. The result are methods which allow equilibrium compositions to be calculated for systems of any degree of complexity whatever, in the ideal as well as real gas state. 

Find the relationship between the error of determining equilibrium composition and the error of determining the equilibrium constant for an ideal system, in which a single chemical reaction is taking place. Apply the results to the second part of Example 10. Extend this procedure to the case of R linearly independent reactions From the equilibrium condition... 

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