When the equilibrium state in a reacting system depends on two or more simultaneous chemical reactions, the equilibrium composition can be found by a direct extension of the methods developed for single reactions. Each reaction will have its own reaction coordinate in which the compositions can be expressed. Some of the products of one reaction may act as reactants in another reaction, but the amount of that substance can still be written in terms of the extents of the reactions. Eventually, the material balances lead us to a system of N nonlinear equations in terms of N unknowns. We illustrate a solution by hand and then demonstrate how numerical solvers can be used.
Example 17.6. Simultaneous reactions that can be solved by hand
We can occasionally come across multiple reactions which can be solved without a computer. These are generally limited to textbook problems, but provide a starting point and test case for applying the general approach. Consider the two series/parallel gas phase reactions:
The reactions are considered series reactions because C is a product of the first reaction, but a reactant in the second. They are parallel because A is a reactant in both reactions. The pressure in the reactor is 10 bar, and the feed consists of 2 moles of A and 1 mole of B. Calculate the composition of the reaction mixture if equilibrium is reached with respect to both reactions.
Solution
The material balance gives:
Note that for a physical solution, 0 ≤ ξ1 ≤ 1, 0 ≤ ξ2 ≤ ξ1 to ensure that all mole numbers are positive. This reaction network is independent of P because Σvi = 0. The equilibrium constants are
Solving the first equation for ξ1 using the quadratic equation,
Similarly, for the second reaction,
We may now solve by trial and error. The procedure is: 1) guess ξ1; 2) solve Eqn. 17.40 for ξ2; 3) solve Eqn. 17.39 for 
; 4) if 
, go to step 1. The iterations are summarized below.
Further iteration results in no further significant change.
These equations were amenable to the quadratic formula, but in general equilibrium criteria can be more complicated. Fortunately, standard programs available that are formulated to solve numerically multiple nonlinear systems of equations, so we can concentrate on applying the program to thermodynamics instead of developing the numerical analysis. Many software packages like Mathematica, Mathcad, MATLAB, and even Excel offer the capability to solve nonlinear systems of equations. Excel provides an especially convenient basis for illustrating the methods presented here.
Example 17.7. Solving multireaction equilibria with Excel
Methanol has a lower vapor pressure than gasoline. That can make it difficult to start a car fueled by pure methanol. One potential solution is to convert some of the methanol to methyl ether in situ during the start-up phase of the process (i.e., automobile). At a given temperature, 1 mole of MeOH is fed to a reactor at atmospheric pressure. It is assumed that only the two reactions given below take place. Compute the extents of the two simultaneous reactions over a range of temperatures from 200°C to 300°C. Also include the equilibrium mole fractions of the various species.
Solution
A worksheet used for this solution is available in the workbook Rxns.xlsx.
Data for reaction (1) have been tabulated by Reactions Ltd.a—at 473.15 K, ΔHT = 96,865 J/mol and lnKa1,473 = 3.8205. Over the temperature range of interest we can apply the shortcut van’t Hoff equation assuming constant heat of reaction using the data at 200°C as a reference.
Data for reaction (2) can be obtained from Appendix E for MeOH and water. For DME, the values are from Reid et al. (1987).b
The shortcut van’t Hoff equation for this reaction gives:
Material balances:
Writing equations for reaction coordinates for reaction 1:
and for reaction 2:
These two equations are solved simultaneously for ξ1 and ξ2. We have rearranged the objective functions to eliminate the ratios of ξ functions and use differences instead because the Excel Solver is much more robust with this mathematical form. The solution is implemented in the worksheet DUALRXN in Rxns.xlsx or Matlab Ex17_07.m. In the example here (see Fig. 17.3), the ΔCP for both reactions is neglected. The equations derived above are entered directly into the cells, and the Solver tool is called.c You will need to designate one of the reaction equations as the target cell, the value of which is set to zero. The other reaction equation should be designated as a constraint (also set to zero). The cells with the reaction coordinates are the variables to be changed to obtain a solution. Under “options,” you may want to specify the “conjugate” method, since that generally seems to converge more robustly for the reacting systems typically encountered. Generally, the Solver tool will require a reasonably accurate initial guess to keep it from converging on absurd results (e.g., yi < 0). The initial guess can be easily developed by varying the values in the reaction-extent cells until the target cells move in the right direction. It sounds difficult, but the given worksheet will get you started, then you can experiment with initial guesses and experience how good your initial guesses need to be.
Rxns.xlsx, Worksheet DUALRXN of MATLAB Ex17_07.m.
a. These data are slightly different from values calculated using tabulated properties from Appendix E, but such variations are common in thermochemical data. The equilibrium compositions are about the same if the example is reworked using data from Appendix E.
b. Reid, R., Prausnitz, J.M., Poling, B. 1987. The Properties of Gases and Liquids, 4th ed. New York: McGraw-Hill.
c. See the online supplement for an introduction to Solver.
Figure 17.3. Worksheet DUALRXN from workbook Rxns.xlsx for Example 17.7 showing converged answers at several temperatures.
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