Spreadsheet simultaneous equations

The most convenient way to set up and solve the equations is to use a spreadsheet but any of the standard procedures and programs available for the solution of linear simultaneous equations can be used Westlake (1968), Mason (1984). 

Most proprietary spreadsheets include a routine for the inversion of matrices and the solution of sets of linear simultaneous equations. By using cell references, with cell copying and cell pointing, it is a simple procedure to set up the split fraction matrices... 

Statistical packages and spreadsheets solve the simultaneous equations in (2.8) to estimate (J rather than computing the matrix inverse in Equation (2.9). 

The absorbance of a mixture is the sum of absorbances of the individual components. At a minimum, you should be able to find the concentrations of two species in a mixture by writing and solving two simultaneous equations for absorbance at two wavelengths. This procedure is most accurate if the two absorption spectra have regions where they do not overlap very much. With a spreadsheet, you should be able to use matrix operations to solve n simultaneous Beer s law equations for n components in a solution, with measurements at n wavelengths. You should be able to use Excel SOLVER to decompose a spectrum into a sum of spectra of the components by minimizing the function (Aca c — Am)2. 

To "solve" this system of simultaneous equations, we want to be able to calculate the value of [A], [B] and [C] for any value of t. For all but the simplest of these systems of equations, obtaining an exact or analytical expression is difficult or sometimes impossible. Such problems can always be solved by numerical methods, however. Numerical methods are completely general. They can be applied to systems of differential equations of any complexity, and they can be applied to any set of initial conditions. Numerical methods require extensive calculations but this is easily accomplished by spreadsheet methods. 

Spreadsheet Summary In some chemical problems, two or more 3 simultaneous equations must be solved to obtain the desired result. Example 12-3 is such a problem. In Chapter 6 of Applications of Microsoft Excel in Analytical Chemistry, the method of determinants and the matrix inversion method are explored for solving such equations. The matrix method is extended to solve a system of 4 equations in 4 unknowns. The matrix method is used to confirm the results of Example 12-3. 

Spreadsheet Summary In Chapter 12 of Applications of Microsoft Excel in Analytical Chemistry, we use spreadsheet methods to determine concentrations of mixtures of analytes. Solutions to sets of simultaneous equations are evaluated using iterative techniques, the method of determinants, and matrix manipulations. I... 

You are free, of course, to pick three characteristic wavelengths, and to solve the resulting three simultaneous equations. The added noise will then (rather strongly) affect your results, but you will have no way of knowing by how much. The method illustrated below is not only much less sensitive to noise, but also provides error estimates and, most importandy, is much easier to implement. Of course it uses matrix algebra, just as you did in section 6.2, but that will be completely invisible to you, the user. The entire analysis comes prepackaged with the spreadsheet. 

Solubility from K p Spreadsheet Exercise, 334 Chapter 10, Problem 43 (Example 10.9) Simultaneous Equations (Example 16.5, spectrophotometric mixtures)... 

The calculation is iterative and can be conveniently carried out using a solver in a spreadsheet to satisfy the Equations 15.65 and 15.66 simultaneously. To reach the two variables simultaneously, the objective can be set up such that the difference between the... 

In this text all numerical problems involve integration of simultaneous ordinary differential equations or solution of simultaneous algebraic equations. You should have no trouble finding ways to solve algebraic equations with a calculator, a spreadsheet, a personal computer, etc. 

This optional chapter provides tools to compute the concentrations of species in systems with many simultaneous equilibria.3 The most important tool is the systematic treatment of equilibrium from Chapter 8. The other tool is a spreadsheet for numerical solution of the equilibrium equations. We will also see how to incorporate activity coefficients into equilibrium calculations. Later chapters in this book do not depend on this chapter. 

Homemade models are often mass and energy balance spreadsheets, simplified kinetic models, or the simultaneous solution of the convection diffusion and heat equations together with nonlinear isotherms. All levels of models have their place. 

Alternatively, all thej model equations may be listed singly, and then solved simultaneously using a standard nonlinear equation solver, such as a spreadsheet program. For the two-component system, the equations include the van Laar equations for both components in each liquid phase ... 

We have seen above that two or three defect species may dominate the defect structure simultaneously, and we have derived analytical expressions for their concentrations, which may be used in modeling fluxes and membrane behavior. Different approaches exist for the increased complexity of cases with more defects Sequential analytical breakdown of the set of equations (equilibria and charge, mass, and site balances) can be carried out by a computer program or spreadsheet... 

Solving these three equations simultaneously in a spreadsheet using Goal Seek, we obtain ( )d = 0.302 and since the ratio > 1, use (j)d = ( d,feed 0.167. 

Traditional Isothermal Kinetics Measurements of conversion-rate of conversion-time data by isothermal method 1 and conversion-time data by isothermal method 2 were described earlier in this section. Isothermal method 1 measurements have the advantage of simultaneously measuring conversion (a) and rate of conversion (daJdt), which allows use of derivative forms of the rate equation such as Eqs. (2.82) and (2.86). Both conversion and rate of conversion data are necessary to model autocatalytic kinetics [e.g., using Eq. (2.86)]. Isothermal method 2 yields both Tg and conversion at a series of times and temperatures (see Rg. 2.70 as an example). However, the measurements are time-consuming, and since the reaction rate is not measured, it must be calculated mathematically, or integrated forms of the rate equation such as Eqs. (2.83)-(2.85) must be used. To perform these analyses generally requires use of a spreadsheet. 

The numerical solution of simultaneous, ordinary differential equations can be accomplished with a number a standard mathematical packages. Appendix 7-A.2 illustrates the use of a spreadsheet to solve these equations, via a fourth-order Runge-Kutta technique. The result is z - 0.0246 gcat-min/1. 

The following spreadsheet shows how the Runge-Kutta technique for simultaneous differential equations can be used to solve Example 7-4. 

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