Suspicious equations

When you re debugging a program and one of the variables is being consistently miscalculated, there s a good chance you re a victim of the hierarchy. The quickest way to fix such an error is to liberally sprinkle parentheses throughout the suspicious equations. Your other choice is to trace through the line step by step to find how the computer is evaluating the equation. 

According to Zeleznik and Gordon, tempers became so heated that a panel convened in 1959 to discuss equilibrium computation had to be split in two. Both sides seemed to have lost sight of the fact that the equilibrium constant is a mathematical expression of minimized free energy. As noted by Smith and Missen (1982), the working equations of Brinkley (1947) and White et al. (1958) are suspiciously similar. As well, the complexity of either type of formulation depends largely on the choice of components and independent variables, as described in Chapter 3. 

The concentrations are then used instead of the activities in various-thermodynamic expressions (e.g., the equilibrium law, and Nernst s equation for the e.m.f.). Some physical chemists are suspicious of this method. However, it is just as thermodynamic as the traditional one, the only difference being that another activity scale is used. 

This is the classical argument introduced by van Heerden in 1953 for the adiabatic stirred tank. It is a most important one to grasp firmly for it can be used in more complicated situations to get some insight into the stability of a system. However, its limitations must be also thoroughly understood. In particular, it can be used to establish instability, but it does not count conclusively for stability because of several reasons. First, we should be suspicious of a single condition for a system in which there are two variables. Second, the diagram for the heat generation was drawn in a rather special way, for the steady state-mass balance equation, f 7), was first... 

Someone tells you that they have an equation that nearly perfectly fits the data, and so that equation is the best description of the phenomenon. You inquire, and find out that parameter values for the equation were obtained from the data set for which the fit is nearly perfect. Why should you be suspicious about the value of the equation What could you do to determine how good the equation is ... 

We started this chapter by delineating the two fundamental types of equations, either nonlinear or linear. We then introduced the few techniques suitable for nonlinear equations, noting the possibility of so-called singular solutions when they arose. We also pointed out that nonlinear equations describing model systems usually lead to the appearance of implicit arbitrary constants of integration, which means they appear within the mathematical arguments, rather than as simple multipliers as in linear equations. The effect of this implicit constant often shows up in startup of dynamic systems. Thus, if the final steady state depends on the way a system is started up, one must be suspicious that the system sustains nonlinear dynamics. No such problem arises in linear models, as we showed in several extensive examples. We emphasized that no general technique exists for nonlinear systems of equations. 

Our new derivative terms (9G/9n )j.p - look suspiciously like our definition of a partial molar property in Equation (2.1), and indeed they are partial molar Gibbs energies. They allow us to deal with compositional changes, and as such... 

Now, clearly each series has quite different implications, even when the correlation coefficient is equally good. The only marginally acceptable function will be that from series I, while the other three are clearly wrong. Therefore, we should always be suspicious of accepting only the regression equation accompanied by the correlation coefficient. We will return to these examples next. 

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