Mathematical equation, solving with calculator

When rearranged, this equation is a cubic equation (an equation in x3), which can be solved with a graphing calculator or mathematical software. However, because K is very small, we suppose that x will turn out to be so small that we can use the approximation procedure ... 

This chapter has introduced Fick s law for dilute solutions and has shown how this law can be combined with mass balances to calculate concentrations and fluxes. The mass balances are made on a small rectangular box. When the box becomes very small, the mass balances become the differential equations used to solve various pharmaceutical transport problems. Thus, this chapter discussed many mathematical equations. Although these mathematical equations are tedious, they are essential to each specific application of mass transfer in pharmaceutical systems to be discussed in the rest of this book. 

We often encounter quadratic or higher-order equations in equilibrium calculations. With modern programmable calculators, solving such problems by iterative methods is often feasible. But frequently a problem can be made much simpler by using some mathematical common sense. 

If variables are intercoimected then the value of one of them may be calculated using another. That is why all variables may be subdivided into dependent and independent ones. The number of dependent parameters is equal to the mmiber of equations, which tie them up with the independent ones. Thus, the set problem mathematically is solved as the difference between the total mmiber of variables, describing the state of the system, and the number of equations, which tie them up. For instance, if 2 components out of 4 are tied up between themselves by equilibrium constants, the number of independent media between them is equal 3. 

Keeping things simple, we will basically use two financial mathematical equivalences and the concepts of net present value (sometimes called net present worth) and annual equivalent benefits (or annual equivalent cost) (Sect. 12.6) to do all calculations for real-Ufe problems and project evaluations. Although it represents a limitation (not using more concepts and equations), you will be amazed at the wide variety of real-life and engineering problems that can be solved with these few rather elementary tools. Thus, we continue to meet our objectives, which are to (a) teach just a few concepts and (b) captivate you with the diversity of problematic situations that arise in process and bioprocess engineering. 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

The relevant calculations are commonly handled poorly, because the equilibrium equations involved are difficult to solve manually (but not with computers). The few calculations that are actually reported in the biochemical literature use simplified methods of limited and frequently unknown validity. Large excesses of magnesium ion are frequently used in experiments, perhaps in an attempt to avoid such calculations. The relevant theory is well worked out and there are excellent reviews. The limitation appears to involve diffusion to the (mathematically) inexpert user, which is one of the motivations of building expert systems. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

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