Operational calculus

Cartesian component notation offers an extremely convenient shorthand representation for vectors, tensors, and vector calculus operations. In this formalism, we represent vectors or tensors in terms of their typical components. For example, we can represent a vector A in terms of its typical component Ait where the index i has possible values 1, 2, or 3. Hence we represent the position vector x as x, and the (vector) gradient operator V as Note that there is nothing special about the letter that is chosen to represent the index. We could equally well write x . x/ . or x , as long as we remember that, whatever letter we choose, its possible values are 1, 2, or 3. The second-order identity tensor I is represented by its components %, defined to be equal to 1 when i = j and to be equal to 0 if z j. The third-order alternating tensor e is represented by its components ,< , defined as... 

The rate laws we have examined so far enable us to calculate the rate of a reaction from the rate constant and reactant concentrations. These rate laws can also be converted into equations that show the relationship between concentrations of reactants or products and time. The mathematics required to accomplish this conversion involves calculus. We do not expect you to be able to perform the calculus operations, but you should be able to use the resulting equations. We will apply this conversion to three of the simplest rate laws those that are first order overall, those that are second order overall, and those that are zero order overall. 

With points on the titration curve closely spaced, obtaining slopes (ApH/AVg) is a numerical differentiation that is a reasonable approximation of the calculus operation of differentiation. Compare the numerical and calculus approaches in titration curves using several different acid concentrations. 

Because values of the critical variable can be listed in very small increments without the customary tediousness, the spreadsheet format is very well suited for performing reasonably accurate calculus operations by numerical differentiation and integration. Useful examples of the former include titration curve slopes, dV /dpH and dpH/dV, which give us important titration curve parameters, namely, the buffer and sharpness indices. Another area of great interest, chemical kinetics, represents an additional topic where numerical differentiation and integration are of great use. 

The Laplace transform is an exceptionally useful computational technique because the equivalents of calculus operations in the Laplace domain are straightforward algebraic ones. 

A rate law tells us how the rate of a reaction changes at a particular temperature as we change reactant concentrations. Rate laws can be converted into equations that tell us what the concentrations of fhe reacfanfs or producfs are af any time during fhe course of a reaction. The mafhematics required involves calculus. We don f expect you to be able to perform fhe calculus operations however, you should be able to use fhe resulting equations. We will apply fhis conversion to fwo of fhe simplesf rate laws fhose fhaf are firsf order overall and fhose fhaf are second order overall. 

Since the rate of this reaction depends on the concentration of N2O5 to the first power, it is a first-order reaction. This means that if the concentration of N2O5 in a flask were suddenly doubled, the rate of production of NO2 and O2 also would double. This rate law can be put into a different form using a calculus operation known as integration, which yields the expression... 

This has major implications in that there is a mathematical (calculus) operator Hop, which represents the total energy of a particle (in only one dimension so far) and a function v[>, which incorporates the De Broglie condition and is an eigenfunction of the total energy operator. Note the left side of the equation must be in energy units since the right side is in terms of tot-... 

What is the cause of this uncertainty We noted earlier that x and P are related and have a mutual effect. Another way to quantify the effect of position and momentum is by using a commutator. In real arithmetic and algebra with real numbers, we are used to interchanging the order of factors as in 2x3 = 3x2 = 6, that is the commutator [3,2] = 0, but when we use calculus operators that interchange of order may not work. Let us define a quantity called the commutator as a bracket that represents the amount by which interchanging the order of two successive operators makes a difference (on some... 

Again N is the normalization constant. On first encounter, one may ask, just what is p The operation denoted by P is a permutation of the order of the orbitals in the basis set performed in the mind of the reader. There is no calculus operator P —it is just the operation you use in your mind when you write down aU the possible permutations that evolve from expanding... 

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

The problems of operations research have stimulated new developments in several mathematical fields various aspects of game theory, stochastic processes, the calculus of variations, graph theory, and numerical analysis, to name a few. 

One of the pleasant aspects of the study of thermodynamics is to find that the mathematical operations leading to the derivation and manipulation of the equations relating the thermodynamic variables we have just described are relatively simple. In most instances basic operations from the calculus are all that are required. Appendix 1 reviews these relationships. 

Thermodynamic derivations and applications are closely associated with changes in properties of systems. It should not be too surprising, then, that the mathematics of differential and integral calculus are essential tools in the study of this subject. The following topics summarize the important concepts and mathematical operations that we will use. 

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

Diffusion operates along well-defined physical principles first described in 1855 by Adolf Fick and now widely known as Fick s Laws of Diffusion. Philbert5 provides a detailed explanation of the laws and a historical account of Fick. While they were designed to describe the behavior of gas molecules under ideal theoretical conditions, Fick s Laws serve reasonably well to describe a wide variety of real diffusion events. Fick wrote the laws as a set of equations in the language of calculus but these can be rephrased in plain English. 

Let us consider what happens when V moves and we perform the operations in Equation (3.1). This is depicted in Figure 3.2. Here we have displayed the drawing in two dimensions for clarity, but our results will apply to the three-dimensional volume. In Figure 3.2 the volume is shown for time t, and a small time increment later, At, as a region enclosed by the dashed lines. The shaded regions Vj and Vn are the volumes gained and lost respectively. For these volumes, we can represent Equation (3.1) by its formal definition of the calculus as... 

With the help of fractional calculus, Dassas and Duby123 have worked on the problem of diffusion towards the fractal interfaces. They have proposed the following generalized diffusion equation involving a fractional derivative operator ... 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

This operation requires differentiating under an integral sign. From the theorems of calculus, if... 

Most chemists immediately utilized the compositional relationships derived from the atomic hypothesis, but for most of the century they continued to dispute the ontological reality of the atoms that rationalized their useful consequences. Even those most doubtful of the reality of the atom found its operational utility indispensible. Humphry Davy was expressing composition by the relative numbers of proportions by i8io, and William Wollaston attempted a calculus of chemical equivalents in 1814. Jons Jakob Berzelius undertook a systematic determination of the most accurate values to assign each of the atomic weights, publishing his first list in 1813. 

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian That is, we wish to find orbitals that minimize (4 hp H I hp). By applying variational calculus, one can show that each such orbital i/f, is an eigenfunction of its own operator hi defined by... 

Finding values of m and b that minimize the sum of the squares of the vertical deviations involves some calculus, which we omit. We will express the final solution for slope and intercept in terms of determinants, which summarize certain arithmetic operations. The... 

---