Additional Mathematical Techniques

In this section we continue the presentation of the mathematics of thermodynamics, but the concepts here are less essential to the understanding of thermodynamics. This is not to imply that the following material is of little use or that the reader should blithely skip ahead to Chapter 3. On the contrary, the next three topics are extremely powerful and may be used to quickly derive and illustrate most of the theoretical 

It is possible to subdivide the properties used to describe a thermodynamic system (e.g., T, P, V,U.) into two main classes termed intensive and extensive variables. This distinction is quite important since the two classes of variables are often treated in significantly different fashion. For present purposes, extensive properties are defined as those that depend on the mass of the system considered, such as volume and total energy content, indeed all the total system properties (Z) mentioned above. On the other hand, intensive properties do not depend on the mass of the system, an obvious example being density. For example, the density of two grams of water is the same as that of one gram at the same P, T, though the volume is double. Other common intensive variables include temperature, pressure, concentration, viscosity and all molar (Z) and partial molar (Z, defined below) quantities.  

Partial molar quantities are very commonly used to describe solutions, or systems containing more than one component. Mathematically, a partial molar quantity Zi is defined as the partial derivative 

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Both the Raman and the infrared spectrum yield a partial description of the internal vibrational motion of the molecule in terms of the normal vibrations of the constituent atoms. Neither type of spectrum alone gives a complete description of the pattern of molecular vibration, and, by analysis of the difference between the Raman and the infrared spectrum, additional information about the molecular structure can sometimes be inferred. Physical chemists have made extremely effective use of such comparisons in the elucidation of the finer structural details of small symmetrical molecules, such as methane and benzene. But the mathematical techniques of vibrational analysis are. not yet sufficiently developed to permit the extension of these differential studies to the Raman and infrared spectra of the more complex molecules that constitute the main body of both organic and inorganic chemistry. 

Clearly, what is required for a reliable recognition of all the peaks during the optimization procedure is information on the pure component spectra and the pure component peaks (elution profiles). A method to obtain both the spectral and the chromatographic data involves the application of a mathematical technique called principal component analysis (PCA) [592]. This method is based on the additivity of spectra according to Beer s law. The absorption (A) at a time / and wavelength A is given by... 

Table IV. Mathematical Techniques Used for MWD in Chain Addition"...

We now describe two other methods of deriving an effective Hamiltonian, both of which are widely used. Although we shall not go into details, the mathematical development will show that the two methods are exactly equivalent and, in addition, that they are very nearly equivalent to the method based on projection operators given in the previous section. The equivalence ofthe three methods is not really very surprising since they are all solutions of the problem by perturbation theory, differing only in the mathematical techniques employed. 

Many mathematical techniques, in addition to the basic approaches already discussed, have been developed for application in various situations that require determination of optimum conditions. A summary of some of the other common and more advanced mathematical techniques, along with selected references for additional information, is presented in the following ... 

Equation 9 represents the IR spectmm (intensity versus wavenumber), which can be derived from expression (8) using a mathematical technique known as Fourier transformation. Needless to say, this requires spectrometer-interfaced computing power, which additionally provides the capacity for spectral manipulation such as deconvolution, smoothing, and subtraction. 

Interfacial rheology is not simple, neither experimentally nor interpretation-ally. In this section we shall therefore discuss the various definitions, elaborations and techniques at some length, attempting to strike a balance between mathematical complexity and physical insight. In particular, in line with previous volumes, we shall sometimes use tensor notation, because it is handy on a descriptive level. However, tensor analysis, which is rigorous, and helps to describe the genered formalism, will be avoided because it tends to be abstract and requires additional mathematical skills. 

In this article an attempt is made to include some new contributions to bring the field of oscillations in the dynamics of chemical reactions up to date. To provide a continuation, the basic structure of the paper follows the outline of the previous article which will be referred to as (G G). Moreover, some recent studies have also been added to the field, and they are included as additional sections. These new sections are L. Oxidation by Chlorite M. Miscellaneous Studies and N. General Models and Mathematical Techniques. 

The hydrogen-supported nickel system has been reported by Schuit and de Boer 48) as having both fast and slow adsorptions occurring. In addition precise data are available 45) on the isothermal variation of pressure with adsorption time on a similar hydrogen-supported nickel system. This classical adsorption data (step response data) can be mathematically transformed to simulate frequency response data by means of suitable mathematical techniques before any experimentation on the system is begun. The results of this simulation will be much the same as though an actual frequency response of Doerner s hydrogen-supported nickel system had been made. Actually any published adsorption isotherm data can be treated. However, the limitations of the simulation method are threefold (1) very accurate adsorption versus time data are required (2) the accuracy and dependability of the result at very fast times are subject to question (3) the adsorptions are not reproducible in the sense that only one real experiment was made for all adsorptions and the sensitivity of the mathematics could distort the result. 

Complex thermal systems cannot usually be optimized using mathematical optimization techniques. The reasons include system complexity opportunities for structural changes not identified during model development incomplete cost models and inability to consider in the model additional important factors such as plant availability, maintenability, and operability. Even if mathematical techniques are applied, the process designer gains no insight into the real thermodynamic losses, the cost formation process within the thermal system, or on how the solution was obtained. 

The first proceeds from direct observations and from a few fundamental laws—the three principal laws supported by general experience—and uses mathematical technique to derive from them additional laws whose consequences are compared with those of experiment. The method combines the advantage of infallibility with the disadvantage that the procedures cannot be materialized through models. 

In a number of the examples discussed in the preceding section, comonomer reactivity ratios were used to predict sequence distributions. A number of procedures exist for deriving reactivity ratios based on copolymer/comonomer composition data. Recently, a new method for determining reactivity ratios, based on in situ NMR measurements has been derived. This method is described. In addition, some of the mathematical techniques available to calculate sequence distributions using reactivity ratios are mentioned briefly, since their use can impinge on a number of the NMR studies of sequence distributions. 

This latter example demonstrates that several mathematical techniques may be required to achieve an analytical solution to a complicated transport model. In addition, the regular perturbation technique is used to reduce the problem to a set of linear problems that we know how to solve. 

An immense amount of experimental data has been accumulated from investigations of the infrared absorption spectra and of the Raman effect in polyatomic molecules. Only an extremely small fraction of this material has been subjected to analysis, although the theoretical tools for such an analysis are quite well developed and the results which could be obtained are of considerable interest. One reason for this situation is the amount of labor required to unravel the spectrum of a complex molecule, but an additional deterrent has been the unfamiliar mathe-rnati( s, such as group theory, in terms of which the most powerful forms of the theory of molecular dynamics have been couched. When only the necessary parts of these mathematical techniques are considered, the difficulty of understanding the theory of the vibrational and rotational spectra of polyatomic molecules is greatly reduced. 

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