Integral calculus properties

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. 

Integration is used frequently in kinetics, thermodynamics, quantum mechanics and other areas of chemistry, where we build models based on changing quantities. Thus, if we know the rate of change of a property, y (the dependent variable), with respect to x (the independent variable), in the form of dy/dx, then integral calculus provides us with the tools for obtaining the form of y as a function of x. We see that integration reverses the effects of differentiation. 

Let s call Z the rate of collisions of gas molecules with a section of wall of area A. A full mathematical calculation of Z requires integral calculus and solid geometry. We present instead some simple physical arguments to show how this rate depends on the properties of the gas. 

For a fuller discussion on the properties and uses of hyperbolic functions, consult G. Chrystal s Algebra, Part ii., London, 1890 and A. G. Greenhill s A Chapter in the Integral Calculus, London, 1888. 

In scientific work, it is often more convenient to measure the rate of change of a quantity rather than the quantity itself. For example, a Geiger counter directly measures the rate of decay of a radioactive material, not the amount of material present. In another example, the speedometer on a car measures the rate of change of the cars position (that is, its velocity) and not its absolute position. The central problem of integral calculus is to determine the value of a particular property given that its derivative (rate of change) is known. 

The d1/2/df1/2 and d-1/2/df1/2 operators are respectively the semidifferentiation and semi-integration operators [81]. These are analogues of the familiar differentiation and integration operators of the calculus. Since they are unfamiliar to many chemists, Table 6 has been included to illustrate some of their definitions and properties. The semi-... 

The formal properties of calculus integrals and the integration by parts formula lead, among others, to the following rules of the Laplace transform ... 

In mathematics the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a variable quantity approaches as closely as one desires. The operations of differentiation and integration from calculus are both based on the theory of limits. The theory of limits is based on a particular property of the real numbers namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more. 

The fact, that macromolecular coil in diluted solution is a fractal object, allows to use the mathematical calculus of fractional differentiation and integration for its parameters description [72-74]. Within the framework of this formalism there is the possibility for exact accounting of such nonlinear phenomena as, for example, spatial correlations [74]. In the last years the methods of ftactional differentiation and integration are applied successfully for pol5mier properties description as well [75-77]. The authors [78-81] used this approach for average distance between polymer chain ends calculation of polycarbonate (PC) in two different solvents. 

The existence of states that are inaccessible to adiabatic processes was shown by Carath odory to be necessary and sufficient for the existence of an integrating factor that converts into an exact differential [2-4]. From the calculus we know that for differential equations in two independent variables, an integrating factor always exists in fact, an infinite number of integrating factors exist. Experimentally, we find that for pure one-phase substances, only two independent intensive properties are needed to identify a thermod)mamic state. So for the experimental situation we have described, we can write SQ gj, as a function of two variables and choose the integrating factor. The simplest choice is to identify the integrating factor as the positive absolute thermodynamic temperature X = T. Then (2.3.3) becomes... 

Another very useful application of the calculus of finite differences is in the derivation of interpolation/extrapolation formulas, the so-called interpolating polynomials, which can be used to represent experimental data when the actual functionality of these data is not known. A very common example of the application of interpolation is in the extraction of physical properties of water from the steam tables. Interpolating polynomials are also used to estimate numerical derivative and integral of the tabulated data (see Chap. 4). The discussion of several interpolating polynomials is given in Secs. 3.7-3.10. 

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