Mathematical Properties of State Functions

Maxwell used the mathematical properties of state functions to derive a set of useful relationships. These are often referred to as the Maxwell relations. Recall the first law of thermodynamics, which may be written as... 

From the standpoint of thermodynamics, the most significant mathematical properties of state functions are summarized in the following statement. The necessary and sufficient condition that the line integral... 

As we have seen from our previous discussions of heat capacities, thermal expansion coefficients, and compressibilities, partial derivatives are the key to discussing changes in thermodynamic systems. In a single-component system of fixed size, the specification of two state variables completely determines the state of the system. Calling one of the molar energy quantities Z, we can write Z = Z(X, Y), where Xand Tare any two state variables, such as Tand I] or Tand V. Using the general mathematical properties of functions of two variables that are discussed in Appendix A,... 

In the previous section, we have shown how it is possible to obtain directly from bulk magnetization data, and only considering the mathematical properties of the general mean-field expression M = f[ H + AM)/T], a direct determination of the molecular field exchange parameter A and its dependence on M, and the mean-field state function /, which will contain information on the magnetic entities in play, and their interactions. 

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems. 

Particulate products, such as those from comminution, crystallization, precipitation etc., are distinguished by distributions of the state characteristics of the system, which are not only function of time and space but also some properties of states themselves known as internal variables. Internal variables could include size and shape if particles are formed or diameter for liquid droplets. The mathematical description encompassing internal co-ordinate inevitably results in an integro-partial differential equation called the population balance which has to be solved along with mass and energy balances to describe such processes. 

The standard wave functions of quantum chemistry are all constmcted from antisymmetric products of MOs. In most applications, these MOs are generated by expansion in a finite set of simple analytical functions - the atomic basis functions. The choice of basis functions for a molecular calculation is therefore an important one, which ultimately determines the quality of the wave function. In this chapter, the mathematical properties of the atomic basis functions are investigated and their usefulness is explored by carrying out simple expansions of the ground-state orbitals of the carbon atom. 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

The second thing to note about the thermodynamic variables is that, since they are properties of the system, they are state functions. A state function Z is one in which AZ = Zi — Z that is, a change in Z going from state (l) to state (2), is independent of the path. If we add together all of the changes AZ, in going from state (1) to state (2), the sum must be the same no matter how many steps are involved and what path we take. Mathematically, the condition of being a state function is expressed by the relationship... 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

Solute equilibrium between the mobile and stationary phases is never achieved in the chromatographic column except possibly (as Giddings points out) at the maximum of a peak (1). As stated before, to circumvent this non equilibrium condition and allow a simple mathematical treatment of the chromatographic process, Martin and Synge (2) borrowed the plate concept from distillation theory and considered the column consisted of a series of theoretical plates in which equilibrium could be assumed to occur. In fact each plate represented a dwell time for the solute to achieve equilibrium at that point in the column and the process of distribution could be considered as incremental. It has been shown that employing this concept an equation for the elution curve can be easily obtained and, from that basic equation, others can be developed that describe the various properties of a chromatogram. Such equations will permit the calculation of efficiency, the calculation of the number of theoretical plates required to achieve a specific separation and among many applications, elucidate the function of the heat of absorption detector. 

Before we can confront the problem of undoing the damage inflicted by spreading phenomena, we need to develop background material on the mathematics of convolution (the function of this chapter) and on the nature of spreading in a typical instrument, the optical spectrometer (see Chapter 2). In this chapter we introduce the fundamental concepts of convolution and review the properties of Fourier transforms, with emphasis on elements that should help the reader to develop an understanding of deconvolution basics. We go on to state the problem of deconvolution and its difficulties. 

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