Definitions thermodynamic functions

Finally, significant advances in the techniques of both thermal and thermochemical measurements have come to fruition in the last decade, notably aneroid rotating-bomb calorimetry and automatic adiabatic shield control, so that enhanced calorimetric precision is possible, and the tedium is greatly reduced by high speed digital computation. Non-calorimetric experimental approaches as well as theoretical ones, e.g., calculation of electronic heat capacity contributions to di- and trivalent lanthanides by Dennison and Gschneidner (33), are also adding to definitive thermodynamic functions. 

It has long been known that the adsorption of a gas on a solid surface is always accompanied by the evolution of heat. Various attempts have been made to arrive at a satisfactory thermodynamic analysis of heat of adsorption data, and within the past few years broad agreement has been achieved in setting up a general system of adsorption thermodynamics. Here we are not concerned with the derivation of the various thermodynamic functions but only with the more relevant definitions and the principles involved in the thermodynamic analysis of adsorption data. For more detailed treatments, appropriate texts should be consulted. " ... 

A class of thermodynamic functions called residual properties is given generic definition by equation 132 ... 

The quantities G, //, and S are called extensive thermodynamic functions because the magnitude of the quantity in each case depends on the amount of substance in the system. The change in Gibbs free energy under addition of unit concentration of component / at constant concentrations of the other components is called the partial Gibbs free energy of the /-component, i.e., the chemical potential of the /-component in the system. The chemical potential is an intensive thermodynamic quantity, like temperature and concentrations. The formal definition is... 

The next important thermodynamic function that we must obtain is the entropy S. The statistical thermodynamic definition of entropy is... 

Why does heat flow from a warm body into a cold one Why doesn t it ever flow in the reverse direction We can see that differences in temperature control the direction of flow of heat, but this observation raises still another question What is temperature Reflection on these questions, and on the interconversion of heat and work, led to the discovery of the second law of thermodynamics and to the definition of a new thermodynamic function, the entropy S. 

Since this formulation includes standard values of thermodynamic functions, we shall define the standard state of a particle adsorbed on a nonuniform surface and the standard state of a free site. A uniform surface with one substance adsorbed is at its standard state at coverage 1/2 (Section IV). In order to apply this definition to each site of a nonuniform surface, we shall restate it as follows a surface site is in its standard state if the probability that it is occupied is 1/2 (in other words, if it is occupied and is vacant equally often). 

There are two interpretations of the statistical quantity A, both being closely related to the geometrical interpretation of the term phase boundary . From a purely macroscopic point of view, the gel of course represents a phase to which thermodynamical functions of state are related. Nevertheless, such a macroscopic image with a sharp boundary can hardly be correct in a PDC-column considering the range of end-to-end distances of the transported coils in the concentration profile, because the transported P-mer and the stationary gel are chemically equal in PDC. The two possible definitions of the quantity A(P) are ... 

Although the choice of standard states is arbitrary, two choices have been established by convention and international agreement. For some systems, when convenient, the pure component is chosen as the substance in the standard state. For other systems, particularly dilute solutions of one or more solutes in a solvent, another state that is not a standard state is chosen as a reference state [19]. This choice determines the standard state, which may or may not be a physically realizable state. The reference state of a component or species is that state to which all measurements are referred. The standard state is that state used to determine and report the differences in the values of the thermodynamic functions for the components or species between some state and the chosen standard state. When pure substances are used in the definition of a standard state, the standard state and the reference state are identical. 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

The definition is completed by assigning a value to m and (f>c in some reference state. To conform with the definitions made in Sections 8.9 and 8.10, the infinitely dilute solution with respect to all molalities or molarities is usually used as the reference state at all temperatures and pressures, and both m and c are made to approach unity as the sum of the molalities or molarities of the solutes approaches zero. The standard state of the solvent is again the pure solvent, and is identical to its reference state in all of its thermodynamic functions. 

In the previous sections concerning reference and standard states we have developed expressions for the thermodynamic functions in terms of the components of the solution. The equations derived and the definitions of the reference and standard states for components are the same in terms of species when reactions take place in the system so that other species, in addition to the components, are present. Experimental studies of such systems and the thermodynamic treatment of the data in terms of the components yield the values of the excess thermodynamic quantities as functions of the temperature, pressure, and composition variables. However, no information is obtained concerning the equilibrium constants for the chemical reactions, and no correlations of the observed quantities with theoretical concepts are possible. Such information can be obtained and correlations made when the thermodynamic functions are expressed in terms of the species actually present or assumed to be present. The methods that are used are discussed in Chapter 11. Here, general relations concerning the expressions for the thermodynamic functions in terms of species and certain problems concerning the reference states are discussed. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

From equations (11)-(15) and from the definitions in the preceding section, all previously defined thermodynamic functions can be related to T, V, and Ni provided that Qi is a known function of T and V. The problem of calculating thermodynamic properties of gases therefore reduces to the problem of evaluating in terms of Tand K... 

We have already introduced in 3 of chap. XX the thermodynamic functions of mixing in the case of perfect solutions. These definitions are easily extended to non-ideal solutions. The results obtained provide a useful basis for the classification of non-ideal solutions which will be made in paragraph 5 of this chapter. 

Attention may be drawn to the fact that although certain restrictions were mentioned in the course of the foregoing deductions, the final results are of general applicability. The Gibbs-Helmholtz equations (25.31), (25.32) and (25.33), for example, will hold for any change in a closed system, irrespective of whether it is carried out reversibly or not. This is because the values of AF and AH (or AA and AH) are quite definite for a given change, and do not depend upon the path followed. The only condition that need be applied is the obvious one that the system must be in thermodynamic equilibrium in the initial and final states of the process, for only in these circumstances can the various thermodynamic functions have definite values ( 4d). 

The chemical potential provides the fundamental criteria for determining phase equilibria. Like many thermodynamic functions, there is no absolute value for chemical potential. The Gibbs free energy function is related to both the enthalpy and entropy for which there is no absolute value. Moreover, there are some other undesirable properties of the chemical potential that make it less than suitable for practical calculations of phase equilibria. Thus, G.N. Lewis introduced the concept of fugacity, which can be related to the chemical potential and has a relationship closer to real world intensive properties. With Lewis s definition, there still remains the problem of absolute value for the function. Thus,... 

From the definition of heat of reaction and the known relations for the thermodynamic functions it is easy to calculate A/f at any pressure and temperature from its value at standard conditions. We have... 

It is sometimes convenient to reformulate the generalized diffusional driving forces dg either in terms of mass or molar functions, using the partial mass Gibbs free energy definition, Gg = hg —Tsg, and the chain rule of partial differentiation assuming that the chemical potential (i.e., /Xg = Gg) is a function of temperature, pressure and concentration (Slattery [89], sect. 8.4). dg can then be expressed in several useful forms as listed below. Expressing the thermodynamic functions on a mass basis we may write ... 

Some other definitions of important thermodynamic functions... 

We could now institute an extensive search of possible thermodynamic functions in the hope of finding a function that is a state variable and also has the property that its rate of internal generation is a positive quantity. Instead, we will just introduce this new thermodynamic property by its definition and then show that the property so defined has the desired characteristics. 

Figure A2.5.4 shows for this two-component system the same thermodynamic functions as in figure A2.5.2, the molar Gibbs free energy G= XjPj + X2P25 the molar enthalpy "w and the molar heat capacity C , again all at constant pressure, but now also at constant composition, x = 1/2. Now the enthalpy is continuous because the vaporization extends over an appreciable temperature range. Moreover, the heat capacity, while discontinuous at the beginning and at the end of the transition, is not a delta function. Indeed the graph appears to satisfy the definition of a second-order transition (or rather two, since there are two discontinuities).

The language of Statistical Mechanics evolved over a considerable period of time. For example, the term "ensemble" is used to denote a statistical population of molecules "partition function" is the integral, over phase-space of a system, of the exponential of -E/kT [where E is the energy of the system, k is Boltzmann s eonstant, and T is the temperature in °K]. From this "function", all of the thermodynamic functions can be derived. The definitions that we shall need are given as follows in 2.5.1. on the next page. 

What does it mean to say that our physical properties must be single-valued It means that our thermodynamic functions (such as equation (3.1)) can deal only with systems at equilibrium where, according to our definition, the properties of the system do not change with time. This equilibrium state may be stable or metastable, but generally speaking for systems more complex than a single component (e.g. a pure mineral) stable equilibrium states are almost always referred to." ... 

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