Extensive thermodynamic function

Although equation 35 is a simple expression, it tends to be confusing. In this equation the enthalpy difference appears as driving force in a mass-transfer expression. Enthalpy is not a potential, but rather an extensive thermodynamic function. In equation 35, it is used as enthalpy pet mole and is a kind of shorthand for a combination of temperature and mass concentration terms. 

The most important new concept to come from thermodynamics is entropy. Like volume, internal energy and mole number it is an extensive property of a system and together with these, and other variables it defines an elegant self-consistent theory. However, there is one important difference entropy is the only one of the extensive thermodynamic functions that has no obvious physical interpretation. It is only through statistical integration of the mechanical behaviour of microsystems that a property of the average macrosystem, that resembles the entropy function, emerges. 

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... 

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

Indeed, since the macroscopic states of a protein are discrete, they are described by discrete surfaces in the phase space of considered variables (Pfeil and Privalov, 1976c). The small globular proteins, or individual cooperative domains, which have only two stable macroscopic states, the native (N) and denatured (D), are described by two surfaces in the phase space, corresponding to their extensive thermodynamic functions. The transition between these states is determined by the differences of... 

In a multicomponent system, the partial molar quantities for a component "i" in a phase can be defined for any extensive thermodynamic function Z (enthalpy, energy, entropy, etc.). The partial molar quantity Z is the change in Z for a change in n, or... 

In thermodynamics, extensive thermodynamic functions of interest to us are homogeneous functions of degree 1. The arbitrary multiplier A will be equal to the mass or moles of the system N (or N). For applying Euler s theorem in thermodynamics, consider the internal eneigy U = U S, V,N). Internal energy U is first order (m = 1) in mass and S,V,N are all proportional to mass, then we have... 

An alternative convention for the thermodynamic treatment of adsorption phenomena was proposed by Verschaffeldt [34] and Guggenheim [35]. In their formulation an interface is treated as a separate surface phase located between two adjacent bulk phases and which has a finite thickness and volume. This phase can be essentially described thermodynamically in a way analogous to bulk phases. By the use of this approach, the adsorption values and all extensive thermodynamic functions are total - not excess quantities as in the Gibbs method. 

The Gibbs convention presented in the previous section can be applied to extensive thermodynamic functions. So, the total internal energy of a system composed of... 

Each term on the right-hand side of the fundamental equation (2.11) has the form x dV where Y, like U, is an extensive thermodynamic function and X is a field. At equilibrium the fields T and p. are the same in each bulk phase, as also is the pressure if the surface boundaries are plane (T = T, p p = p ) and o- is uniform over the whole of the -surface. We need not ask what are the values of p, T, and p at the interface indeed the question has no meaning within the description of the system now being used. We return to this question, however, when we discuss the point-thermodynamic description in 2.S. 

In fluid mechanics it might be natural to employ mass based thermodynamic properties whereas the classical thermodynamics convention is to use mole based variables. It follows that the extensive thermodynamic functions (e.g., internal energy, Gibbs free energy, Helmholtz energy, enthalpy, entropy, and specific volume) can be expressed in both ways, either in terms of mass or mole. The two forms of the Gibbs-Duhem equation are ... 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

The extensive thermodynamic variables are homogeneous functions of degree one in the number of moles, and Euler s theorem can be used to relate the composition derivatives of these variables. 

The process we have followed Is Identical with the one we used previously for the uranium/oxygen (U/0) system (1-2) and Is summarized by the procedure that Is shown In Figure 1. Thermodynamic functions for the gas-phase molecules were obtained previously (3) from experimental spectroscopic data and estimates of molecular parameters. The functions for the condensed phase have been calculated from an assessment of the available data, Including the heat capacity as a function of temperature (4). The oxygen potential Is found from extension Into the liquid phase of a model that was derived for the solid phase. Thus, we have all the Information needed to apply the procedure outlined In Figure 1. 

For a closed system of fixed composition, the extensive thermodynamic properties such as y, U, S, A, Y, and G are functions of any pair of convenient independent variables. For example. Equation (7.38) suggests that G is a natural function of T and P. That is G =f(T, P). The total differential of G would be... 

Extensive thermodynamic properties at constant temperature and pressure are homogeneous functions of degree 1 of the mole numbers. From Euler s theorem [Equation (2.33)] for a homogeneous function of degree n... 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

A lot of thermodynamics makes use of the important concept of state function, which is a property with a value that depends only on the current state of the system and is independent of the manner in which the state was prepared. For example, a beaker containing 100 g of water at 25°C has the same temperature as 100 g of water that has been heated to 100°C and then allowed to cool to 25°C. Internal energy is also a state function so the internal energy of the beaker of water at 25°C is the same no matter what its history of preparation. State functions may be either intensive or extensive temperature is an intensive state function internal energy is an extensive state function. 

The internal energy is homogeneous of degree 1 in terms of extensive thermodynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All extensive variables are homogeneous functions of the first degree of other extensive properties. All intensive properties are homogeneous functions of the zeroth degree of the extensive properties. 

Sixth, the extensive thermodynamic and kinetic studies on dimer-tetramer assembly using various hybrid and mutant hemoglobins carried out by Ackers and co-workers clearly indicate that there are at least three molecular functional states for Hb A during the transition from the deoxy to the oxy state (Smith and Ackers, 1985 Ackers and Smith, 1987 Smith et al., 1987 Daugherty et al., 1991 Ackers et al., 1992). These results are not consistent with a two-structure allosteric description for the oxygenation of Hb A. 

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

Upper values data from most extensive calculations lower values best experimental data available, thermodynamic functions calculated from partition functions by means of the Sackur-Tetrode equation... 

These facts mean that extensive thermodynamic properties are not truly homogeneous (11) in the sense in which that word is used in mathematics and thermodynamics. Experiments with relatively large particles show that thermodynamic functions are homogeneous within the limits of precision ordinarily attained in the laboratory. The fact is simply that we usually do not detect deviations from homogeneity. It is not true that such differences do not exist. Most texts and treatises dismiss the problem so quickly that students are likely to conclude that even the mention of the word homogeneity is somewhat pedantic. 

In this Section the internal energy function has been introduced in the form E - E(T,V), whereas in Section 1.18 it has been formulated as E - E(S,V). Considering that thermodynamic functions of state should be useful in deriving various intensive and extensive variables, are the two formulations equivalent If not, which one is more fundamental In a similar vein discuss the relation between H... 

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