Extensive property free energy

One of the most important consequences of Euler s integral theorem, as applied to stability criteria and phase separation, is the expansion of the extensive Gibbs free energy of mixing for a multicomponent mixture in terms of partial molar properties. This result is employed to analyze chemical stability of a binary mixture. 

Many extensive properties can be normalized by the size of the system to give specific properties, that is, properties per unit mass, or per mole, or per volume. For example, consider the extensive Gibbs free-energy change (AG xn) involved in a chemical reaction, which can be normalized to an intensive quantity (AGjjjj) via... 

The lubricant properties of alkanethiols and fluorinated alkanes have been studied extensively by scanning probe techniques [163]. In agreement with experiments on LB monolayers it was found that the fluorocarbon monolayers show considerably higher friction than the corresponding hydrocarbon monolayers [164, 165 and 166] even though the fluorocarbons are known to have the lowest surface free energy of all organic materials. 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

The energy of a system can be changed by means of thermal energy or work energy, but a further possibility is to add or subtract moles of various substances to or from the system. The free energy of a pure substance depends upon its chemical nature, its quantity (AG is an extensive property), its state (solid, liquid or gas), and temperature and pressure. Gibbs called the partial molar free heat content (free energy) of the component of a system its chemical potential... 

In addition to the fundamental variables p, V, T, U, and S that we have described so far, three other thermodynamic variables are commonly encountered enthalpy Helmholtz free energy and Gibbs free energy. They are extensive variables that do not represent fundamental properties of the... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

The analogue to one-component thermodynamics applies to the nature of the variables. So Ay S, U and V are all extensive variables, i.e. they depend on the size of the system. The intensive variables are n and T -these are local properties independent of the mass of the material. The relationship between the osmotic pressure and the rate of change of Helmholtz free energy with volume is an important one. The volume of the system, while a useful quantity, is not the usual manner in which colloidal systems are handled. The concentration or volume fraction is usually used ... 

This simple relationship allows us to express all the thermodynamic variables in terms of our colloid concentration. The Helmholtz free energy per unit volume depends upon concentration of the colloidal particles rather than the size of the system so these are useful thermodynamic properties. If we use a bar to symbolise the extensive properties per unit volume we obtain... 

The concept of the similarity of molecules has important ramifications for physical, chemical, and biological systems. Grunwald (7) has recently pointed out the constraints of molecular similarity on linear free energy relations and observed that Their accuracy depends upon the quality of the molecular similarity. The use of quantitative structure-activity relationships (2-6) is based on the assumption that similar molecules have similar properties. Herein we present a general and rigorous definition of molecular structural similarity. Previous research in this field has usually been concerned with sequence comparisons of macromolecules, primarily proteins and nucleic acids (7-9). In addition, there have appeared a number of ad hoc definitions of molecular similarity (10-15), many of which are subsumed in the present work. Difficulties associated with attempting to obtain precise numerical indices for qualitative molecular structural concepts have already been extensively discussed in the literature and will not be reviewed here. 

So far, we have seen several ways of calculating the Gibbs free energy of a two-component mixture. To extend calculations to ternary and higher-order mixtures, we use empirical combinatory extensions of the binary properties. We summarize here only some of the most popular approaches. An extended comparative appraisal of the properties of ternary and higher-order mixtures can be found in Barron (1976), Grover (1977), Hillert (1980), Bertrand et al. (1983), Acree (1984), and Fei et al. (1986). 

Any characteristic of a system is called a property. The essential feature of a property is that it has a unique value when a system is in a particular state. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the size of a system, such as temperature T and pressure p. Extensive properties are those that are dependent on the size of a system, such as volume V, internal energy U, and entropy S. Extensive properties per unit mass are called specific properties such as specific volume v, specific internal energy u, and specific entropy. s. Properties can be either measurable such as temperature T, volume V, pressure p, specific heat at constant pressure process Cp, and specific heat at constant volume process c, or non-measurable such as internal energy U and entropy S. A relatively small number of independent properties suffice to fix all other properties and thus the state of the system. If the system is composed of a single phase, free from magnetic, electrical, chemical, and surface effects, the state is fixed when any two independent intensive properties are fixed. 

The properties of a substance can be classified as either intensive or extensive. Intensive properties, which include density, pressure, temperature, and concentration, do not depend on the amount of the material. Extensive properties, such as volume and weight, do depend on the amount. Most thermodynamic properties are extensive including energy (E), enthalpy (H), entropy (5), and free energy (G). 

The standard free-energy change, A G°, for a reaction is the change in free energy that occurs when reactants in their standard states are converted to products in their standard states. As with AH° (Section 8.10), the value of AG° is an extensive property that refers to the number of moles indicated in the chemical equation. For example, AG° at 25°C for the reaction... 

The reason why E° values are independent of the amount of reaction can be understood by looking at the equation AG° = -nFE°. Free energy is an extensive property (Section 1.4) because it depends on the amount of substance. If we double the amount of Ag+ reduced, the free-energy change, A G°, doubles. The number of electrons transferred, n, also doubles, however, so the ratio E° = —AG°/nF is constant. Electrical potential is therefore an intensive property, which does not depend on the amount of substance. 

The ESPS method draws on and synthesizes a number of ideas in the extensive free-energy literature, including the importance of representations and space transformations between them [63, 68, 69], the utility of expanded ensembles in turning virtual transitions into real ones [23], and the general power of multicanonical methods to seek out macrostates with any desired property [27],... 

The thermodynamic treatment of an interface generally considers a system composed of the interface (y) and two adjacent homogeneous phases (a and / ). The extensive properties of the systems must be ascribed to these three regions, for example, the Gibbs free energy G and the number of moles of a species in the system fulfill... 

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