Variables extensive

Generally speaking, intermolecular forces act over a short range. Were this not the case, the specific energy of a portion of matter would depend on its size quantities such as molar enthalpies of formation would be extensive variables On the other hand, the cumulative effects of these forces between macroscopic bodies extend over a rather long range and the discussion of such situations constitutes the chief subject of this chapter. 

The individual reactions need not be unimolecular. It can be shown that the relaxation kinetics after small perturbations of the equilibrium can always be reduced to the fomi of (A3.4.138t in temis of extension variables from equilibrium, even if the underlying reaction system is not of first order [51, fil, fiL, 58]. 

Thus the extensive variables characterizing the lamellar system are entropy [Pg.6]

Before describing these thermodynamic variables, we must talk about their properties. The variables are classified as intensive or extensive. Extensive variables depend upon the amount while intensive variables do not. Density is an example of an intensive variable. The density of an ice crystal in an iceberg is the same as the density of the entire iceberg. Volume, on the other hand, is an extensive variable. The volume of the ocean is very different from the volume of a drop of sea water. When we talk about an extensive thermodynamic variable Z we must be careful to specify the amount. This is usually done in terms of the molar property Zm, defined as... 

As with other extensive variables, we will usually work with the molar function Um defined as... 

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 second observation is that the equations we have derived for extensive variables Z can be applied to the difference AZ. For example ... 

The substitutions can be made because the extensive thermodynamic variables in the equations are homogeneous of degree one.d Thus, dividing the equation by n converts the extensive variable to the corresponding molar intensive variable. For example, to prove that equation (3.48) follows from equation... 

The relationships summarized in Table 3.1. expanded to include differences and molar properties, serve as the starting point for many useful thermodynamic calculations. An example is the calculation of AZ for a variety of processes in which p, V, and T are changed.e For any of the extensive variables Z = S, U, H, A or G, we can write... 

First, we note that all of the thermodynamic equations that we have derived for the total extensive variables apply to the partial molar properties. Thus, if... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

The extensive variable Q associated with the electrical potential + in Eqs. (15), (17), and (21) is the thermodynamic surface excess charge density, which is defined by... 

This equation is valid for all species Ah a fact that is a consequence of the law of definite proportions. The molar extent of reaction is a time-dependent extensive variable that is measured in moles. It is a useful measure of the progress of the reaction because it is not tied to any particular species A. Changes in the mole numbers of two species j and k can be related to one another by eliminating between two expressions that may be derived from equation 1.1.4. 

Another advantage of using the concept of extent is that it permits one to specify uniquely the rate of a given reaction. This point is discussed in Section 3.0. The major drawback of the concept is that the extent is an extensive variable and consequently is proportional to the mass of the system being investigated. 

The specific heat is the amount of heat required to change one mole of a substance by one degree in temperature. Therefore, unlike the extensive variable heat capacity, which depends on the quantity of material, specific heat is an intensive variable and has units of energy per number of moles (n) per degree. 

The magnitude of the entropy flux is the entropy transported through unit area per unit time, which is the divergence V Js- It is convenient to define all extensive variables per unit volume (denoted here by lower case symbols)... 

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

In thermodynamics the state of a system is specified in terms of macroscopic state variables such as volume, V, temperature, T, pressure,/ , and the number of moles of the chemical constituents i, tij. The laws of thermodynamics are founded on the concepts of internal energy (U), and entropy (S), which are functions of the state variables. Thermodynamic variables are categorized as intensive or extensive. Variables that are proportional to the size of the system (e.g. volume and internal energy) are called extensive variables, whereas variables that specify a property that is independent of the size of the system (e.g. temperature and pressure) are called intensive variables. 

Type of work Intensive variable Extensive variable Differential work in dU... 

In general, dw is written in the form (intensive variable)-d(extensive variable) or as a product of a force times a displacement of some kind. Several types of work terms may be involved in a single thermodynamic system, and electrical, mechanical, magnetic and gravitational fields are of special importance in certain applications of materials. A number of types of work that may be involved in a thermodynamic system are summed up in Table 1.1. The last column gives the form of work in the equation for the internal energy. 

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 ... 

Details77 have appeared of the synthesis of fiuorophosphoranes containing four-membered rings,78 and extensive variable-temperature n.m.r. studies have been described. Thus, for the phosphorane (82), the most stable conformations are (83) and (84), which inconvert to a minor conformer (85), via the intermediate (86).77... 

In the study of thermodynamics we can distinguish between variables that are independent of the quantity of matter in a system, the intensive variables, and variables that depend on the quantity of matter. Of the latter group, those variables whose values are directly proportional to the quantity of matter are of particular interest and are simple to deal with mathematically. They are called extensive variables. Volume and heat capacity are typical examples of extensive variables, whereas temperature, pressure, viscosity, concentration, and molar heat capacity are examples of intensive variables. 

A normalization parameter used in treating ligand binding equilibria to convert two extensive variables, dissociation sud substrate concentration, into a parameter whose value is related to the fractional saturation of ligand binding sites. For the simple Michaelis-Menten treatment, v = + i m/[S], if R is the reduced... 

Various properties into Intensive and Extensive variables, listed ... 

Extensive Variable Heat capacity. Enthalpy, Entropy, Gibbs free energy. Volume, Mass, No. of moles. 

The advantage of the chemical potential over the other thermodynamic quantities, U, H, and G, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such as p., T, and P are useful in defining equilibrium. 

Although gibbsite and kaolinite are important in quantity in some soils and hydrothermal deposits, they have diminishing importance in argillaceous sediments and sedimentary rocks because of their peripheral chemical position. They form the limits of any chemical framework of a clay mineral assemblage and thus rarely become functionally involved in critical clay mineral reactions. This is especially true of systems where most chemical components are inert or extensive variables of the system. More important or characteristic relations will be observed in minerals with more chemical variability which respond readily to minor changes in the thermodynamic parameters of the system in which they are found. However, as the number of chemical components which are intensive variables (perfectly mobile components) increases the aluminous phases become more important because alumina is poorly soluble in aqueous solution, and becomes the inert component and the only extensive variable. 

Figure 36. Representation of the zeolite-clay mineral assemblages found in a systeirf at 25°C and atmospheric pressure where Na is an intensive variable (perfectly mobile component) whereas A1 and Si are extensive variables or inert components of the system. G = gibbsite kaol = kaolinite Mo = montmorillonite Si = amorphous silica Anal = analcite.

Na and Ca to play equivalent roles in zeolites, as well as K, and if we consider A1 and Si as the major variables combined with K and Na in the phyllosilicates, we can adequately represent the phases in a (Ca-Na)-K-Al-Si system where H O is in excess in the fluid phase. If the system has four chemical variables and the natural assemblages are frequently found to contain four authigenic minerals, we must assume that most chemical variables are inert or extensive variables of the chemical system which controlled the crystallization of the zeolite-clay mineral containing sediments. ... 

Figure 40. Representation of Mg-Si-i O system in terms of the activity of Mg, H+ and aqueous silica (25°C and atmospheric pressure). Solid line boundaries are taken from Wollast, et al., (1968), dashed lines are deduced boundaries based upon the data of Siffert (1962). The appearance of sepiolite is found only above pH 8 and thus the log Mg +/H+ ratio is not valid for all Mg +-H+ values. There are no specified extensive variables or inert components in the system described. Br = brucite M03 = trioctahedral montmorillonites Sep = sepiolite T = talc.

Given the chemical data for natural mineral compositions, it should be possible to construct a phase diagram including those phases that are likely to occur with sepiolite and palygorskite in a system where the mass of the components is an extensive variable of the system (a closed system). 

Figure 41. Phase diagram for the extensive variables R -R -Si combining the data for synthetic magnesian chlorites and the compositional series of natural sepiolites and palygorskites. Numbers represent the major three-phase assemblages related to sepiolite-palygorskite occurrence in sediments. Chi = chlorite M03 = trioctahedral montmorillonites M02 = dioctahedral montmorillonite Sep = sepiolite Pa = palygorskite Kaol = kaolinite T = talc.

The first representation (Figure 46a) shows a portion of Al-Si-K space where H O is present in great abundance and where pH is determined by the phases present. The mineral assemblages are determined by the relative proportions of Al, Si and K, which are extensive variables. No intensive variables are considered. The maximum number of phases which can be present at a given composition is three three phases will be present in a phase-field. 

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