Thermodynamics intensive variables

We therefore need four equations to calculate these four concentrations as functions of the thermodynamic intensive variables of the system MOi+, T, P and These equations are formulated from the requirements that the system has to meet ... 

If we choose P as the thermodynamic pressure P, then oSf(E) is referred to as the generalized PF. It is clear from (4.5.81) that the integral in this case diverges. The reason is that E(T, L, X) is a function of the single extensive variable L. Transforming L into the thermodynamic intensive variable P gives a partition function which is a function of the intensive variables T, P, X only. However, the Gibbs-Duhem relation states that... 

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... 

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... 

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. 

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. 

For obvious reasons, we need to introduce surface contributions in the thermodynamic framework. Typically, in interface thermodynamics, the area in the system, e.g. the area of an air-water interface, is a state variable that can be adjusted by the observer while keeping the intensive variables (such as the temperature, pressure and chemical potentials) fixed. The unique feature in selfassembling systems is that the observer cannot adjust the area of a membrane in the same way, unless the membrane is put in a frame. Systems that have self-assembly characteristics are conveniently handled in a setting of thermodynamics of small systems, developed by Hill [12], and applied to surfactant self-assembly by Hall and Pethica [13]. In this approach, it is not necessary to make assumptions about the structure of the aggregates in order to define exactly the equilibrium conditions. However, for the present purpose, it is convenient to take the bilayer as an example. 

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 ... 

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. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

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. 

The conditions for eqnilibrinm have not changed, and application of the phase rnle is conducted as in the previous section. The difference now is that composition can be counted as an intensive variable. Composition is accounted for through direct introduction into the thermodynamic quantities of enthalpy and entropy. The free energy of a mixtnre of two pure elements, A and B, is still given by the definition... 

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. 

However, before considering such a complex system of four independent variables, which is represented in planar perspective, let us first take the variables as they can be represented in a sequence of change from inert components which, one by one, become "perfectly mobile" or intensive variables of a thermodynamic system. We will first assume that the phases which will be present in some portion of the system are gibbsite, kaolinite, crystalline or amorphous silica, mica, illite, mixed layered illite-montmorillonite (beidellite), K-feldspar (no pure potassium zeolite is present). Initially we will simplify the mineralogy in the following way ... 

Consider next the energy equation, neglecting kinetic and gravitational-potential energy. Here the extensive variable is the internal energy of the gas E and the intensive variable is the specific internal energy e. The first law of thermodynamics provides the system energy balance... 

Equation (6.35b) shows that the new intensive variable, chemical potential pi, as introduced in this chapter, is actually superfluous for the case c = 1, because its variations can always be expressed in terms of the old variations dT dP. Thus, as stated in Inductive Law 1 (Table 2.1), only two degrees of freedom (independently variable intensive properties) suffice to describe the thermodynamic variability of a simple c = 1 system. This confirms (as expected) that chemical potential pu only becomes an informative thermodynamic variable when chemical change is possible, that is, for c > 2 chemical components. 

A curious feature of the space Ms of thermodynamic variables in an equilibrium state S is that its dimensionality varies with the number of phases, p, even though the values of the intensive variables (which might be used to parametrize the state S) do not. The intensive-type ket vectors R/ of (10.8) can actually be defined for all c + 2 intensities (T, —P, fjL, pi2, , pic) arising from the fundamental equation of a c-component system, U(S, V, n, ri2,. .., nc), even if only /of these remain linearly independent when p phases are present. 

An intrinsic feature of the thermodynamic formalism is the freedom to consider general combinations of extensive or intensive variables [cf. (8.70), (8.75)] as alternatives to standard choices. This freedom is used, for example, in considering the Gibbs free energy G = U — (T)S + (P)V as a linear combination of standard (U, S, V) extensities, or the phase-coexistence coordinate a [cf. (7.27), (7.28)] as a linear combination of standard (T, P) intensities. 

An application of (11.10) was already seen in (11.16), where a specified linear combination of intensive variables was found to be associated with variations of an extensive coordinate. Such linear combinations are also necessary to represent variations along a coexistence curve, or along other paths in a phase diagram that are not parallel to one of the axes. Additional incentives to describe more general variations may arise from purely experimental considerations, where the variables under practical experimental control may involve simultaneous changes of two or more reference variables. It is therefore desirable that general expressions be available to allow easy transformation from one thermodynamic coordinate system to another. 

With the help of (12.89), it is now possible to establish some simple theorems concerning the possibility of stationary points (e.g., maxima, minima, or horizontal inflections) in thermodynamic phase diagrams. In each case, we suppose that Rh Rj are chosen from any set of/+l intensive variables (spanning at least/ — 1 dimensions), and that iy are... 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

Consider a material or system that is not at equilibrium. Its extensive state variables (total entropy number of moles of chemical component, i total magnetization volume etc.) will change consistent with the second law of thermodynamics (i.e., with an increase of entropy of all affected systems). At equilibrium, the values of the intensive variables are specified for instance, if a chemical component is free to move from one part of the material to another and there are no barriers to diffusion, the chemical potential, q., for each chemical component, i, must be uniform throughout the entire material.2 So one way that a material can be out of equilibrium is if there are spatial variations in the chemical potential fii(x,y,z). However, a chemical potential of a component is the amount of reversible work needed to add an infinitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. 

Buffer Capacities of Natural Waters. Natural waters are buffered in different ways and to varying degrees with respect to changes in pH, metal ion concentrations, various ligands, and oxidation-reduction potential. The buffer capacity is an intensive variable and is thermodynamic in nature. Hydrogen-ion buffering in natural waters has recently been discussed in detail by Weber and Stumm (38). Sillen (32) has doubted... 

Finally, the thermodynamic properties of a system considered as variables may be classified as either intensive or extensive variables. The distinction between these two types of variables is best understood in terms of an operation. We consider a system in some fixed state and divide this system into two or more parts without changing any other properties of the system. Those variables whose value remains the same in this operation are called intensive variables. Such variables are the temperature, pressure, concentration variables, and specific and molar quantities. Those variables whose values are changed because of the operation are known as extensive variables. Such variables are the volume and the amount of substance (number of moles) of the components forming the system. 

Two methods may be used, in general, to obtain the thermodynamic relations that yield the values of the excess chemical potentials or the values of the derivative of one intensive variable. One method, which may be called an integral method, is based on the condition that the chemical potential of a component is the same in any phase in which the component is present. The second method, which may be called a differential method, is based on the solution of the set of Gibbs-Duhem equations applicable to the particular system under study. The results obtained by the integral method must yield... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

C) In many experimental studies, all of the intensive variables are determined, giving a redundancy of experimental data. However, Equations (10.70) and (10.73) afford a means of checking the thermodynamic consistency of the data at each experimental point for the separate cases. Thus, for Equation (10.70), the required slope of the curve of P versus ylt consistent with the thermodynamic requirements of the Gibbs-Duhem equations, can be calculated at each experimental point from the measured values of P, xt, and at the experimental temperature. This slope must agree within the experimental error with the slope, at the same composition, of the best curve... 

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