Thermodynamic description of equilibrium

To this point we have used a number of terms familiar to geochemists without giving the terms rigorous definitions. We have, for example, discussed thermodynamic components without considering their meaning in a strict sense. Now, as we begin to develop an equilibrium model, we will be more careful in our use of terminology. We will not, however, develop the basic equations of chemical thermodynamics, which are broadly known and clearly derived in a number of texts (as mentioned in Chapter 2). 

The ancient categories of water, earth, and air persist in classifying the phases that make up geochemical systems. For purposes of constructing a geochemical model, we assume that our system will always contain a fluid phase composed of water and its dissolved constituents, and that it may include the phases of one or more minerals and be in contact with a gas phase. If the fluid phase occurs alone, the system is homogeneous the system when composed of more than one phase is heterogeneous. 

Species are the molecular entities, such as the gases CO2 and O2 in a gas, or the electrolytes Na+ and SO4 in an aqueous solution, that exist within a phase. Species, unlike phases, do not have clearly identifiable boundaries. In addition, species may exist only for the most fleeting of moments. Arriving at a precise definition of what a species is, therefore, can be less than straightforward. 

The overall composition of any system can be described in terms of a set of one or more chemical components. We can think of components as the ingredients in a recipe. A certain number of moles of each component go into making up a system, just as the amount of each ingredient is specified in a recipe. By combining the components, each in its specified mass, we reproduce the system s bulk composition. 

Whereas species and phases exist as real entities that can be observed in nature, components are simply mathematical tools for describing composition. Expressed another way, a component s stoichiometry but not identity matters water, ice, and steam serve equally well as component H2O. Since a component needs no identity, it may be either Active or a species or phase that actually exists in the system. When we express the composition of a fluid in terms of elements or the composition of a rock in terms of oxides, we do not imply that elemental sodium occurs in the fluid, or that calcium oxide is found in the rock. These are Active components. If we want, we can invent components that exist nowhere in nature. 

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Recall that grain boundaries are sinks in which impurities or solute might become concentrated. The thermodynamic description of equilibrium segregation involves redistribution of alloyed solute such that the total free energy of the system is minimized. For a two-component system such as a binary alloy, if one considers a dilute solution of component B dissolved as solute in solvent A, the surface excess of B at surfaces per unit area, Fb, is given by [7, 9]... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

First, the simple thermodynamic description of pe (or Eh) and pH are both most directly applicable to the liquid aqueous phase. Redox reactions can and do occur in the gas phase, but the rates of such processes are described by chemical kinetics and not by equilibrium concepts of thermodynamics. For example, the acid-base reaction... 

Unsually short NMR T, relaxation values were observed for the metal-bonded H-ligands in HCo(dppe)2, [Co(H2)(dppe)]+ (dppe = l,2-bis(diphenylphosphino)ethane), and CoH(CO) (PPh3)3.176 A theoretical analysis incorporating proton-meta) dipole-dipole interactions was able to reproduce these 7) values if an rCo H distance of 1.5 A was present, a value consistent with X-ray crystallographic experiments. A detailed structural and thermodynamic study of the complexes [H2Co(dppe)2]+, HCo(dppe)2, [HCo(dppe)2(MeCN)]+, and [Co(dppe)2(MeCN)]2+ has been reported.177 Equilibrium and electrochemical measurements enabled a thorough thermodynamic description of the system. Disproportionation of divalent [HCo(dppe)2]+ to [Co(dppe)2]+ and [H2Co(dppe)2]+ was examined as well as the reaction of [Co(dppe)2]+ with H2. 

Statistical thermodynamic descriptions of these transitions in substitutional alloys have been developed for the cases of both binary and ternary alloys , using a simple nearest neighbor bond model of the surface segregation phenomenon (including strain energy effects). Results of the model have been evaluated here using model parameters appropriate for a Pb-5at%Bi-0.04at%Ni alloy for which experimental results will be provided below. However, the model can be applied in principle to the computation of equilibrium surface composition of any ternary solution. 

It is indeed somewhat surprising that the quantity of each phase is in some sense irrelevant to thermodynamic description of the phase-transition phenomenon. Consider, for example, a 1 kg sample of pure water in equilibrium with its own vapor at, say, the normal boiling point (T = 100°C, P = 1 atm), initially with rcvap moles of vapor and nnq moles of liquid, as shown at the left ... 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

Equilibrium and nonequilibrium thermodynamics can be combined to give a complete thermodynamic description of a process or process step. Equilibrium thermodynamics allows us to calculate the changes in thermodynamic properties with the change in process conditions. Nonequilibrium thermodynamics allows us to calculate unambiguously the work that is lost associated with the process taking place. It relates this loss to the process s flows and forces driving these flows and identifies the various friction factors as a function of the relationship between flows and forces. 

The intensive variables T, P, and nt can be considered to be functions of S, V, and dj because U is a function of S, V, and ,. If U for a system can be determined experimentally as a function of S, V, and ,, then T, P, and /q can be calculated by taking the first partial derivatives of U. Equations 2.2-10 to 2.2-12 are referred to as equations of state because they give relations between state properties at equilibrium. In Section 2.4 we will see that these Ns + 2 equations of state are not independent of each other, but any Ns+ 1 of them provide a complete thermodynamic description of the system. In other words, if Ns + 1 equations of state are determined for a system, the remaining equation of state can be calculated from the Ns + 1 known equations of state. In the preceding section we concluded that the intensive state of a one-phase system can be described by specifying Ns + 1 intensive variables. Now we see that the determination of Ns + 1 equations of state can be used to calculate these Ns + 1 intensive properties. 

With the establishment of conventions for the Standard State and for the reference zero value of the chemical potential, it is possible to develop fully the thermodynamic description of chemical reactions. This development relies on the concept of thermodynamic activity, introduced in Section 1.2, and on the condition for chemical equilibrium in a reaction 1,15... 

We have seen that, for each equilibrium state, the adsorptive has the same chemical potential in the gas phase and in the adsorbed phase, and that, for a given system, only two variables are required to provide a thermodynamic description of each of these phases. Consequently, there is a relation between the two intensive variables 77 andp. 

Based on the above calculations, both the free energy function (EEE) and the standard enthalpy of formation AH° (298) of NiCp were obtained, the first from its structure and normal mode of vibration by applying the harmonic-oscillator, rigid-rotator approximation, and the second from a Born-Haber thermodynamic cycle.The thermodynamic description of NiCp, together with that of the other participating compounds, permitted a second series of partial equilibrium calculations of the... 

Here we discuss the thermodynamic description of electromagnetic radiation in equilibrium with the walls of an evacuated cavity that contains a tiny porthole for experimental observations. 

For each of these idealized models there is a stationary state. For a continuous open system, this is the steady state. Rate laws and steady material flows arc required to define the steady state. For a closed system, equilibrium is the stationary state. Equilibrium may be viewed as simply the limiting case of the stationary state when the flows from the surroundings approach zero. The simplicity of closed-system models at equilibrium is in the rather small body of information required to describe the time-invariant composition. We now turn our attention to the principles of chemical thermodynamics and the development of tools for the description of equilibrium states and energetics of chemical change in closed systems. 

All these questions are answered by the thermodynamic description of the equilibrium constant, provided in the next section. Readers who have already studied thermodynamics should continue to Section 14.3. 

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