Fundamental equation, thermodynamics

With j (besides p and T) the constancy of the chemical composition is expressed. From the thermodynamic fundamental equation the following now follows for H and U ... 

The last of Eqs. (51) defines a thermodynamic fundamental equation for G = G(N, P, 7) in the Gibbs energy representation. Note that passing from one ensemble to the other amounts to a Legendre transformation in macroscopic thermodynamics [39]. Vq is just an arbitrary volume used to keep the partition function dimensionless. Its choice is not important, as it just adds an arbitrary constant to the free energy. The NPT partition function can also be factorized into the ideal gas and excess contributions. The configurational integral in this case is ... 

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

Thermodynamic Analyses of Cycles The thermodynamic quahty measure of either a piece of equipment or an entire process is its reversibility. The second law, or more precisely the entropy increase, is an effective guide to this degree of irreversibility. However, to obtain a clearer picture of what these entropy increases mean, it has become convenient to relate such an analysis to the additional work that is required to overcome these irreversibihties. The fundamental equation for such an analysis is... 

One of the fundamental equations of thermodynamics concerns systems at equilibrium and relates the equilibrium constant K to the difference in standard free energy (A6°) between the products and the reactants. 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

In the thermodynamic models (FIAM and BLM), the internalisation flux is assumed to be rate-limiting, and the concentration of carriers or sensitive sites bound by the solute of interest negligible with respect to the total number of carriers (i.e. free carrier concentration constant). The fundamental equations describing the equilibrium models can be summarised as ... 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

Isotherm Subtraction. A second method (7) of determining the net proton coefficient from adsorption data is an adaptation of the thermodynamics of linked functions as applied to the binding of gases to hemoglobin (19). The net proton coefficient determined by this method is designated, Xp- The computational procedure makes a clear distinction between the influence of adsorption density and pH on the magnitude of the net proton coefficient. The fundamental equation used in the calculation of Xp is... 

A fundamental equation combines the first and second laws of thermodynamics and, in this manner, addresses the behavior of matter. For a reversible change in a closed system of constant composition and without nonexpansion work, one can write... 

From a thermodynamic point of view the most important reaction characteristic for practical application is its free enthaply change AG°. According to the fundamental equation AG°=-RTlnK, the equilibrium constant of the reaction is determined by AG°. A high negative value (-20 kJ/mol or even less) usually imphes that the reaction results in high yield and quantitative transformation of substrate to product... 

In any discussion of the thermodynamic stability and rcduction-oxidatiun (redox) properties of ions the fundamental equation used is ... 

This is the fundamental equation for the thermodynamic treatment of polarizable interfaces. It is a relation among interfacial tension y, surface excess 1 -, applied potential V, charge density qM, and solution composition. It shows that interfacial tension varies with the applied potential and with the solution composition. This is in fact the relation that was desired. Its implications will now be analyzed. 

Statistical mechanics is, obviously, a course unto itself in the standard chemistry/physics curriculum, and no attempt will be made here to introduce concepts in a formal and rigorous fashion. Instead, some prior exposure to the field is assumed, or at least to its thermodynamical consequences, and the fundamental equations describing the relationships between key thermodynamic variables are presented without derivation. From a computational-chemistry standpoint, many simplifying assumptions make most of the details fairly easy to follow, so readers who have had minimal experience in this area should not be adversely affected. 

As first shown by J. W. Gibbs, the analytical characterization of thermodynamic equilibrium states can be expressed completely in terms of such first and second derivatives of a certain fundamental equation (as described in Section 5.1). 

However, Gibbs demonstrated the equal importance of a second fundamental equation that reveals a beautiful duality of the thermodynamic formalism the deep symmetry between entropy (5.28) and internal energy U (5.29) ... 

In order to better understand the physical nature of the chemical potential jxt of a chemical substance, let us first review the major mathematical features of the Gibbsian thermodynamics formalism. The starting point is the Gibbs fundamental equation for the internal energy function... 

The functional relationship that expresses how U depends on other extensities Xj is called the fundamental equation of equilibrium thermodynamics ... 

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. 

The starting point for thermodynamic description, whether in the calculus-based or the geometry-based formalism, is the Gibbs fundamental equation for a given equilibrium state S. In the energy representation, this is expressed as... 

In addition, several alternative formulations of thermodynamic geometry have been presented, starting from entropy-based (or other) fundamental equations (see Sections 5.4 and 5.5). From the equilibrium thermodynamics viewpoint, these alternative formulations are completely equivalent, and each could be considered a special case of the general transformations outlined in Section 11.4. Nevertheless, each alternative may suggest distinct statistical-mechanical origins, Riemannian paths, or other connotations that make it preferable for applications outside the equilibrium thermodynamics framework. 

These statements are "laws of experience . That is, no one has been able to find exceptions to them (although many have tried). If one assumes that these two laws are valid, then four fundamental equations, referred to as the Four Fundamental Equations of Gibbs, can be obtained. From these four, more than 50,000,000 equations relating the thermodynamic properties of the system can be derived using relatively simple mathematics. The derivations are rigorous. Thus, if the two laws are true,... 

The fundamental equation for the internal energy U of a thermodynamic system is... 

These two laws can be combined for a system involving only pressure-volume work to obtain dU = TdS — PdV This so-called fundamental equation shows two things (1) thermodynamic properties of a system obey the rules of calculus and (2) the choice of independent variables (in this case S and V) plays a very important role in thermodynamics. The second law can be used to show that when S and V are held constant, the internal energy U of a system must decrease... 

The beauty of the fundamental equation for U (equation 2.2-8) is that it combines all of this information in one equation. Note that the Ns + 2 extensive variables S, V, and n, are independent, and the Ns + 2 intensive variables T, P, and /uj obtained by taking partial derivatives of U are dependent. This is wonderful, but equations of state 2.2-10 to 2.2-12 are not very useful because S is not a convenient independent variable. Fortunately, more useful equations of state will be obtained from other thermodynamic potentials introduced in Section 2.5. 

Fundamental equation 2.2-8 has been presented as the equation resulting from the first and second laws, but thermodynamic treatments can also be based on the entropy as a thermodynamic potential. Equation 2.2-8 can alternatively be written as... 

Taking the differentials of the seven thermodynamic potentials defined above and substituting equation 2.6-1 yields the fundamental equations for these seven additional thermodynamic potentials ... 

In this chapter we will find that when isomers are in chemical equilibrium, it is convenient to treat isomer groups like species in order to reduce the number of terms in the fundamental equation. We will also discuss the effect of ionic strength and temperature on equilibrium constants and thermodynamic properties of species. More introductory material on the thermodynamics of chemical reactions is provided in Silbey and Alberty (2001). 

In treating the fundamental equations of thermodynamics, chemical potentials of species are always used, but in making calculations when T and P are independent variables, chemical potentials are replaced by Gibbs energies of formation AfG . Therefore, we will use equation 3.1-10 in the form... 

When the pH is specified, we enter into a whole new world of thermodynamics because there is a complete set of new thermodynamic properties, called transformed properties, new fundamental equations, new Maxwell equations, new Gibbs-Helmholtz equations, and a new Gibbs-Duhem equation. These new equations are similar to those in chemical thermodynamics, which were discussed in the preceding chapter, but they deal with properties of reactants (sums of species) rather than species. The fundamental equations for transformed thermodynamic potentials include additional terms for hydrogen ions, and perhaps metal ions. The transformed thermodynamic properties of reactants in biochemical reactions are connected with the thermodynamic properties of species in chemical reactions by equations given here. 

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