Generalized thermodynamic formalism

To circumvent the above problems with mass action schemes, it is necessary to use a more general thermodynamic formalism based on parameters known as interaction coefficients, also called Donnan coefficients in some contexts (Record et al, 1998). This approach is completely general it requires no assumptions about the types of interactions the ions may make with the RNA or the kinds of environments the ions may occupy. Although interaction parameters are a fundamental concept in thermodynamics and have been widely applied to biophysical problems, the literature on this topic can be difficult to access for anyone not already familiar with the formalism, and the application of interaction coefficients to the mixed monovalent-divalent cation solutions commonly used for RNA studies has received only limited attention (Grilley et al, 2006 Misra and Draper, 1999). For these reasons, the following theory section sets out the main concepts of the preferential interaction formalism in some detail, and outlines derivations of formulas relevant to monovalent ion-RNA interactions. Section 3 presents example analyses of experimental data, and extends the preferential interaction formalism to solutions of mixed salts (i.e., KC1 and MgCl2). The section includes discussions of potential sources of error and practical considerations in data analysis for experiments with both mono- and divalent ions. 

A general thermodynamic formalism for steric stabilization has been presented by Everett (1978c). This is an elaboration of that set forth by Ash et al. (1973), which will be discussed in some detail in Section 17.5 in connection with the stabilization by polymers free in solution. As Everett (1978c) has pointed out, the procedure propounded by Ash et al. (1973) is inapplicable to steric stabilization because equilibrium is not maintained between adsorbed polymer and polymer in the bulk solution. An alternative approach must therefore be devised. 

Interpenetration and flocculation A general thermodynamic formalism for steric stabilization... 

The fundamental relationship may now be used to rigorously derive an expression for the chemical potential of species Bj in the presence of field-dipole interactions. Consistent with general thermodynamic formalism we obtain from Eq. (4.13)... 

A final observation is in order the quantitative application of the equilibrium thermodynamical formalism to living systems and especially to ecosystems is generally inadequate since they are complex in their organisation, involving many interactions and feedback loops, several hierarchical levels may have to be considered, and the sources and types of energy involved can be multiple. Furthermore, they are out-of-equilibrium open flow systems and need to be maintained in such condition since equilibrium is death. Leaving aside very simple cases, in the present state of the art we are, therefore, limited to general semiquantitative statements or descriptions (e.g. ecosystem narratives ). 

The path thermodynamics formalism allows us to extract some general conclusions on the relation between and W p. Let us consider the CFT... 

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. 

This derivation confirms that Ic and are thermodynamic, rather than mechanical quantities. We shall generalize the thermodynamic formalism in sec. 4.7. 

In the previous section, we derived a general and formal relationship between thermodynamics of solvation and structural changes induced in the water. Now, we present an approximate relationship between the structure of water,... 

We are concerned here with the thermodynamic stability of carbocations. For the sake of generality, we formally consider that they derive from a neutral molecule R-H, which can either lose a hydride or be protonated in the gas phase, reactions (1) and (2) ... 

The general thermodynamics of polymerization of cyclic acetals and the influence of substitution are discussed in Chapt. 2 of this volume (Thermodynamics). It may suffice to state here, that the monomers used to date for polymer synthesis are mostly derivatives of 1,2-glycols or 1,4-glycols and formaldehyde (i.e. 5- and 7-membered formals). 6-membered formals (1,3-dioxane and its derivatives) are nonpolymeri-zable due to the thermodynamic restrictions. 

This simplified equilibration process holds strictly for a cubic crystal in a general non-cubic case, the volume, static pressure and bulk modulus have to be replaced by the strain components (related to unit-cell parameters), stress components and elastic constants, respectively. The computer program PARAPOCS[17] performs the lattice-dynamical, thermodynamical and quasi-harmonic calculations in the general tensorial formalism, and has been used to obtain all results reported below for calcite and aragonite. 

The formalism and general thermodynamic relations that we have seen in the previous chapters have a wide applicability. In this chapter we will see how thermodynamic quantities can be calculated for gases, liquids and solids. We will also study some basic features of equilibrium between different phases. 

Details of Fermi level dependence of the diffusion coefficient, summarized in (77) can be obtained in the framework of a more general kinetic-thermodynamic formalism [13, 49, 154, 159]. 

The principal thermodynamic properties of interfaces have been widely discussed. However, one property that is pertinent to adhesion has not been treated adequately, up to the present. This is the heat capacity per unit area of a layer of small but arbitrary thickness. (We do not refer, here, to the Gibbsian surface excess heat capacity of an interface between two condensed phases, which is, in general, small.) Formally, we define this property as follows Ca(A) is the heat capacity per unit interfacial area, for a... 

The two-parameter formalism makes no stipulations concerning the temperature dependence of the radius of gyration and of the second virial coefficient, except that both quantities must reflect any temperature dependence of the excluded volume integral P through the variable z. Without an explicit theory of the behavior of j , we can only assert general thermodynamic requirements. We recall that the osmotic pressure is given by —(RT/Vi) In, with the activity of the solvent and Fj its molar volume. Then using equation (63), we can write the chemical potential of the solvent as a... 

It is the purpose of this paper to establish a statistical thermodynamical formalism for dilute polyelectrolyte solutions which may serve to discuss the assumptions which are usually introduced in the theoretical treatment of such systems. Use is made of a model which is kept as general as possible, and certainly is more realistic than a Kuhn-like chain, without being too complicated to be handled by ordinary statistical procedures. Canonical ensemble statistics are used, the external thermodynamic variables being the volume V, temperature T and composition of the system. Of course, for the problem thus outlined no exact solution of practical nature is presented which is in the present stage still beyond reach. It is hoped that such a formal treatment may help a better understanding of the problems underlying the theoretical approach to polyelectrolyte systems. It will also help to discuss the generality of theoretical description as presented by Marcus [7]. 

Ottinger FI C 1997 General pro]ection operator formalism for the dynamics and thermodynamics of complex fluids Phys. Rev. E 57 1416... 

The thermodynamic transition between different forms as the above described is formally discontinuous. The difference between polymorphs is shown in general also by a different metrical description of the corresponding lattices. 

In a formal sense, Equation (2.38) applies to all batch reactor problems. So does Equation (2.42) combined with Equation (2.40). These equations are perfectly general when the reactor volume is well mixed and the various components are quickly charged. They do not require the assumption of constant reactor volume. If the volume does vary, ancillary, algebraic equations are needed as discussed in Section 2.6.1. The usual case is a thermodynamically imposed volume change. Then, an equation of state is needed to calculate the density. 

These two seemingly distinct approaches of thermodynamic integration and perturbation can be seen as the limiting cases of a more general formalism in which the transformation between the two states proceeds at a finite rate. Seen in this light, one might also hope to obtain free energies from a transformation that converts the initial to the final state neither infinitely slowly (as in thermodynamic... 

Several forms are imaginable for the [Ni°(butadiene)2L] and [Ni°(butadiene)J active catalysts, depending on the monodentate (p2) or the bidentate (p4) coordination mode of butadiene from either its s-cis or its s-trans configuration. The two butadienes can be coordinated in bis(p2), p4, p2, and bis(p4) modes for the PR3/P(OR)3-stabilized catalyst complex, giving rise to formal 16e, 18e, and 20e species. On the other hand, bis(p4)- and p4,p2-butadiene species and also tris(p2)- and p4,bis(p2)-butadiene compounds are possible species for the [Ni°(butadiene)2] and [Ni°(butadiene)3] forms for the [Ni°(butadiene)J active catalyst. In general, for butadiene to coordinate in a bidentate fashion, the p4-cis mode is thermodynamically favorable relative to the p4-trans mode, while the p2-trans mode prevails for monodentate coordination. 

The formalism shown above is in general easily extended to multi-component systems. All thermodynamic mixing properties may be derived from the integral Gibbs energy of mixing, which in general is expressed as... 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

The first chair of theoretical physics in France was the professorship established for Pierre Duhem in 1894 at the Bordeaux Faculty of Sciences. 1 Duhem was well known in French scientific circles not only as a physicist but as a physicist of exceptional mathematical skills who addressed himself early in his scientific studies to chemical problems. He wrote a controversial doctoral thesis (1886) in which he developed the concept of thermodynamic potential for chemistry and physics, and he later developed a treatment of equilibrium processes formally analogous to the mechanics of Lagrange. The goal was to make mechanics a branch of the more general science of thermodynamics, a science that embraces "every change of qualities, properties, physical state, chemical constitution. "2... 

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