Thermodynamic treatment

For the thermodynamic characterization of enzyme stability, the denaturation process is also considered a one-step, reversible transition between the native and denatured states  

For the thermodynamic characterization of enzyme stability, the most critical step is the determination of the equilibrium constant of denat-uration. The equilibrium constant can be calculated from knowledge of the relative proportions of native and denatured enzymes at a particular temperature. The equilibrium constant can thus be calculated as 

Obviously, the larger the equilibrium constant of denaturation at a particular temperature, the less stable the enzyme. The enthalpy, entropy, and 

The standard-state enthalpy of denaturation (A// ) can be calculated from the slope of the natural logarithm of the equiUbrium constant versus inverse temperature plot (Fig. 12.96) using the van t Hoff equation  

The standard-state entropy of denaturation can also be determined easily by realizing that at the transition midpoint temperature (T ), where /d = /n, A d = 1, and thus InKo =0, AG is equal to zero (Fig. 12.8)  

Phase equilibrium is the most general thermodynamic approach to evaluate the phase state of a complex system. Polymer-plasticizer systems are described by the principles of phase equihbrium for reversible systems. 

Let s consider the system at temperature T where the volume fraction of polymer is ( )2A- This point lies in the phase separation field. If the separation of this system into two phases is complete, then the volume fractions of polymer in each of the coexisting phases are ( ) 2 and ( ) 2. If phase separation is incomplete, the volume fractions of polymer in each of the coexisting phases are and (these phases appear when a system is in a metastable condition zone). One phase is enriched with plasticizer and the other phase is enriched with polymer. 

When the temperature of the system increases, incompatibility decreases and the system becomes a single-phase system at temperature Tao- The temperature of this transition may vary depending on the system composition and it reaches its maximum at some point Tj ,. Above Tjj, called the upper critical solution temperature, UCST, both components are compatible in any proportion. The diagrams of polymer-plasticizer blends differ from dia- 

The reason for ne tive tem-peramre dependence and LCST is explained by free volume theories developed by Prigogine and Patterson.The monomeric liquid expands on heating more than does the polymer. On mixing there is a decrease in volume that causes a decrease in entropy, which may explain incompatibility at the higher temperatures. 

Phase diagrams for crystallizing polymer-plasticizer systems are complicated by the influence of the crystallization processes. Steiic restrictions of polymer chains limit the completeness of the crystallization process. The rates of ctystallization can significantly change depending on pressure, temperature, and the presence of impurities. 

The aim of the thermodynamic treatment is to relate the elastic force opposing the deformation of the elastomer to changes in energy and entropy occurring during the process. 

Let us consider an elastic solid of initial length Iq under a uniaxial tensile force / that causes an infinitesimal deformation dl. The work done on the solid is 

Equation (3.1) has been simplified by omitting the contribution to the work from the change in volume, dV, accompanying the extension dl. This simplification is justified because the work of expansion —PdV (P = atmospheric pressure) is three or four orders of magnitude smaller than the term / dl being considered. Equation (3.1) would be totally exact if the process were carried out under constant-volume conditions. 

For a reversible change at constant volume, the work done is equal to the change in the Helmholtz function, F  

With the aim of evaluating the change in internal energy accompanying the deformation process, the entropy term in Eq. (3.3) needs to be stated as a function of properties that can be determined experimentally. For this, we will make a simple thermodynamic deduction. According to the first law of thermodynamics, 

The analysis of the experimental results begins with the combined first and second laws of thermodynamics, applicable to reversible processes. As shown in the Appendix 2, it reads 

Recent theory described in Section E of Chapter 5 suggests that true equilibrium will be difficult to achieve by stress relaxation of an elastomer, but provides a relationship for extrapolation to equilibrium. 

In equation (6-1) the increment of work, dW, refers to all of the work (i.e., electrical, mechanical, pressure-volume, chemical, etc.) performed by the system (the sample) on its surroundings. The development of thermodynamics given in most physical chemistry texts is confined to gases where d W becomes simply pressure-volume work, PdV, where P is the external environment. In the case of an elastomer deformed by an amount dL in tension and exerting a restoring force f the mechanical work performed on the system to accomplish the deformation, namely fdL, must also be included in dW. Thus, for an elastomer strained uni axially in tension, 

Because the experiments discussed were performed under constant pressure conditions, it is appropriate to use the thermodynamic entity H, the enthalpy. H is defined as 

the restoring force exerted by the elastomer when it has undergone a deformation by the amount dL at constant temperature and pressure is  

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It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

A. Further Development of the Thermodynamic Treatment of the Surface Region... 

We now come to a very important topic, namely, the thermodynamic treatment of the variation of surface tension with composition. The treatment is due to Gibbs [35] (see Ref. 49 for an historical sketch) but has been amplified in a more conveniently readable way by Guggenheim and Adam [105]. 

While a thermodynamic treatment can be developed entirely in terms of f(P,T), to apply adsorption models, it is highly desirable to know on a per square centimeter basis rather than a per gram basis or, alternatively, to know B, the fraction of surface covered. In both the physical chemistry and the applied chemistry of the solid-gas interface, the specific surface area is thus of extreme importance. 

However, a body of thermodynamic treatment has been developed on the basis that the adsorbent is inert and with attention focused entirely on the adsorbate. The abbreviated presentation given here is based on that of Hill (see Refs. 65 and 113) and of Everett [114]. First, we have the defining relationships ... 

Thermodynamic treatments may, of course, be extended to multicomponent systems. See Ref 117 for an example. 

The thermodynamic treatment that was developed for physical adsorption applies, of course, to chemisorption, and the reader is therefore referred to Sec-... 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

Other conventions for treating equiUbrium exist and, in fact, a rigorous thermodynamic treatment differs in important ways. Eor reactions in the gas phase, partial pressures of components are related to molar concentrations, and an equilibrium constant i, expressed directiy in terms of pressures, is convenient. If the ideal gas law appHes, the partial pressure is related to the molar concentration by a factor of RT, the gas constant times temperature, raised to the power of the reaction coefficients. 

In the thermodynamic treatment of electrode potentials, the assumption was made that the reactions were reversible, which implies that the reactions occur infinitely slowly. This is never the case in practice. When a battery deUvers current, the electrode reactions depart from reversible behavior and the battery voltage decreases from its open circuit or equiUbrium voltage E. Thus the voltage during battery use or discharge E is lower than the voltage measured under open circuit or reversible conditions E by a quantity called the polari2ation Tj. 

Other cell variables such as sound speed and heat capacities can be calculated using similar techniques. Some codes allow a variety of multimaterial element thermodynamic treatments. For example, CTH allows all materials in an element to have the same or different pressures or temperatures [44], Material interfaces in multimaterial elements do not coincide with element boundaries, as shown in Fig. 9.14 [45]-[49]. The interfaces must be constructed using pattern matching or some other technique. 

If 8j and 82 are identical then AH will be zero and so AF is bound to be negative and the compounds will mix. Thus the intuitive arguments put forward in Section 5.3 concerning the solubility of amorphous polymers can be seen to be consistent with thermodynamical treatment. The above discussion is, at best, an oversimplification of thermodynamics, particularly as applied to solubility. Further information may be obtained from a number of authoritative sources." ... 

The distribution coefficient is an equilibrium constant and, therefore, is subject to the usual thermodynamic treatment of equilibrium systems. By expressing the distribution coefficient in terms of the standard free energy of solute exchange between the phases, the nature of the distribution can be understood and the influence of temperature on the coefficient revealed. However, the distribution of a solute between two phases can also be considered at the molecular level. It is clear that if a solute is distributed more extensively in one phase than the other, then the interactive forces that occur between the solute molecules and the molecules of that phase will be greater than the complementary forces between the solute molecules and those of the other phase. Thus, distribution can be considered to be as a result of differential molecular forces and the magnitude and nature of those intermolecular forces will determine the magnitude of the respective distribution coefficients. Both these explanations of solute distribution will be considered in this chapter, but the classical thermodynamic explanation of distribution will be treated first. 

Supply air Treated or untreated air entering the space. For the purpose of drawings it is color-coded to show the various thermodynamic treatments. 

Hall, D.G., 1972. Thermodynamic treatment of some factors affecting the interaction between colloidal particles. Journal of the Chemical Society Faraday Transactions, 68(2), 2169-2182. 

A system is any part of external reality that can be subjected to thermodynamic treatment the material with which the system is in contact forms the surroundings, e.g. an electrochemical cell could be the system and the external atmosphere the surroundings. 

Treatment of Solutions by Statistical Mechanics. Since the vapor pressure is directly connected with the free energy, in the thermodynamic treatment the free energy is discussed first, and the entropy is derived from it. In the treatment by statistical mechanics, however, the entropy is discussed first, and the free energy is derived from it. Let us first consider an element that consists of a single isotope. When the particles share a certain total energy E, we are interested in the number of recog-... 

In discussing the experimental data, we shall wish to make use of the equilibrium constants that are to be found in the literature. We must therefore inquire into the relation that the thermodynamic treatment bears to the treatment that has been given above. When the expression for any reaction, such as (66) for example, has been written down, the species that have been written on the left-hand side are called the reactants, and those on the right-hand side are called the products. The... 

The general thermodynamic treatment of binary systems which involve the incorporation of an electroactive species into a solid alloy electrode under the assumption of complete equilibrium was presented by Weppner and Huggins [19-21], Under these conditions the Gibbs Phase Rule specifies that the electrochemical potential varies with composition in the single-phase regions of a binary phase diagram, and is composition-independent in two-phase regions if the temperature and total pressure are kept constant. 

Hence, the main aim of the technological process in obtaining fibres from flexible-chain polymers is to extend flexible-chain molecules and to fix their oriented state by subsequent crystallization. The filaments obtained by this method exhibit a fibrillar structure and high tenacity, because the structure of the filament is similar to that of fibres prepared from rigid-chain polymers (for a detailed thermodynamic treatment of orientation processes in polymer solutions and the thermokinetic analysis of jet-fibre transition in longitudinal solution flow see monograph3. ... 

The thermodynamic treatment of sublimation is exactly analogous to that of evaporation. Ramsay and Young (1884) have proved experimentally that during sublimation the temperature remains constant, and heat is absorbed for unit mass this is the latent-heat of sublimation, Ls. 

In the previous sections, we emphasized that at constant temperature, the liquid-phase activity coefficient is a function of both pressure and composition. Therefore, any thermodynamic treatment of gas solubility in liquids must consider the question of how the activity coefficient of the gaseous solute in the liquid phase varies with pressure and with composition under isothermal conditions. 

We are interested in understanding how this relative motion of two parts of the molecule contributes to its thermodynamic properties. It turns out that the thermodynamic treatment of this internal rotation depends upon the relative... 

The thermodynamics treatment followed in this volume strongly reflects our backgrounds as experimental research chemists who have used chemical thermodynamics as a base from which to study phase stabilities and thermodynamic properties of nonelectrolytic mixtures and phase properties and chemical reactivities in metals, minerals, and biological systems. As much as possible, we have attempted to use actual examples in our presentation. In some instances they are not as pretty as generic examples, but real-life is often not pretty. However, understanding it and its complexities is beautiful, and thermodynamics provides a powerful probe for helping with this understanding. 

Energy expended by living cells for maintenance is expressed quantitatively in appropriate units, for example kJ Kg s, and in animals it is largely provided as ATP. In this chapter, we outline how this is achieved, although our thermodynamic treatment lacks formal rigor. Further information on classical thermodynamics is given in textbooks of physical chemistry. 

Solvent effects also play an important role in the theory separating enthalpy and entropy into external and internal parts (134-136) or, in other terms, into reaction and hydration contributions (79). This treatment has been widely used (71, 73, 78, 137-141). The most general thermodynamic treatment of intermolecular interaction was given by Rudakov (6) for various states of matter and for solution enthalpy and entropy as well as for kinetics. A particular case is hydrophobic interaction (6, 89, 90). 

Every interface is more or less electrically charged, unless special care is exercised experimentally [26]. The energy of the system containing the interface hence depends on its electrical state. The thermodynamics of interfaces that explicitly takes account of the contribution of the phase-boundary potential is called the thermodynamics of electrocapillarity [27]. Thermodynamic treatments of the electrocapillary phenomena at the electrode solution interface have been generalized to the polarized as well as nonpolarized liquid liquid interface by Kakiuchi [28] and further by Markin and Volkov [29]. We summarize the essential idea of the electrocapillary equation, so far as it will be required in the following. The electrocapillary equation for a polarized liquid-liquid interface has the form... 

In the case of a solution with a previously known aH+ (see below), we could determine 2°H+-.H2(iatm)> provided that a reference electrode of zero potential is available however, experiments, especially with the capillary electrometer of Lippmann, did not yield the required confirmation about the realization of such a zero reference electrode16. Later attempts to determine a single electrode potential on the basis of a thermodynamic treatment also were not successful17. For this reason, the original and most practical proposal by Nernst of assigning to the standard 1 atm hydrogen potential a value of zero at any temperature has been adopted. Thus, for F2H+ H2(iatm) we can write... 

Simple thermodynamic treatment of gas compression stipulates that the maximum temperature (Tmax) attained is dependent on the adiabatic index (y) of the gas ... 

For the cell above, this thermodynamic treatment does, in fact, give the correct predictions provided that the value of Eai/Cl remains close to i2/c,-. However, there are a number of problems with the treatment that must be addressed in a more thorough-going account ... 

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