Gibbs free energy pressure dependence

For an open system of variable snrface area, the Gibbs free energy must depend on composition, temperatnre, T, pressure, p, and the total snrface area. A ... 

The Gibbs free energy change of a system will depend not only on temperature and pressure but upon the chemical potentials of the species involved, and this statement may be expressed in the form of the partial differential... 

Whether a reaction is spontaneous or not depends on thermodynamics. The cocktail of chemicals and the variety of chemical reactions possible depend on the local environmental conditions temperature, pressure, phase, composition and electrochemical potential. A unified description of all of these conditions of state is provided by thermodynamics and a property called the Gibbs free energy, G. Allowing for the influx of chemicals into the reaction system defines an open system with a change in the internal energy dt/ given by ... 

Thermodynamics predicts under which conditions a catalyst can be reduced. As with every reaction, the reduction will proceed when the change in Gibbs free energy, AG, has a negative value. Equation (2-2) shows how AG depends on pressures and temperature ... 

The free energy functions are defined by explicit equations in which the variables are functions of the state of the system. The change of a state function depends only on the initial and final states. It follows that the change of the Gibbs free energy (AG) at fixed temperature and pressure gives the limiting value of the electrical work that could be obtained from chemical transformations. AG is the same for either the reversible or the explosively spontaneous path (e.g. H2 -I- CI2 reaction) however, the amount of (electrical) work is different. Under reversible conditions... 

This relationship identifies the surface energy as the increment of the Gibbs free energy per unit change in area at constant temperature, pressure, and number of moles. The path-dependent variable dWs in Eq. (2.60) has been replaced by a state variable, namely, the Gibbs free energy. The energy interpretation of y has been carried to the point where it has been identified with a specific thermodynamic function. As a result, many of the relationships that apply to G also apply to y ... 

Before discussing all these biopolymer applications, we first take this opportunity to remind the reader that, in general, any thermodynamic variable can be expressed as the sum of two functions, one of which depends only on the temperature and pressure, and another which depends on the system composition (expressed as the mole fraction xt of the /-component). Therefore, for example, the chemical potential fM of the /-component of the system at constant temperature T and pressure p (the general experimental conditions), /. e., partial molar Gibbs free energy (dG/dn TtP may be expressed as (Prigogine and Defay, 1954) ... 

The subject of interest is a gel swollen by solvent. Let F be the Gibbs free energy change after mixing of solvent and an initially unstrained polymer network [1]. When the gel is isotropic and is immersed in a pure solvent with a fixed pressure Po, F is a thermodynamic potential dependent on the temperature T, the pressure p inside the gel, and the solvent particle number Ns inside the gel. It satisfies... 

The pressure dependence of the Gibbs free energy is needed to calculate G at conditions other than the standard state. From the definition of the free energy (Eq. 9.1) the total differential of G is... 

Equation 9.20 gives the pressure dependence of the Gibbs free energy of a pure substance. More generally, for a mixture one should consider the chemical potential /r, which is defined as the partial molar free energy of species k ... 

We have assumed that the molar volume remains constant, which is certainly a reasonable assumption because most liquids are practically incompressible for the pressures considered. For a spherical drop in its vapor, we simply have AGm = 2-yVm/r. The molar Gibbs free energy of the vapor depends on the vapor pressure / o according to... 

Surfactants form semiflexible elastic films at interfaces. In general, the Gibbs free energy of a surfactant film depends on its curvature. Here we are not talking about the indirect effect of the Laplace pressure but a real mechanical effect. In fact, the interfacial tension of most microemulsions is very small so that the Laplace pressure is low. Since the curvature plays such an important role, it is useful to introduce two parameters, the principal curvatures... 

Flow imparts both extension and rotation to fluid elements. Thus, polymer molecules will be oriented and stretched under these circumstances and this may result in flow-induced phenomena observed in polymer systems which include phase-changes, crystallization, gelation or fiber formation. More generally, the Gibbs free energy of polymer blends or solutions depends under non-equilibrium conditions not only on temperature, pressure and concentration but also on the conformation of the macromolecules (as an internal variable) and hence, it is sensitive to external forces. 

Since the equilibrium distribution coefficient Kassoc l is related to the overall energy change in Gibbs free energy AGassoc, for the separation process carried out at constant pressure P and constant molar volume V of the solvent, then the capacity factor K, also takes on the well-known fundamental thermodynamic dependency through the relationships ... 

Ordinary diffusion depends on the partial Gibbs free energy and the concentration gradient. The pressure diffusion is considerable only for a high-pressure gradient, such as centrifuge separation. The forced diffusion is mainly important in electrolytes and the local electric field strength. Each ionic substance may be under the influence of... 

As may be seen from Eq. 6.8, the Gibbs free energy, G, is the most important thermodynamic parameter in describing chemical reactions, because it represents the moles of the constituents participating in the reaction, and how a chemical reaction changes the number of moles. Dependence of a chemical reaction on thermodynamic parameters, such as the temperature and pressure, is best represented by AG in an Arrhenius type of equation for the dissolution constant K... 

As discussed in Chapters 4 and 5, CBPC formation is governed by the oxide solubility. The solubility, in turn, is related to the Gibbs free energy, which is a function of temperature and pressure. As a result, the CBS formulation depends on the downhole temperature and pressure. The effect of the temperature on the solubility has already been discussed in Section 6.4. The pressure effect can be assessed in a similar manner, but as we shall see, it is negligibly small and can be ignored for all practical purposes. 

By comparing Equation IV.5 with the equilibrium condition expressed by Equation IV.3, we see that dG for a system equals zero at equilibrium at constant temperature and pressure. Moreover, G depends only on U, P, V, T, and S of the system. The extremum condition, dG = 0, actually occurs when G reaches a minimum at equilibrium. This useful attribute of the Gibbs free energy is strictly valid only when the overall system is at constant temperature and pressure, conditions that closely approximate those encountered in many biological situations. Thus our criterion for equilibrium shifts from a maximum of the entropy of the universe to a minimum in the Gibbs free energy of the system. 

Using Equation IV.2 we can readily determine the pressure dependence of the Gibbs free energy as needed in the last bracket of Equation IV. 11—namely, dG/dP)TEhn.n is equal to Fby Equation IV.12. Next, we have to consider the partial derivative of this V with respect to rij (see the last equality of Eq. IV. 11). Equation2.6 indicates that dV/dnj)TPEkn. is Vj, the partial molal volume of species j. Substituting these partial derivatives into Equation IV. 11 leads to the following useful expression ... 

Figure 15. (a) Temperature dependence of the Gibbs free energy for ice Ih (solid line) and ice Ic (dotted line) at atmospheric pressure, where contributions from the configurational entropy and anharmonic vibrations are omitted, (b) Temperature dependence of the energy which is defined as the sum of the interaction energy at its minimum structure and the vibrational energy for ice Ih (solid line) and ice Ic (dotted line). 

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