Gibbs energy pressure dependence

The application of the second law of thermodynamics is useful for understanding complex protein sorption phenomena (Haynes and Norde, 1994 Quiquampoix et al., 2002). It assumes that the spontaneous adsorption of a protein at constant temperature and pressure leads to a decrease in the Gibbs energy of the system. The Gibbs energy (G) depends on enthalpy [H], which is a measure of the potential energy (energy that has to be supplied to separate the molecular constituents from one another), and entropy (S), which is related to the disorder of the system. 

The entropy and Gibbs energy of an ideal gas do depend on pressure. By equation 85 (constant T),... 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

This is a 4 2 reaction, and is thus pressure dependent. However, it is necessary to compute the equilibrium partial pressure of some alternative gaseous species, such as SiCls, and other hydrocarbons such as C2H2 and for this a Gibbs energy minimization calculation should be made. 

At constant temperature and pressure the excess Gibbs energy of the surface layer depends on surface area S and on the composition of the layer (i.e., on the excess amounts of the components). When there are changes in surface area and composition (which are sufficiently small so that accompanying changes in parameters a... 

Figure 3.1 (a) (b) The efficiency of 1 1 and 1 4 vapour phase transport reactions, showing the marked dependence of the optimum for the 1 4 reaction on the pressure, (c) The dependence of the 1 2 and 1 4 reaction Gibbs energy on temperature and pressure, showing that the formation of nickel carbonyl is favoured by high pressure, and that of zirconium tetra-iodide, which is much more stable, is favoured by low pressures... 

ArV is not necessarily positive, and to compare the relative stability of the different modifications of a ternary compound like AGSiOs the volume of formation of the ternary oxide from the binary constituent oxides is considered for convenience. The pressure dependence of the Gibbs energies of formation from the binary constituent oxides of kyanite, sillimanite and andalusite polymorphs of A SiOs are shown in Figure 1.10. Whereas sillimanite and andalusite have positive volumes of formation and are destabilized by pressure relative to the binary oxides, kyanite has a negative volume of formation and becomes the stable high-pressure phase. The thermodynamic data used in the calculations are given in Table 1.7 [3].1... 

A homogeneous open system consists of a single phase and allows mass transfer across its boundaries. The thermodynamic functions depend not only on temperature and pressure but also on the variables necessary to describe the size of the system and its composition. The Gibbs energy of the system is therefore a function of T, p and the number of moles of the chemical components i, tif. 

The mutual solubility of two liquids A and B depends, in general, on how much the molecules of each liquid tend to attract those of its own kind, relative to their tendency to attract those of the other. This tendency is measured by the excess Gibbs energy of mixing of the two liquids (see section 2.4), Am gL, which is related to the partial vapor pressures p/ and of the two liquids A and B in the mixture. If the composition of the system is given by and Wb moles of the respective components in a given phase, their mole fractions in this phase are... 

EMF and vapour pressure measurements are dependent on the temperature, the number of phases involved and, importantly, the reference state of the component in question. The problem with the reference state is important as experimentally stated values of partial Gibbs energies will be dependent on this value. The standard states are fixed before optimisation and may actually have values different from those used by the original author. Therefore, as far as possible like should be compared with like. 

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 ... 

Here the pre-exponential factor At is the product of a temperature-dependent constant (ksT/h) = 2 X 10 °Ts where ke and h are the Boltzmann and Planck constants, and a solvent-specific coefficient that relates to both the solvent viscosity and to its orientational relaxation rate. This coefficient may be near unity for very mobile solvent molecules but may be considerably less than unity for viscous or orientationally hindered highly stractured solvents. The exponential factor involves the activation Gibbs energy that describes the height of the barrier to the formation of the activated complex from the reactants. It also describes temperature and pressure dependencies of the reaction rate. It is assumed that the activated complex is in equilibrium with the reactants, but that its change to form the products is rapid and independent of its environment in the solution (de Sainte Claire et al., 1997). 

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 ... 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

The similarity to the Gibbs-Duhem equation is quite apparent, and indeed this equation is the Gibbs-Duhem equation if X refers to the Gibbs energy. We should note that the differential dXt, the differential that appears in Equations (6.13) and (6.14), depends upon the differential quantities of the temperature, the pressure, and the mole fractions as expressed in Equation (6.7). At constant temperature and pressure Equation (6.12) becomes a special case of Equation (6.14). 

We turn our attention in this chapter to systems in which chemical reactions occur. We are concerned not only with the equilibrium conditions for the reactions themselves, but also the effect of such reactions on phase equilibria and, conversely, the possible determination of chemical equilibria from known thermodynamic properties of solutions. Various expressions for the equilibrium constants are first developed from the basic condition of equilibrium. We then discuss successively the experimental determination of the values of the equilibrium constants, the dependence of the equilibrium constants on the temperature and on the pressure, and the standard changes of the Gibbs energy of formation. Equilibria involving the ionization of weak electrolytes and the determination of equilibrium constants for association and complex formation in solutions are also discussed. 

The dependence of the equilibrium constant on pressure for a chemical reaction is easily determined from Equation (11.4) with the aid of Equation (4.38), which gives the change of the Gibbs energy with pressure at constant temperature and constant number of moles. Thus, we have... 

The change in Gibbs energy, G for one mole, with pressure, P at constant temperature T is dependant on volume, V (equation (20.3) Frame 20) ... 

Activity coefficients yk have traditionally been calculated from correla equations for GE/RT by application of Eq. (11.62). The excess Gibbs energy a function of Tt P, and composition, but for liquids at low to moderate, pressi it is a very weak function of P. Under these conditions, its pressure dependen and therefore the pressure dependence of the activity coefficients are usual neglected. This is consistent with our earlier omission of the Poynting factor fr... 

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 ... 

---