Gibbs energy-temperature sections

It is clear from Table 17.5 that in terms of the gasification stage, both enthalpy and Gibbs energy values are significantly smaller than those of the conventional CTL route. The lost work associated with this first stage is also comparatively smaller. In terms of the synthesis section, we note that it operates close to its Carnot temperature (TCarnot = 480 K), and thus the lost work from this process is reduced significantly. Overall, the lost work amounts to 19 kJ/mol, as compared to the 112 kJ/mol for the conventional route. 

In the ideal case of an electrochemical converter, such as a fuel cell, the change in Gibbs free energy, AG, (Section 2.2.3) of the reaction is available as useful electric energy at the temperature of the conversion. The ideal efficiency of a fuel cell, operating irreversibly, is then... 

The thermodynamic quantity 0y is a reduced standard-state chemical potential difference and is a function only of T, P, and the choice of standard state. The principal temperature dependence of the liquidus and solidus surfaces is contained in 0 j. The term is the ratio of the deviation from ideal-solution behavior in the liquid phase to that in the solid phase. This term is consistent with the notion that only the difference between the values of the Gibbs energy for the solid and liquid phases determines which equilibrium phases are present. Expressions for the limits of the quaternary phase diagram are easily obtained (e.g., for a ternary AJB C system, y = 1 and xD = 0 for a pseudobinary section, y = 1, xD = 0, and xc = 1/2 and for a binary AC system, x = y = xAC = 1 and xB = xD = 0). 

In this section two prediction techniques are discussed, namely, the gas gravity method and the Kvsi method. While both techniques enable the user to determine the pressure and temperature of hydrate formation from a gas, only the KVSI method allows the hydrate composition calculation. Calculations via the statistical thermodynamics method combined with Gibbs energy minimization (Chapter 5) provide access to the hydrate composition and other hydrate properties, such as the fraction of each cavity filled by various molecule types and the phase amounts. 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Close to T = 0 K the most stable phase to exist is the solid state, a point discussed further in Frame 22, section 22.1. Thus at T = 0 K, Gs, the Gibbs energy of the solid phase, would be expected to be lower than that for any (hypothetical) liquid phase, Gi, which in turn would be anticipated to be lower than that for any (hypothetical) gaseous phase, Gg at this temperature. 

Whatever the mechanism is, particles adhere spontaneously if, at constant temperature and pressure, the Gibbs energy G of the system decreases. The main contributions to the Gibbs energy of particle adhesion A Gad are from electrostatic, hydrophobic and dispersion forces,1 5 and, furthermore, in case of protein adsorption, from rearrangements in the structure of the protein molecule.6 9 When the sorbent surface is not smooth but hairy , additional, mainly steric, interactions come into play.4,10 12 Hairy surfaces are often encountered in nature as a result of adsorbed or grafted natural polymers, such as polysaccharides, that reach out in the surrounding medium with some flexibility. Interaction of particles with such hairy surfaces will be dealt with in section 3. 

As mentioned in Sections 1.1 and 2.9, the third law of thermodynamics makes it possible to obtain the standard Gibbs energy of formation of species in aqueous solution from measurements of the heat capacity of the crystalline reactant down to about 10 K, its solubility in water and heat of solution, the heat of combustion, and the enthalpy of solution. According to the third law, the standard molar entropy of a pure crystalline substance at zero Kelvin is equal to zero. Therefore, the standard molar entropy of the crystalline substance at temperature T is given by... 

Gibbs energy curves have been calculated for the hexagonal and cubic phases for various hypothetical (Ti,Al)N deposition temperatures (Figure 9). The point of intersection of these curves at each temperature defines the composition at which there is a transition from one structure to the other. It can be seen that this composition is nearly temperature independent and has a calculated value of around 0.7 mol fraction AIN, Experimental studies of the extent of the metastable cubic range in the section AlN-TiN have been carried out by Knotek and Leyendecker and more recently, with... 

Al temperatures below about 1600 C, the Gibbs energy change for the MgO reaction is more negative than for the AI2O3 reaction. This means that under these conditions MgO is more stable with respect to its constituent elements than is Ai2O3, and that Mg will react with ALOj to form MgO and Al. However, above about 1600 C the situation reverses, and Al will react with MgO to reduce it to Mg with the concomitant formation of AUOj. This is a rather high temperature, achievable in an electric arc furnace (compare the extraction of silicon from its oxide, discussed in Section 5.16). 

If y is measured at constant temperature and pressure in a one component system, then it is equal to the Gibbs energy of the interface, G . A more precise definition of is given in the following section, but the idea that work done can be equated to Gibbs energy is familiar from thermodynamics. It follows that the temperature derivative of y can be related to the entropy of the interface as follows ... 

The second comment concerns the choice of standard states. Clearly, in defining the process of solvation, one must specify the thermodynamic variables under which the process is carried out. Here we used the temperature T, the pressure P, and the composition N1 ..., Nc of the system into which we added the solvaton. In the traditional definitions of solvation, one needs to specify, in addition to these variables, a standard state for the solute in both the ideal gas phase and in the liquid phase. In our definition, there is no need to specify any standard state for the solvaton. This is quite clear from the definition of the solvation process yet there exists some confusion in the literature regarding the standard state involved in the definition of the solvation process. The confusion arises from the fact that Ap is determined experimentally in a similar way as one of the conventional standard Gibbs energy of solvation. The latter does involve a choice of standard state, but the solvation process as defined in this section does not. For more details, see the next two sections. 

Third, the equality of the Gibbs energies for the p-process and the solvation process do not imply equality between any other thermodynamic quantities pertaining to these two processes. One must exercise extreme care in deriving the relations between, say, the standard entropy of the p-process and the entropy of solvation these cannot be obtained by taking the temperature derivative of equation (7.54). As we shall see in the next section, this is a tricky point which has been overlooked even by experts working in this field. 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

The temperature of a liquid mi.xture is reduced so that solids form. However, unlike the illustrations in Section 12.3, on solidification, a solid mixture (rather than pure solids) is formed. Also, the liquid phase is not ideal. Assuming that the nonideality of the liquid and solid mixtures can be described by the same one-constant Margules excess Gibbs energy expression, derive the equations for the compositions of the coexisting liquid and solid phases as a function of the freezing point of the mixlure and the pure-component propenies. 

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