Free-energy function

The left-hand side of the diagram represents the standard free energies of formation at room temperature. The relative order tells us that only Ag20 could be reduced by carbon under these conditions. 

The capacity of carbon to reduce oxides varies with temperature since the free energies of formation vary with temperature as 

We find that whereas copper can be reduced at any temperature greater than room temperature, iron can only be extracted at temperatures above 1200 K, Zinc and magnesium can only be reduced at temperatures above 1300 K and 2100 K respectively, under which conditions they exist as vapours. A1203 is stable up to 2300 K. 

We can also see from the diagram that above about 500 K AgzO is unstable and will dissociate into its elements since its free energy of formation becomes positive above this temperature. CuO is unstable above 1800 K. 

Similar diagrams can be constructed for sulphides but carbon has only a low affinity for sulphur and is rarely able to reduce sulphide ores. 

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Ben]amin I, Barbara P F, Gertner B J and Hynes J T 1995 Nonequilibrium free energy functions, recombination dynamics, and vibrational relaxation of tjin acetonitrile molecular dynamics of charge flow in the electronically adiabatic limit J. Phys. Chem. 99 7557-67... 

Free Energy Function The conformational free energy was estimated by the following energy expression ... 

Chemical systems spontaneously react in a fashion that lowers their overall free energy. At a constant temperature and pressure, typical of many bench-top chemical reactions, the free energy of a chemical reaction is given by the Gibb s free energy function... 

G King, A Warshel. Investigation of the free energy functions for electron transfer reactions. J Chem Phys 93 8682-8692, 1990. 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

Fig. 10(b)). One of the reasons for the differences between both theories is a different form of a hard sphere part of the free energy functional. Segura et al. have used the expression resulting from the Carnahan-Starhng equation of state, whereas the Meister-Kroll-Groot approach requires the application of the PY compressibility equation of state, which produces higher oscillations. 

In the limit of infinite system size in the y direction, the free energy functional can be represented as a sum of bulk (0, ), surface (0 and O ), and line (Oi) contributions as... 

When applied to the grand canonical free energy functional (47) the above condition leads to the following integral equation... 

The first step in studying phenomenological theories (Ginzburg-Landau theories and membrane theories) has usually been to minimize the free energy functional of the model. Fluctuations are then included at a later stage, e.g., using Monte Carlo simulations. The latter will be discussed in Sec. V and Chapter 14. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

When the solid phase 0+ at x = -f oo coexists with the gas phase 0 at X = -oo, the stationary profile of the phase field is determined so as to minimize the free energy functional F (56). The functional derivative gives... 

In the CVM, the free energy of a given alloy is approximated in terms of probabihties for a selected set of finite clusters. The largest cluster explicitly considered in the free energy functional specifies the level of the approximation. The common practice for an fcc-based system is the tetrahedron approximation [26] in which nearest neighbor tetrahedron cluster is taken as the largest cluster. Hence, within the tetrahedron approximation, the free energy expression, F,is symbolically expressed as... 

Later on Cahn and Hilliard presented some thermodynamic estimates for the nucleation of liquid in vapour. Values of AO and the composition profiles c(r) of the embryos have been estimated using the mean-field and gradient expansion approximations for the free energy functional F c(7 ). A number of qualitative features in variation... 

The critical hydrogen content for the ductility loss increased with increasing hydrogen solubility in the alloy. The fracture surfaces were not characteristic of those found under conditions of SCC. In terms of hydrogen and deuterium solubility in a similar series of bcc alloys, the equilibrium constants were determined at infinite dilution as a function of temperature The free energy function was expressed in terms of the bound-proton model. 

The effect of temp on chemical equilibria is conventially determined via the free energy function AG°/RT and the ideal equilibrium constant K. Table 1 gives the free, energy function G°/RT for the important detonation products of CHNO expls. From these data A G°/(RT) can be obtained for different temps for the reactions of interest, and ideal equilibrium constants computed according to ... 

Ideal Gas Free Energy Functions [G°/(RT)] for Detonation Products... 

Free energy functions, (G°, T — H°m 0), can also be obtained by starting with the equation... 

Thus, values for C°p m T, S°m T, (H°m T - H°m 0) and (G°mT H°m0) can be obtained as a function of temperature and tabulated. Figure 4.16 summarizes values for these four quantities as a function of temperature for glucose, obtained from the low-temperature heat capacity data described earlier. Note that the enthalpy and Gibbs free energy functions are graphed as (// , T - H°m 0)/T and (G T — H q)/T. This allows all four functions to be plotted on the same scale. Figure 4.16 demonstrates the almost linear nature of the (G°m T H°m 0)/T function. This linearity allows one to easily interpolate between tabulated values of this function to obtain the value at the temperature of choice. 

Enthalpy and free energy functions can also be tabulated using T = 298.15 K as a reference temperature by making use of the relationships... 

The free energy function can also be calculated from 5 r and (H°m T - H°m 298) directly from the relationship... 

Table 4.3 summarizes values taken from the JANAF tables for the Gibbs free energy functions and standard enthalpies of formation for a few common substances. The JANAF tables provide a more complete tabulation. 

Example 4.1 Use the Gibbs free energy functions from Table 4.3 to calculate ArG° at 1000 K for the reaction... 

FIGURE 3.9. A schematic description of the free-energy functionals at the range where AG0 > a. 

Free energy functions, 89, 90, 94 Free energy perturbation method (FEP), 81-82,146, 186-187 computer program for, 97-98 Free energy relationships, linear, 92-96, 148-149... 

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