System free energy

Transitioning towards a carbon-free energy system is all the more timely as the production of fossil fuels is anticipated to peak in the 21st century owing to the steadily rising production rate and unavoidable resource limitations peak-oil or plateau around 2015-2020, peak-gas around 2030 and peak-coal around 2060 (if exploited with no restriction, which would lead to an unacceptable C02 concentration of 600 ppm in the atmosphere). 

In spite of the emergency of the transition required to a carbon-free energy system, learnings from the past history of hydrogen and nuclear energy give several reasons to be optimistic ... 

A longer-term need is to eliminate unbumed hydroearbon (UHC) and CO2 emissions. Gas turbines aeeount for 20% of U.S. CO2 emissions, whieh is a signifieant fraction of the total current CO2 emissions. This number will increase as natural gas turbines replaee older coal-fired steam generation plants. The use of hydrogen-blended hydrocarbon fuels thus provides both a solution to the immediate need for NOx reduction, and also provides a transition strategy to a carbon free energy system in the future. 

Systems seek to minimize their total free energy G. Since surfaces and interfaces carry excess free energy, systems will seek to minimize the amount of surface/interface area per unit volume. 

Helmholtz free energy The maximum amount of energy available to do work resulting from changes in a system at constant volume. See free energy and Gibbs-Helmholtz equation. 

Figure III-l depicts a hypothetical system consisting of some liquid that fills a box having a sliding cover the material of the cover is such that the interfacial tension between it and the liquid is zero. If the cover is slid back so as to uncover an amount of surface dJl, the work required to do so will he ydSl. This is reversible work at constant pressure and temperature and thus gives the increase in free energy of the system (see Section XVII-12 for a more detailed discussion of the thermodynamics of surfaces).

The total free energy of the system is then made up of the molar free energy times the total number of moles of the liquid plus G, the surface free energy per unit area, times the total surface area. Thus... 

At constant temperature and pressure a small change in the surface free energy of the system shown in Fig. IV-1 is given by the total differential... 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

Referring to Section V-2, the double-layer system associated with a surface whose potential is some value j/o requires for its formation a free energy per unit area or a t of... 

Of these the last eondition, minimum Gibbs free energy at eonstant temperahire, pressure and eomposition, is probably the one of greatest praetieal importanee in ehemieal systems. (This list does not exhaust the mathematieal possibilities thus one ean also derive other apparently ununportant eonditions sueh as tliat at eonstant U, S and Uj, Fisa minimum.) However, an experimentalist will wonder how one ean hold the entropy eonstant and release a eonstraint so that some other state fiinetion seeks a minimum. 

We have seen that equilibrium in an isolated system (dt/= 0, dF= 0) requires that the entropy Sbe a maximum, i.e. tliat dS di )jjy = 0. Examination of the first equation above shows that this can only be true if. p. vanishes. Exactly the same conclusion applies for equilibrium under the other constraints. Thus, for constant teinperamre and pressure, minimization of the Gibbs free energy requires that dGId Qj, =. p. =... 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

Evaluating its contribution to the free energy of the system requires taking the themiodynamic limit (N x) for the four-particle distribution fiinction. Lebowitz and Percus [75] and Hiroike [76] showed that the... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

The reference free energy in this case is an upper bound for tlie free energy of the electrolyte. A lower bound for the free energy difference A A between the charged and uncharged RPM system was derived by Onsager... 

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