In the free energy formula of Eq (7.36), nonideality is expressed by the general form gex , the excess free energy. The simplifications used in the prior analyses of ideal melting and phase separation, namely neglecting sex and confining hex to the regular-solution model, are not valid for most binary systems. In order to construct phase diagrams by the common-tangent technique, more elaborate solution models are needed to relate free energy to composition for all likely phases. Figure 8.9 shows plots of g Vs xB for the three phases at six temperatures, with T6 the highest and T1 the lowest. In the six graphs, the curves for each phase keep approximately the same shape but shift relative Fig. 8.9 Free energy composition curves for an A-B binary system with two solid phases (a and P) and a liquid phase (From Ref. 1) Fig. 8.9 Free energy composition curves for an A-B binary system with two solid phases (a and P) and a liquid phase (From Ref.
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As in Eq (7.21) for the enthalpy, the molar Gibbs free energy of a solution (g) can be written in terms of pure-component contributions (gA and gB ) and an excess value (gex ). However, an important contribution needs to be added. For a binary solution, the terms contributing to g are
Even though the thermochemical database need contain only AGo (or, equivalently, AHo and ASo ), the number of reactions that would have to be included in such a compilation is intractably large. The key to reducing data requirements to manageable size is to provide the standard free energy changes of forming the individual molecular species from their constituent elements. Particular reactions are constructed from these so-called formation reactions. For molecular compounds containing two or more elements, the basic information is the Free energy change for reactions by which the compound is created from its constituent elements, the latter in their normal state at the particular temperature. These reaction free energy changes are called standard free energies of formation of the compound. For example, the methane combustion reaction of Eq (9.1) involves one elemental compound (O2 ) and three molecular compounds (CH4 , CO2 , and H2 O).
Another notable solid-solid equilibrium is the graphite-to-diamond transition in the element carbon. Graphite is fairly common in the earth's crust but the rarity of diamond is the origin of its value. Under normal terrestrial conditions (300 K, 1 atm) the two forms of carbon are not in equilibrium and so, thermodynamically speaking, only one form should exist. The stable form is the one with the lowest Gibbs free energy. At 300 K, the enthalpy difference between diamond and graphite is Ahd-g 1900 J mole, with diamond less stable than graphite in this regard. Being a highly ordered structure, diamond has a molar entropy lower than that of graphite, and Asd-g -3.3 J mole-K (see Fig. 3.6). This difference also favors the stability of graphite. The combination of the enthalpy and entropy effects produces a free-energy difference of Since the phase with the lowest free energy (graphite) is stable, diamond is a metastable phase.
The definition has been chosen so that the activity tends to unity for pure i that is, gi , the molar free energy of pure i. Activity varies monotonically with concentration. Therefore, when component i approaches infinite dilution ai 0 and - ot. This inconvenient behavior of the chemical potential at zero concentration is avoided by using the activity in practical thermodynamic calculations involving species in solution. Another reason for the choice of the mathematical form of the relation between and ai embodied in Eq (7.29) is that the activity is directly measurable as the ratio of the equilibrium pressure exerted by a component in solution to the vapor pressure of the pure substance. This important connection is discussed in Chap. 8. Problem 7.7 shows how this equation can be used to assess the validity of formulas for hex . In an equally important application, the above equation can be integrated to give the Gibbs free energy analog of Eq (7.19) for the enthalpy
The equilibrium criterion of minimum Gibbs free energy (Sect. 1.11) can be applied to any of the phase transitions described in the previous section. At fixed pressure and temperature, let the system contain nI moles of phase I and nII moles of phase II, with molar Gibbs free energies of gI and gII, respectively. The total Gibbs free energy of the two-phase mixture is This is an expression of chemical equilibrium. It complements the conditions of thermal equilibrium (TI TII) and mechanical equilibrium (pI pII). Since the Gibbs free energy is defined by g h - Ts, another form of Eq (5.2) is
To Matthews that Tesla had entrusted two of his greatest inventions prior to his death - the Tesla interplanetary communications set and the Tesla anti-war device. Tesla also left special instructions to Otis T. Carr of Baltimore, who used this information to develop free-energy devices capable of 'powering anything from a hearing aid to a spaceship.' (73) Tesla's technology, through Carr's free-energy device, will revolutionize the world. Instead of purchasing power from the large corporations, which is then delivered to our homes via wires and cables, the new technology consists of nothing more than a small antenna that will be attached to the roof of every building
As in any system constrained to constant temperature and pressure, the equilibrium of a chemical reaction is attained when the free energy is a minimum. Specifically, this means that dG 0, where the differential of G is with respect to the composition of the mixture. In order to convert this criterion to an equation relating the equilibrium concentrations of the reactants and products, the chemical potentials are the essential intermediaries. At equilibrium, Eq (7.27) provides the equation
Irrespective of the complexity of the nonideal behavior of the phases involved, the phase diagram can always be constructed if the free energy Vs composition curves for each phase can be drawn. The link between the two graphical representations is the common-tangent rule. Because of the wide variations in the shapes of free-energy curves, the types of phase diagrams deduced from them reaches zoological proportions. In this section, a common variety called the eutectic phase diagram5 is developed by the graphical method.
The structure of a phase diagram is determined by the condition of chemical equilibrium. As shown in Sect. 8.2, this condition can be expressed in one of two ways either the total free energy of the system (Eq (8.1)) is minimized or the chemical potentials of the each component (Eq (8.2)) in coexisting phases are equated. The choice of the manner of expressing equilibrium is a matter of convenience and varies with the particular application. Free-energy minimization is usually used with the graphical method and chemical-potential equality is the method of choice for the analytic approach.
The chemical potential is directly related to the Gibbs free energy of a system. For a one-component system, the chemical potential is identical to the molar Gibbs free energy of the pure substance. In solutions or mixtures, the chemical potential is simply another name for the partial molar Gibbs free energy. The discussion in Sect. 7.3, in which enthalpy was used to illustrate partial molar and excess properties, applies to the Gibbs free energy one need only replace h everywhere by g. The reason that the partial molar Gibbs free energy (g) is accorded the special name chemical potential is not only to shorten a cumbersome five-word designation. More important is the role of the chemical potential in phase equilibria and chemical equilibria when the restraints are constant temperature and pressure. Instead of the symbol g, the chemical potential is designated by The connection between the Gibbs free energy of a system at fixed T and p and the equilibrium state is shown in Fig. 1.18.
By the time I arrived at Princeton in the fall of 1962, I was thoroughly pumped up to join the quest for controlled fusion at PPL. My under-grad senior thesis on an obscure plasma instability led to working in Jim Drummond's Plasma Physics Group at the Boeing Scientific Research Laboratories in my hometown, Seattle, during the year following graduation. In fact, this millennial dream of realizing a virtually limitless source of pollution-free energy was largely my motivation for applying to grad school, and only to Princeton.
That an under-ice ocean exists on Europa is remarkable. It is especially remarkable when it is realized that Jupiter sits well outside of the habitable zone (defined in Chapter 5, and see Figure 5.9) and given that the surface temperature of the moon is not much greater than 100 K. How, indeed, can this ocean exist There is not enough solar energy to warm Europa above the freezing point of water, and the moon is so small that it should have cooled off relatively rapidly after formation.19 The possibility of terraforming Europa and, indeed, the other Galilean moons has been discussed by numerous researchers, but in all cases, bar the stellifying of Jupiter option, the biggest hurdle to overcome is that of supplying enough surface heat.
The term on the left is A , the chemical-potential difference of overall reaction (10.21). It is the aqueous equivalent of the free-energy difference AG used in describing nonaqueous cells. The electric potential difference on the right is the cell EMF, s, so the equation is
Propellant to generate electricity is not an efficient way to power a satellite that could use free solar energy instead (by means of solar cells). Nevertheless, electrodynamic tether power generation could be useful for generating short bursts of electrical energy, for instance when needed for high-energy but short duration experiments involving powerful lidars (instruments similar to radar but using laser light instead of short wavelength radio waves). 2ff7e9595c
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