Helmholtz

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The partial molar enthalpy for every component i is found from an appropriate form of the Gibbs-Helmholtz equation... 

Gibbs-Helmholtz equation This equation relates the heats and free energy changes which occur during a chemical reaction. For a reaction carried out at constant pressure... 

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. 

The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > )... 

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

According to the Helmholtz theorem, the two-dimensional vector field can be represented as a sum of an irrotational field and of a solenoidal one... 

We start with the Helmholtz integral and we use the Kirchhoff treatment, as Beckmarm and Spizzichino did [10]. Likewise, we shall make almost the same assumptions about the statistics of the radii h(cp,z) in order to find a way to deal with the integrals involved in the calculation. 

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

By analogy with the Helmholtz condenser formula, for small potentials the diffuse double layer can be likened to an electrical condenser of plate distance /k. For larger yo values, however, a increases more than linearly with o, and the capacity of the double layer also begins to increase. 

The compact layer can be structured into what is called an inner Helmholtz plane... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

Fig. V-14. Energy level diagram and energy scales for an n-type semiconductor pho-toelectrochemical cell Eg, band gap E, electron affinity work function Vb, band bending Vh, Helmholtz layer potential drop 0ei. electrolyte work function U/b, flat-band potential. (See Section V-9 for discussion of some of these quantities. (From Ref. 181.)...

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Thus, for spontaneous processes at constant temperature and volume a new quantity, the Helmholtz free energy A, decreases. At equilibrium under such restrictions cL4 = 0. 

If there are other kinds of work, similar expressions apply. For example, with electromagnetic work (equation (A2.1.8)1 instead of pressure-volume work, one can write for the Helmholtz free energy... 

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...

There is, or may be, an iimer layer of specifically adsorbed anions on the surface these anions have displaced one or more solvent molecules and have lost part of their iimer solvation sheath. An imaginary plane can be drawn tlirough the centres of these anions to fomi the inner Helmholtz plane (IHP). 

The layer of solvent molecules not directly adjacent to the metal is the closest distance of approach of solvated cations. Since the enthalpy of solvation of cations is nomially substantially larger than that of anions, it is nomially expected that tiiere will be insufBcient energy to strip the cations of their iimer solvation sheaths, and a second imaginary plane can be drawn tlirough the centres of the solvated cations. This second plane is temied the outer Helmholtz plane (OHP). 

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

If one assumes that tlie singular part A of the Helmholtz free energy is such a fimction... 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

For many practically relevant material/environment combinations, thennodynamic stability is not provided, since E > E. Hence, a key consideration is how fast the corrosion reaction proceeds. As for other electrochemical reactions, a variety of factors can influence the rate detennining step. In the most straightforward case the reaction is activation energy controlled i.e. the ion transfer tlrrough the surface Helmholtz double layer involving migration and the adjustment of the hydration sphere to electron uptake or donation is rate detennining. The transition state is... 

Helmholtz theorem [70], into a longitudinal part and a transverse... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

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