Free Helmholtz

Fa, f7 Interfacial free (Helmholtz) energy in Gibbs convention (J) and interfacial free (Helmholtz) energy per unit area (J m-2), respectively / Force per unit area (N m-2)... 

The advantage of this approach over kinematically extended continua is twofold first, the additional parameter is abstract and does not require special assumptions of the local kinematics as it is necessary in micropolar continua, for example. As a consequence, the corresponding boundary conditions can be formulated more flexibly. Second, the coupling between the macroscopic balance of momentum and the additional balance of equilibrated forces is obtained consti-tutively, via the free Helmholtz energy function and not via the free energy function in combination with measures of deformation assumed a priori. Therefore, such an approach allows for size effects in both ways - either smaller is stiffer" or smaller is weaker depending on the boundary conditions. 

Note that according to Eqs. (15) and (16) the free Helmholtz energy serves as a potential for the stresses T and for the microstructural flux S. According to the assumption of elastic material behavior, results of this type have to be expected. The additional balance equation for k [Eq. (17)] possesses the same structure as the balance of equihbrated forces obtained in Refs. [14, 20, 33] and appHed, e.g., in Ref [36]. Following the MuUer-Liu approach, Svendsen [39] also derived a generalization of Eq. (17) for a model with scalar-valued stractural parameters. 

The values of m and n, as notified also previously, are experimentally determined, through a calculation based on the relation (4.43), and then compared with the measured values, e.g. the 1iiermod5mamic parameters, as the isotherm compressibility and the free Helmholtz energy. Thus, empirically, there was established that, most often, there are obtained good results based on the choice ... 

Once a liquid film is formed, the most important issue is whether the film will stay as a film or will broke into droplets. Consider a solid surface of total area part of which A) is covered by a liquid film of uniform thickness, h. The free Helmholtz... 

In conclusion, only the specific internal energy part, u, of the specific excess free Helmholtz energy for a given D-face... 

At this point is must be emphasized again that a surface or interfacial tension is a tensor by its physical origin (cf O Chap. 4) and hence it may not be split into additive scalar components in general. Such additivity holds for the energy, and referring again to OEq. 6.11 for the specific excess free Helmholtz energy for a surface of phase A in equilibrium with its vapor phase... 

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... 

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...

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 ... 

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... 

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

Our discussion so far has considered the calculation of Helmholtz free energies, which a obtained by performing simulations at constant NVT. For proper comparison with expe inental values we usually require the Gibbs free energy, G. Gibbs free energies are obtaini trorn a simulation at constant NPT. 

The chemical potential p, of the adsorbate may be defined, following standard practice, in terms of the Gibbs free energy, the Helmholtz energy, or the internal energy (C/,). Adopting the last of these, we may write... 

Helmex Helmezine Helmholtz energy Helmholtz free energy... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

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