Adsorbate Helmholtz free energy

Application of Equation (17) to Equation (24) gives the Helmholtz free energy of the adsorbed molecules ... 

Helmholtz free energy of the adsorbed film is given as the sum of the van der Waals attraction potential of all molecules in the film with all atoms in the adsorbent, the vapor-liquid surface free energy and the free energy of all molecules in the bulk liquid. This leads to the following relation between the adsorbed amount Nand the relative pressure p/p0 [100, 101] ... 

The molar Helmholtz free energy of the adsorbed phase is simply the sum of the intrinsic Helmholtz free energy and the solid-fluid potential averaged over the adsorbed phase. Assuming a liquid-like behavior of the adsorbed phase, this free energy is given by... 

An expression for the Helmholtz free energy relates it with the canonical partition function and the partition function of adsorbed molecules and the transition state ... 

Just as with a bulk phase, the fundamental thermodynamic equations representing the combined first and second laws may be written in four equivalent ways in terms of the internal energy, enthalpy, Helmholtz free energy, or Gibbs free energy. For an adsorbed phase... 

Helmholtz free energy (of adsorbed phase) (3) dispersion force constants (2)... 

Various attempts have been made to include interactions between adsorbed species. As pointed out by Conway et al. [1984], the correct way to handle interactions is to include the appropriate pairwise or long-range interaction term into the partition function, which allows calculation of the Helmholtz free energy and the chemical potential. These quantities are a function of due to (a) the configurational term, as included in the Langmuir case and (b) the interaction or deviation from ideality. 

In the absence of adsorbate, the Helmholtz free energy of a clean surface becomes... 

Combining Eqs. (10.7) to (10.12) gives the Helmholtz free energy of the complete emulsion. The free energy is minimised with respect to a change in the interfacial area A. This involves transfer of adsorbed components to or from the interface, thereby changing the bulk concentration and thus y the result is. 

The chemical potential of the adsorbent/adsorbate is give by the derivative of the Helmholtz free energy with respect to the number density ... 

Xp = the value of 7 at which the capillary filling takes place, the mean is (x a = the cross-sectional area of the adsorbate molecule A = molar Helmholtz free energy =a/(1.84-1.92)... 

The most frequent errors are connected with the exact interpretation of II present in Eq. (10). In order to define II exactly and with general validity let us eonsider the Helmholtz free energy of the adsorbate ... 

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... 

Most cations are strongly solvated, since their radii are small, and the free energy of solvation is approximately proportional to z2/r +, where ze0 is the cation charge in coulombs and r+ its ionic radius. The result of this is that even if the charge on the electrode is negative, there is usually little tendency for these cations to shed their water molecules and adsorb directly on the metal surface. Thus, the distance of closest approach of cations is determined by the radius of the inner solvent coordination sphere, and if the metal surface itself constitutes a plane, then the cation nuclei, at the distance of closest approach, will also constitute a plane termed the outer Helmholtz plane (OH P). 

In Eq. (83), all terms are either known or measurable, except the quantity of interest . Moreover, it should be noted that Eq. (83) is very interesting since it takes into account a small adsorbate, i.e., —CH2— group, whose surface area and surface free energy are slightly affected by temperature. This means that the variations in area and surface entropy of an adsorbed —CH2— group are negligible with temperature. Therefore, it is possible to determine the surface enthalpy derived from the Gibbs-Helmholtz equation... 

Here p(z) is the local density of the adsorbed fluid at a distance z from one of the walls of the pore, f(z) is the intrinsic molecular Helmholtz energy of the adsorbate phase, p is the chemical potential. The free energy f(z) comprises tire ideal, mean-field attractive terms, and the excess fine energy (repulsive) term as a function of so-called smoothed weighted average density. 

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