Helmholtz energy defined

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... 

The Helmholtz and Gibbs energies on the other hand involve constant temperature and volume and constant temperature and pressure, respectively. Most experiments are done at constant Tandp, and most simulations at constant Tand V. Thus, we have now defined two functions of great practical use. In a spontaneous process at constant p and T or constant p and V, the Gibbs or Helmholtz energies, respectively, of the system decrease. These are, however, only other measures of the second law and imply that the total entropy of the system and the surroundings increases. 

In order to focus on the driving force for phase transitions induced by a magnetic field it is advantageous to use the magnetic flux density as an intensive variable. This can be achieved through what is called a Legendre transform [12], A transformed Helmholtz energy is defined as... 

In Section 9.2 we have defined the Gibbs energy of solvation AGo in the T, P, N ensemble. In the T, V, N (canonical) ensemble the appropriate quantity is A4a, the Helmholtz energy of solvation. It can be shown that the two are equal for macroscopic systems, provided the volume V in the T, V, N ensemble is equal to the average volume of a system in the T, P, N ensemble. 

Although adsorption exists as a subject of scientific investigation independent of its role in heterogeneous catalysis, it requires particular attention here because of its central role in heterogeneous catalysis. Most or all catalytic reactions involve the adsorption of at least one of the reactants. Many terms related to adsorption have already been defined in Appendix II, Part I, 1.1. These include surface, interface, area of surface or interface, and specific surface area. Appendix II, Part I, recommends A or S and a or s as symbols for area and specific area, respectively. As and as may be used to avoid confusion with Helmholtz energy A or entropy S where necessary. 

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter. 

In this discussion of indifferent states we have always used the entropy, energy, and volume as the possible extensive variables that must be used, in addition to the mole numbers of the components, to define the state of the system. The enthalpy or the Helmholtz energy may also be used to define the state of the system, but the Gibbs energy cannot. Each of the systems that we have considered has been a closed system in which it was possible to transfer matter between the phases at constant temperature and pressure. The differentials of the enthalpy and the Helmholtz and Gibbs energies under these conditions are... 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

An important excess property is the excess Gibbs energy GE. Many models have been developed to describe and predict GE from the properties of the molecules in the mixture and their mutual interactions. GE models often refer to the condensed state, the solid and liquid phases. In case significant changes in the volume take place upon mixing, or separation, the Helmholtz energy A, defined as... 

In this section we will consider the Legendre transforms that define the enthalpy H, Helmholtz energy A, and Gibbs energy G. 

This is referred to as a Gibbs-Helmholtz equation, and it provides a convenient way to calculate H if G can be determined as a function of 71 P, and ,. There is a corresponding relation between the internal energy U and the Helmholtz energy, which is defined by equation 2.5-2 ... 

We then introduce two new energy functions called free energy / (Helmholtz energy) for the independent variables temperature T and volume V, and free enthalpy G (Gibbs energy) for the independent variables temperature T and pressure p as defined, respectively, in Eqs. 3.21 and 3.22 ... 

Two additional properties, also defined for convenience, are the Helmholtz energy,... 

The physical meaning of the Helmholtz free energy is similar to that of the Gibbs energy, both being criteria to define - equilibrium. The equilibrium criterion in a closed system, which is only capable of doing P-V work and held at constant temperature and volume, is the minimum of Helmholtz energy. See also Helmholtz. 

The Gibbs representation provides a simple, clear-cut mode of accounting for the transfer of adsorptive associated with the adsorption phenomenon. The same representation is used to define surface excess quantities assumed to be associated with the GDS for any other thermodynamic quantity related with adsorption. In this way, surface excess energy (U°), entropy (Sa) and Helmholtz energy (Fa) are easily defined (Everett, 1972) as ... 

As indicated in Chapter 2, the adsorbent surface is characterized by a surface tension y whose magnitude depends on the nature of the surrounding medium (liquid, gas or vacuum) with which the adsorbent is in equilibrium. The isothermal extension of the surface area A, with no other change in the thermodynamic state of the system of adsorption, results in an increase d F of the Helmholtz energy of the system. Thus, the surface tension, y, is defined as ... 

This quantity is defined in (1.1.61 and, for SG interfaces, is obtainable from the adsorption Isotherm as in [1.1.7]. This pressure is directly related to the change in excess Helmholtz energy of adsorption by using (1.3.3] twice once with, and once without, adsorbate, followed by subtraction ... 

It should also be stressed that there is some arbitrariness in [2.4.401 in that surface excess Helmholtz energies can be defined in ways differing from [1.3.71, as discussed in sec. 1.2.10, but that this can never lead to alternative relations between measurable quantities. 

Basically, all the methods for measuring interfacial tensions described so far have in common that the Helmholtz energy for extending an interface is determined. Upon this extension, the interfacial tension should not vary, otherwise the quantity y would become ill-defined. One of the changes that might be incurred could result from strong curving of the interface. In the present chapter this issue was avoided because we have only considered macroscopic interfaces with radii of curvatures above 0(10-100 nm). Already in sec. 1.2.23c we showed that y is then still independent of curvature. 

Usually it is more appropriate to analyze interfacial tensions in terms of Helmholtz energies than to work with Gibbs energies because (i) in [2.2.2] both the bulk and surface work are made explicit and (ii) there is some arbitrariness in defining G in effect in our convention (which is as recommended by lUPAC) G° does not even contain the interfacial tension. See also the text following [4.2.25]. 

For the chemical potentials the situation is different. For the surfactant the chemical potential in the monolayer p is not defined by that in the subphase because of the absence of transport. Nevertheless, this quantity is well-defined as the molar Helmholtz energy needed to add more surfactant to the layer ... 

We have defined the solvation process as the process of transfer from a fixed position in an ideal gas phase to a fixed position in a liquid phase. We have seen that if we can neglect the effect of the solvent on the internal partition function of the solvaton s, the Gibbs or the Helmholtz energy of solvation is equal to the coupling work of the solvaton to the solvent (the latter may be a mixture of any number of component, including any concentration of the solute s). In actual calculations, or in some theoretical considerations, it is often convenient to carry out the coupling work in steps. The specific steps chosen to carry out the coupling work depend on the way we choose to write the solute-solvent interaction. 

Consider a solute s with internal rotational degrees of freedom. We assume that the vibrational, electronic, and nuclear partition functions are separable and independent of the configuration of the molecules in the system. We define the pseudo-chemical potential of a molecule having a fixed conformation Ps as the change in the Helmholtz energy for the process of introducing s into the... 

Finally, we look for the natural energy function for the variables T and p. The Helmholtz energy of the natural variables T and V offers a starting point. We define the state function Gibbs free energy G by... 

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