Helmholtz energy definition

Equation 54 implies that U is a function of S and I. a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Helmholtz energy,. 1. and Gibbs energy, G ... 

TEMPERATURE. The thermal state of a body, considered, with reference to its ability to communicate heat to other bodies (J. C. Maxwell). There is a distinction between temperature and heat, as is evidenced by Helmholtz s definition of heat, [energy that is transferred from one body to another by a thermal process), whereby a thermal process is meant radiation, conduction, and/or convection. 

Two other important derivatives of the Gibbs and Helmholtz energies with respect to the temperature can be derived from these simpler relations. From the definition of the Gibbs energy and Equation (4.37) we obtain... 

From their definitions (Eq. (4.39)) based on the energy, the enthalpy, and the Gibbs and Helmholtz energies, we may set the chemical potentials to be functions of other variables, as follows ... 

We can obtain expressions for the differentials of the enthalpy and the Gibbs and Helmholtz energies from the usual definitions. In such expressions M is an independent variable. Because it is more convenient to use H as an independent variable, the new functions (H — /j0HM), (A — /i0HM), and (G — /i0HM) are used. The differential expressions for these functions are... 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

Reference to Eq. (6.2) shows that the left side of this equation is by definition the Helmholtz energy A. Therefore,... 

On the basis of these definitions, one easily obtains expressions for the Helmholtz energy... 

In the aforementioned examples, the conditional solvation Helmholtz energy includes the direct interaction between the two solute particles, as well as the effect of the solvent. In some applications it is found useful to exclude the direct interaction. This occurs whenever we want to estimate the contributions to the solvation Helmholtz energy of each part of a combined solute. In our definitions of both AA (Ri) and AA (R2/Ri ), we transferred one solute s from a fixed position in an ideal gas into the liquid. Now suppose that we are given a pair of solutes at a distance R = R2 — J i in an ideal gas. This pair of solutes can be viewed as a single molecule. We wish to know the contribution of each particle (1 and 2) to the Helmholtz energy of solvation of the pair. The latter is... 

The natnral variables S and V of [/ are transformed to T and V by Legendre transformation of U to the energy fnnction A, called the Helmholtz free energy, or Helmholtz energy. By definition. 

Equation (3.4.1) follows directly from the definition of the process of solvation (see also Appendix G). The expression for the solvation, Gibbs or Helmholtz energy in terms of the process of inserting a particle at a fixed position is quite old, probably due to Kirkwood (1935) and later used in the scaled particle theory [see Sec. 3.8 and also Hill (1960) and Widom (1963,1982)]. 

It might seem useful to have a function related to the total work available from a system, but in fact A is little used in this sense. It (that is, Aj-y) is also not much used as a thermodynamic potential, despite the fact that replacement processes in weathering, metamorphism and metasomatism are commonly interpreted to occur at constant volume (Nahon and Merino 1997, Carmichael 1986). However, replacement processes do not, by definition, take place in a closed system, so that the Helmholtz energy is not the appropriate potential. So what is the appropriate potential quantity in open systems We consider this in 4.14. 

By transforming only S we obtain the definition of the Helmholtz energy, and by transforming only V we get the definition of enthalpy. For more information, see Alberty (2001). 

The enthalpy, Helmholtz energy, and Gibbs energy are important functions used extensively in thermodynamics. They are state functions (because the quantities used to define them are state functions) and are extensive (because U, S, and V are extensive). If temperature or pressure are not uniform in the system, we ean apply the definitions to eonstituent phases, or to subsystems small enough to be essentially uniform, and sum over the phases or subsystems. 

We shall very briefiy present the definitions of these quantities. We use a shorthand notation whenever possible. For instance, we shall use 2(1) rather than the more detailed notation 2(U 0, 0, 0) as we have done in previous sections. Thus the intrinsic binding constants and the corresponding Helmholtz energy changes for binding the zth ligand are... 

According to the definition of the Helmholtz energy function, we can easily deduce that the pressure is given by the opposite of the derivative of this Helmholtz energy with regard to volume ... 

The Gibbs and Helmholtz energies, both named after prominent thermodynam-icists, are the last energies that will be defined. Their definitions, coupled with the appropriate use of partial derivation, allow us to derive a rich set of mathematical relationships. Some of these mathematical relationships let the full force of thermodynamics be applied to many phenomena, like chemical reactions and chemical equilibria and—importantly—predictions of chemical occurrences. These relationships are used by some as proof that physical chemistry is complicated. Perhaps they are better seen as proof that physical chemistry is widely applicable to chemistry as a whole. 

We now define two more energies. The definition of the Helmholtz energy, A, is... 

Equation 22.6 defines surface tension in terms of Gibbs energy. Borrowing an analogy from chemical potential, we submit that surface tension can also be defined in terms of enthalpy, internal energy, or Helmholtz energy. Write partial derivatives for those definitions. 

In a similar manner, we construct the Helmholtz energy by applying the definition A=E— TS, introduced in Section 1.9, which sensibly involves the temperature Tof the system. This changes the independent variable from S to T. Using Eq. (1.10.2b), we then find that... 

Because adsorbates are discrete, we can approximate the differential by finite difference in the Helmholtz energy before and after the addition of one adsorbate to the surface. Applying the definition and simplifications of Helmholtz energy given in eqns (2.17)-(2.19), the chemical potential becomes... 

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 ... 

But it was not until J. P. Joule published a definitive paper in 1847 that the ealorie idea was abandoned. Joule eonelusively showed that heat was a form of energy. As a result of the experiments of Rumford, Joule, and others, it was demonstrated (explieitly stated by Helmholtz in 1847), that the various forms of energy ean be transformed one into another. 

Conditions for equilibrium and the definition of Helmholtz and Gibbs energies... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

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