Helmholtz energy description

In 1958 Cahn and Hilliard proposed a phenomenological theory for surface and interfacial tensions that was based on a general formalism for heterogeneous systems. It has a certain analogy with the descriptions of non-uniformities in magnetic and ferro-electric domains in solids. The basic idea was that the loeal Helmholtz energy density per molecule / is expanded in a Taylor series about, the corresponding quantity in a uniform phase. Mathematically,... 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

In order to complete the thermodynamic description, the Helmholtz energy A is now calculated. On the basis of equation (1.3.15), one may write... 

The following sections present descriptions of Helmholtz energy equations of state used to represent the thermodynamic properties of a number of fluids over wide ranges of temperature and pressure. Although the selection of examples is arbitrary, those given here are selected to illustrate both the theoretical basis and the empirical nature of each. 

The Helmholtz energy change due to the formation of hydrogen bonds (association) is captured by A detailed description as well as the corresponding... 

In Chapter 1, we describe the fundamental thermodynamic variables pressure (p), volume (V), temperature (T), internal energy ((/), entropy (5), and moles (n). From these fundamental variables we then define the derived variables enthalpy (//), Helmholtz free energy (A) and Gibbs free energy (G). Also included in this chapter is a review of the verbal and mathematical language that we will rely upon for discussions and descriptions in subsequent chapters. 

The formula for the adiabatic Gibbs potential (Equation 6.51) is exact within the framework of the mean held description. As with the adiabatic Helmholtz potential, we examine it numerically following the Carlson theory of elliptic integrals [15-21], In order to make clear comparisons between the two types of free energy, the width D of the container is set to be 200 nm throughout (except for Figure 6.21). 

Gibbs free energy, G = H -TS, or Helmholtz free energy, F = U - TS, are the key quantities in the description of chemical equilibrium. For a given chemical process, consider the following equations ... 

In following the same mathematical description from Eqs. (2)-(5) it can be shown that the Helmholtz free energy in the event of two competing solutes is equal to... 

MFLG DESCRIPTION OF BINARY SYSTEMS A two component system is in the MFLG approximation treated as a pseudo-ternary mixture of constituents 1, 2 and holes (index 0). The appropriate thermodynamic function for the description of fluid phase equilibria is the Helmholtz free energy of mixing vacant and occupied sites, and reads in the simplest version of the model ). 

---