Helmholtz free energy, thermodynamics

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, > )... 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to... 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

In most applications, thermodynamics is concerned with five fundamental properties of matter volume (V), pressure (/ ), temperature (T), internal energy (U) and entropy (5). In addition, three derived properties that are combinations of the fundamental properties are commonly encountered. The derived properties are enthalpy (//). Helmholtz free energy (A) and Gibbs free energy ) ... 

In addition to the fundamental variables p, V, T, U, and S that we have described so far, three other thermodynamic variables are commonly encountered enthalpy Helmholtz free energy and Gibbs free energy. They are extensive variables that do not represent fundamental properties of the... 

The Helmholtz free energy A is the second of the three derived thermodynamic properties. It is defined as... 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

Consider a thermodynamic system with an external parameter (or constraint) A that can be used to control the state of the system. When changing the control parameter A a certain amount of work is performed on the system. According to the second law of thermodynamics the average work necessary to do that is smaller than the Helmholtz free energy difference between the two equilibrium states corresponding to the initial and final values of the constraint [33]... 

Table 8 gives the results of this thermodynamic analysis for the spreading of film types I and II from the bulk, and the direct transition from film types I and II. It is obvious that the Helmholtz free energies, entropies, and enthalpies are differentiated stereochemically. 

As in the classical expression (25) the quantity ip can be inferred directly as representing the statistical analogue of the Helmholtz free energy. The average behaviour of the canonical ensemble thus obeys the laws of thermodynamics. 

Reciprocal molten salt systems are those containing at least two cations and two anions. We shall deal with the simplest member of this class, that containing the ions A+, B+, X-, and Y-. The four constituents of the solution, AX, BX, AY, and BY, will be designated by 1, 2, 3, and 4 respectively. There are four ions in the system and one restriction of electroneutrality. Consequently, of the four constituents, there are only three which are independent components. In order to calculate the Helmholtz free energy of mixing conveniently, we must (arbitrarily) choose the three components. Here we choose BX, AY, and BY. This choice requires that in order to make mixtures of some compositions a negative quantity of BY must be used. This presents no difficulty in the theory and is thermodynamically self-consistent. One mole of some arbitrary composition (XA, XB, Yx, XY) can be made by mixing Arx moles of BX (component 2), XA moles of AY (component 3), and (XY — XA) moles of BY (component 4). ... 

The analogue to one-component thermodynamics applies to the nature of the variables. So Ay S, U and V are all extensive variables, i.e. they depend on the size of the system. The intensive variables are n and T -these are local properties independent of the mass of the material. The relationship between the osmotic pressure and the rate of change of Helmholtz free energy with volume is an important one. The volume of the system, while a useful quantity, is not the usual manner in which colloidal systems are handled. The concentration or volume fraction is usually used ... 

This simple relationship allows us to express all the thermodynamic variables in terms of our colloid concentration. The Helmholtz free energy per unit volume depends upon concentration of the colloidal particles rather than the size of the system so these are useful thermodynamic properties. If we use a bar to symbolise the extensive properties per unit volume we obtain... 

Define, now, a new thermodynamic function, the Helmholtz free energy A,... 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Thermodynamic properties for explosion calculations are presented for major organic chemical compounds. The thermodynamic properties include enthalpy of formation, Gibbs free energy of formation, internal energy of formation and Helmholtz free energy of formation. The major chemicals include hydrocarbon, oxygen, nitrogen, sulfur, fluorine, chlorine, bromine, iodine and other compound types. 

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

The potential I speak of is usually called the potential of average force. Insofar as it is to be identified to a thermodynamic potential it is a local Helmholtz free energy as a function of the coordinate positions of all the atoms (or radicals) that must change relative positions in the reaction it may be defined by... 

OTHER THERMODYNAMIC POTENTIALS GIBBS AND HELMHOLTZ FREE ENERGY... 

---