Systems Helmholtz free energy

Statistical mechanics enables one to express the chemical potential i, for an ideal gas phase system in terms of the spectroscopic properties of individual gas phase molecules. The reader is referred to standard statistical mechanics texts (e.g. D. A. McQuarrie Statistical Mechanics , reading list) for the development of the relationship between the system Helmholtz free energy, A , and the corresponding canonical partition function Qi... 

A = Helmholtz free energy A0 = reference system Helmholtz free energy Ai = the ith order term in perturbation of A Ax = Helmholtz free energy of a hypothetical pure reference fluid... 

The reference system Helmholtz free energy is given as follows ... 

Helmholtz free energy The maximum amount of energy available to do work resulting from changes in a system at constant volume. See free energy and Gibbs-Helmholtz equation. 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

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 ... 

In the case of an associating fluid with the repulsive-attractive reference system potential, the attractive van der Waals forces between molecules may also be considered in a perturbational manner [114]. The Helmholtz free energy can be written as a sum of three terms... 

The difference in the Helmholtz free energy between the target and the reference systems, AA, can be written in terms of the ratio of the corresponding partition... 

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]... 

Fig. 11.1. The Helmholtz free energy as a function of /3 for the three free energy models of the harmonic oscillator. Here we have set h = uj = 1. The exact result is the solid line, the Feynman-Hibbs free energy is the upper dashed line, and the classical free energy is the lower dashed line. The classical and Feynman-Hibbs potentials bound the exact free energy, and the Feynman-Hibbs free energy becomes inaccurate as the quantum system drops into the ground state at low temperature...

The discussion will be limited to systems at temperatures which are high enough so that we need only consider the classical partition function Q in calculating the Helmholtz free energy A, where, of course... 

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... 

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 ]. 

Lasaga, 1981b), where N-q is the total number of B molecules in the system. Because the chemical potential is related to the partial derivative of the Helmholtz free energy at constant volume ... 

In a finite, one component, system defined by the number of molecules, N, the volume, V, and the temperature, T, the Helmholtz free energy of formation, of a stable droplet is written... 

Doi [4,100,114,117] formulated a(E) for rodlike polymer solutions by noticing that (E> is related to the change 8(AF) in the dynamic Helmholtz free energy of the system due to a virtual deformation k St in a short time 8t by... 

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