Helmholtz and Gibbs Free Energies

They are defined through the internal energy U and entropy S (Section 

The Helmholtz free energy is of more theoretical importance, used often in statistical mechanics. 

The Gibbs free energy, on the other hand, is very important from the practical point of view, for it represents the basic quantity in chemical and/or phase equilibrium calculations. 

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Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U enthalpy, H and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows ... 

The first and second laws of thermodynamics and the Helmholtz and Gibbs free energies are rearranged to obtain the relationships between the state functions (i.e., E, H, A, and G) and temperature, pressure, and volume. For an infinitesimal process the first law is given by ... 

The other thermodynamic functions are also easily found. If we confine ourselves to low pressures, of the order of atmospheric pressure, we can neglect the term PV in the Gibbs free energy of liquid or solid. Then the Helmholtz and Gibbs free energies are approximately equal and are given by... 

To apply the preceding concepts of chemical thermodynamics to chemical reaction systems (and to understand how thermodynamic variables such as free energy vary with concentrations of species), we have to develop a formalism for the dependence of free energies and chemical potential on the number of particles in a system. We develop expressions for the change in Helmholtz and Gibbs free energies in chemical reactions based on the definition of A and G in terms of Q and Z. The quantities Q and Z are called the partition functions for the NVT and NPT systems, respectively. 

Before setting out on the exact mean field theory solution to the one-dimensional colloid problem, I wish to emphasize that the existence of the reversible phase transition in the n-butylammonium vermiculite system provides decisive evidence in favor of our model. The calculations presented in this chapter are deeply rooted in their agreement with the experimental facts on the best-studied system of plate macroions, the n-butylammonium vermiculite system [3], We now proceed to construct the exact mean field theory solution to the problem in terms of adiabatic pah-potentials of both the Helmholtz and Gibbs free energies. It is the one-dimensional nature of the problem that renders the exact solution possible. 

To clarify the different roles played by the Helmholtz and Gibbs free energies of ionic solutions, it is relevant to reconsider the derivation of these thermodynamic quantities in the original Debye-Hiickel theory [1—4],... 

Equations (5.8.6a) through (5.8.6d) represent straightforward extension of results cited in Section 1.18. The new relations specify electric and magnetic polarizations in terms of Helmholtz and Gibbs free energies. Note, however, that it is the total pressure, inclusive of magnetic effects, that is held fixed. 

Derivation of Helmholtz and Gibbs Free-Energy Departure Functions from the Peng-Robinson Equation of State at Infinite Pressure... 

We use the terms Helmholtz and Gibbs energies for what has previously been referred to as Helmholtz and Gibbs free energies, respectively. 

For both Helmholtz and Gibbs free energies the term free indicates that only the portion (U - TS) of the internal energy is free to perform work. For differential changes in Gibbs free energy and enthalpy, it becomes... 

We note that different forms of the first law have been promulgated by various authors. Equation (14-11) is the correct form in the general case in which the dielectric is deformable. We define the Helmholtz and Gibbs free energies of the system by the relations... 

A thermodynamic analysis (2) shows that the Helmholtz and Gibbs free energies of the system are... 

A and G are referred to as the Helmholtz and Gibbs free energy, respectively. Note that terms enthalpy and free energy allude to heat and work, respectively. The terminology subscribes to the circumstances where the potentials are most often directed. For instance ... 

The method used to obtain a relation (6.2.7) between I/ideai and I/reai can also be used to relate the corresponding Helmholtz and Gibbs free energies. The main idea is that as p 0 or y —> oo, the thermodynamic quantities for a real gas approach those of an ideal gas. Let us consider the Helmholtz free energy F. Since (0F/0y)y = —p (5.1.6) we have the general expression... 

H is the chemical potential of component i. The subscripts v and rij, indicate differentiation with total volume and all n s other than m, held constant. A and F are the molar Helmholtz and Gibbs free energies, respectively. 

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