Helmholtz s free energy

This equation, Keenan s equation, can also be used for Helmholtz s free energy. It clearly shows that we can discuss any process in terms of the available, usable, useful or free energy, effectively used, rather than in terms of entropy changes times the temperature, which correspond to the left overs of such process. 

By using the concept of Helmholtz s free energy per unit volume hi = Pii i — GiVi) an(J the balances of mass (2) and energy (6), we can recast the entropy equation (7) in the following form ... 

The corresponding change of Helmholtz s free energy, F, will be... 

The evaluation of the integral requires one to know the changes of pressure and volume during the explosion. Therefore it is difficult to be evaluated. It is easier to relate the thermodynamic states before and after the reaction with each other. This is achieved with the help of Helmholtz s free energy f. We have (cf. [5])... 

Helmholtz s energy represents the maximum amount of work which a system can exert on its surroundings. Hence, it constitutes an upper bound for the work done by the explosion. Helmholtz s free energy is not normally found in tables. Therefore calculations based on Eq. (2.46) make use of the internal energy and entropy differences. We then have... 

A more rigorous derivation of Equation 2.1 for a solid surface can be found using interfadal thermodynamics. Small changes in Helmholtz s free energy can be written by generalizing Equation 1.14 for multiple interfaces. If temperature and the individual phase volumes are constant, the result is... 

However, the specific entropy s is not a convenient independent variable as it is intuitively difficult to comprehend and practically difficult to control. The classical approach consists of introducing another state function, the specific Helmholtz s free energy, via the Legendre transform ... 

In the latter case, the internal consistency requirement resulted in a universal value of three for the proportionality factor between the interfacial tension and the integral of the Helmholtz s free-energy density difference, Argo- In the third case [107], we have also incorporated density gradient contributions into, both, the equation for the chanical potential and the equation of state. The striking result was that a universal value of four was then obtained for the aforementioned proportionality factor. In the following text, we will present the basic formalism, starting from the second case [7], and we will discuss some representative applications. 

EoS models can also be used in the frame of the gradient approximation, such as the Cahn-Hilliard theory [100] of inhomogeneous systems, for the description of surface properties. In the frame of this theory, the Helmholtz s free-energy density r in a one-component inhomogeneous system can be expressed as an expansion of density p and its derivatives ... 

Once solved Laplace s equation for the bounded problem, we are able to determine the potentials within the cavity, c, and in the bulk, e. The charges within the cavity induce a polarisation in the dielectric, giving rise to a reaction potential, OR(ri), which acts back on the dissolved charges. Once determined the analytical expression for R(rt), Helmholtz s free energy of this interaction is just the difference between the reversible work of assembling the charge distribution in the presence of the dielectric and under vacuum, and simply reads ... 

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as... 

The variation of the Helmholtz energy during dispersion of a homogeneous metal with volume V to particles with radius r can be expressed in Eq. (8-2) where a is the increase in the system s free energy due to the formation of a unit of new interface surface. 

For example, imagine a liquid droplet, the volume of which is fixed at a given temperature, while the only way to lower the droplet s free energy is to change its shape. In this case, the Helmholtz free energy of the liquid droplet is merely given as... 

The preceding free energy functional is a trivial generalization of van der Waal s free energy functional. In a similar spirit, one can also write the Helmholtz free energy functional for solids (metastable and stable) as... 

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

N, Number of particles P, Pressure V, Volume T, Temperature E, Energy fi. Chemical potential A, Helmholtz free energy S, Entropy G, Gibbs free energy. 

A = work function (Helmholtz free energy), Btu/lb or Btu C = heat capacity, Btu/lb °R Cp = heat capacity at constant pressure = heat capacity at constant volume F= (Gibbs) free energy, Btu/lb or Btu g = acceleration due to gravity = 32.174 ft/s ... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

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... 

---