Reversible processes Helmholtz energy

For a system at constant temperature, this tells us that the work done is less than or equal to the decrease in the Helmholtz free energy. The Helmholtz free energy then measures the maximum work which can be done bv the system in an isothermal change For a process at constant temperature, in which at the same time no mechanical work is done, the right side of Eq. (3.5) is zero, and wo see that in such a process the Helmholtz free energy is constant for a reversible process, but decreases for an irreversible process. The Helmholtz free energy will decrease until the system reaches an equilibrium state, when it will have reached the minimum value consistent with the temperature and with the fact that no external work can be done. 

For a vstem at constant pressure and temperature, we see that the Gibbs free energy is constant for a reversible process but decreases for an irreversible process, reaching aminimum value consistent with the pressure and temperature for the equilibrium state just as for a system at constant volume the Helmholtz free energy is constant for a reversible process but decreases for an irreversible process. As with A, we can get the equation of state and specific heat from the derivatives of <7, in equilibrium. We have... 

One should not conclude from Eq 4.2-7 that the reversible work for any process is equal to the change in Helmholtz energy, since this result was derived only for an isothermal, constant-volume process. The value of VK , and the thermodynamic functions to which it is related, depends on the constraints placed on the system during the change of state (see Problem 4.3). For example, consider a process occurring in a closed system at fi.xed temperature and pressure. Here we have... 

Since a reversible change provides the maximum (minimum) amount of work for a given expansion (compression), the change in Helmholtz energy provides a bound on the work associated with an isothermal process. 

For isothermal processes on closed systems, the reversible work is given by the change in Helmholtz energy, as already noted in (3.2.15). 

The above treatment can be applied to the calculation of the energetics of nucleation [100]. To this end, one considers the formation of an embryo within a bulk phase of initial density p. At constant temperature and volume, the reversible work of embryo formation is simply the change in Helmholtz energy associated with the above process. 

The minimization of Helmholtz free energy is a very useful principle. Many interesting features such as phase transitions and formation of complex patterns in equilibrium systems [2] can be understood using this principle. One can also show that the Helmholtz free energy F is the energy that is free, available to do work in a reversible process (Example 5.1) — hence the name free energy. ... 

The relations (b)...(e) are called the fundamental equations of thermodynamics. Derivation of these relations had to be limited to reversible processes during which only volume work occurred. However, since internal energy U, enthalpy H, the Gibbs free energy G, and the Helmholtz free energy A are all state functions, the relations derived apply to any change - reversible as well as irreversible - connecting states of equilibrium in a system. Thus, in conclusion we have... 

Consider a film of liquid on a wire balance as shown in Fig. 1. A force is applied to a frictionless, movable wire, stretching the film, which adheres to the wire. The process occurs at constant temperature and volume of the liquid. The reversible work done as the movable wire moves a distance dx (at constant T and V) is equal to the increase in the Helmholtz free energy of the system ... 

In the above we have invented a new function of state, A = E — TS, involving state functions that had been previously introduced. A is called the Helmholtz (free) energy function. The right-hand side follows from Eq. (1.12.7d). As is seen, changes in A are tracked by the reversible performance of work at constant T. If no work is involved, but irreversible (and therefore, uncontrollable) processes are allowed to occur at constant temperature, we find from the above that... 

Gibbs free energy is a potential for reversible work in constant T-P processes, and always decreases in spontaneous processes. By comparison with (5.26) it is clear that G is a measure of the net work or non-PAV work. This function therefore contrasts with the Helmholtz work function, which measures total work, including mechanical PY work. The Gibbs free energy is a particularly useful measure of the electrical or chemical work attainable from a process and is used a great deal with chemical systems where PY work is often unimportant. 

We consider first the nucleation under an applied simple shear stress a without the presence of an accompanying mean normal stress We consider the nucleation as a reversible experiment in which the transformation strain in the volume element Qf can be built up by an imposed external shear strain y. As is clear, and as discussed by Orowan (1954), this results in a rise in the Helmholtz free energy AF(y) that initially is given by a quadratic function and for the whole process can be taken as being given by a cosine potential (Kocks et al. 1975) (see also Johnson and Samwer (2005)),... 

The minus in (7.22) implies that the work performed by an external force is expended for increase in the Helmholtz free energy of dF. Since for a reversible and equilibrium process dF = —dW the surface contribution to the free energy must be proportional to the increment in the surface area,... 

The Helmholtz free energy (or Helmholtz fimctlon),f, is defined by f/-7S, where f/is the "internal energy. For a reversible isothermal process, AFrepresents the useful work available. 

The energy change for any equimolar process occurring at constant temperature is a work process. If the isothermal, equimolar process is carried out reversibly at constant pressure, the work is Gibbs free energy. If the isothermal process is carried out reversibly at constant volume, the work is Helmholtz free energy. 

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