The Free Energy and Work Functions

The Work Function.— Although the entropy concept is the fundamental consequence of the second law of thermodynariics, there are two other functions, which utilize the entropy in their derivation, that are more convenient for use in many instances. One of these, the free energy, will be employed extensively in subsequent portions of this book in connection with the study of equilibria, both chemical and physical, and the direction of chemical change. 

In order to obtain some understanding of the physical significance of the work function, consider an isothermal change from the initial state indicated by the subscript 1 to the final state indicated by 2 thus, 

If AS in equation (25.2) is replaced by Qny./T, where Or v. is the heat taken up when the given change is carried out in a reversible manner, then 

Since AAt and AEt are completely defined by the initial and final states of the system, the results obtained above, e.g., equations (25.3) and (25.5), provide proof of the statement made earlier ( 8b), that in any isothermal reversible process the values of W and Q are definite, depending only on the initial and final states. The work term W here includes all the forms of work performed on or by the system. 

The Free Energy.— The second, and more generally useful, function derived from the entropy is called the free energy, and is defined by 

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Theoretically, once the free energy and the dissipation function are specified, the stress-strain relation is fully defined. This is therefore one possible way to construct a constitutive model. However the above reasoning does not work for plasticity. 

The work of adsorption is determined from Eq. (1) by using the Hnear section of the surface pressure curve as a function of the bulk concentration of the surfactant (the surface pressure does not exceed 3mN/M). The values obtained for the work of adsorption are usually compared with the values obtained for certain fractional surface coverages. Thus, it becomes possible to estimate the change in the free energy as a function of the molecular interaction in the monolayer. (The relation between the values of the work of adsorption determined from different isotherms is discussed in Refs. [37-39]). The adsorption entropy and enthalpy are determined from the temperature dependence of the work of adsorption [36-39]. 

The Helmholtz energy has been known to physicists as the free energy and as the Helmholtz function. It has been known to chemists as the work function and as the Helmholtz free energy. The differential of the Helmholtz energy is... 

The minimum free energy approach to a network has been used to describe the stress-strain behavior of hydrogen-bonded solids at small strains [9]. The present forms of the stress tensor and work function (in two dimensions) have been applied to the fiber network problem [7,8], for which the minimizing constraints were taken to be the stress tensor components. In the case of an elastomer network, the stress constraints must be replaced by deformation constraints if the admissable solutions to the problem are to be restricted to constant volume deformations. 

This expression shows that the maximum possible useful work (i.e., reversible work) that can be obtained from any process occurring at constant temperature and pressure is a function of the initial and final states only and is independent of the path. The combination of properties U + PV - TS or H - TS occurs so frequently in thermodynamic analysis that it is given a special name and symbol, F, the free energy (sometimes called the Gibbs Free Energy). Using this definition, Equation 2-143 is written... 

In the investigation of the properties of the free energy no assumption has been made as to the nature of the external work At. Let us now assume that there is some function 12 of the variables defining the physical and chemical state of the system, such that ... 

Since the bias function should enhance the sampling of pathways with important work values it can be made to depend on the work only, ir[z 2 ) = n W( (. Z))]. To minimize the statistical error in the free energy difference the bias function needs to be selected such that both the statistical errors of the numerator and the denominator of (7.44) are small. Ideally, the bias function should have a large overlap with both the unbiased work distribution P(W) and the integrand of (7.36), P (W) exp (—j3W). Just as Sun s work-biased ensemble Pa[z( ), the biased path ensemble )] can... 

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