Helmholtz energy derivatives

Equations (6.37) through (6.39) for the Gibbs energy have as their counterparts analogous equations for the Helmholtz energy. Derived from Eqs. (6.9) and (6.2), they are... 

The molar Helmholtz energy derived from Equation (2.33) is... 

All thermodynamic properties can be derived from the partition function. It can be shown that the Helmholtz energy, A, is related to Zby the simple expression... 

Two other important derivatives of the Gibbs and Helmholtz energies with respect to the temperature can be derived from these simpler relations. From the definition of the Gibbs energy and Equation (4.37) we obtain... 

Every coefficient in Equation (5.112) except that of (ST)2 can be expressed more simply as a second derivative of the Helmholtz energy. We take only the coefficient of (SV)2 as an example. Both (dE/dV)Sn and (cM/5K)r are equal to — P. The differential of (dE/dV)Sn is expressed in terms of the entropy, volume, and mole numbers and the differential of (dA/dV)Tmole numbers, so that at constant mole numbers... 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

Derivation of the expression for the minimum production of S in the systems with constant T and V (volume) differs from the one above only by replacement of enthalpy by internal energy (U) and the Gibbs energy by the Helmholtz energy in the equations. When we set S and P or S and V dissipation turns out to be zero according to the problem statement. In the case of constant U and V or H and P, the interaction with the environment does not hinder the relaxation of the open subsystem toward the state max Sos. 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

Because of the landmark nature of van der Waals work we shall now discuss some important aspects of his theory. In doing so a selection has to be made (the German version of van der Waals paper runs to over a hundred pages ). We shall use FICS-nomenclature and follow as much as possible van der Waals own arguments and derivations, although parts of the latter can nowadays be carried out more efficiently. For instance, the minimization of the Helmholtz energy as a function of the profile shape can nowadays be elegantly done by variational calculus, the principles of which will be outlined in appendix 3. 

Probably this equation remains valid for small deviations from equilibrium ). For colloid scientists, [4.5.30] is not unfamiliar, because in three dimensions the modulus is related to the derivative of the disjoining pressure /7(h) with respect to the distance h between interacting particles and where /7(h) is, in turn, the derivative of the Gibbs or Helmholtz energy of interaction. There is a formed difference... 

Functionals are functions of functions. In this Volume we met functionals in van der Waals theory for the interfacial tension (sec. 2.5a) and in the mean field theory for the surface pressure of polymeric monolayers (sec. 3.4e). In these two cases equations were derived in which the excess interfacial Helmholtz energy had to be minimized as a function of a density distribution across the interface and of the spatial derivative of this profile [density Junctionals). The technique of finding the function that minimizes the Helmholtz energy is called variational calculus, or calculus of variations. 

Activity coefficient models are equations representing the Gibbs or the Helmholtz energy of solutions. Activity coefficients and related properties are derived form these energy functions by proper differentiation (Equation (1)). 

From the quantities derived above, one may construct the internal energy of solvation (AE = AH — PAV ) and the Helmholtz energy of solvation (AA = AG — PAV ). As noted in section 7.2, the difference between AE and AH and between AA and AG is usually very small and may be neglected for most systems of interest discussed in this book. For more details see Ben-Naim (1987). 

We will begin the derivation with Helmholtz energy, as it is the natural energy function for the independent variables T and V of equations of state. By the fundamental differential equation for A, Equation (4.81)... 

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... 

Note that according to Eqs. (15) and (16) the free Helmholtz energy serves as a potential for the stresses T and for the microstructural flux S. According to the assumption of elastic material behavior, results of this type have to be expected. The additional balance equation for k [Eq. (17)] possesses the same structure as the balance of equihbrated forces obtained in Refs. [14, 20, 33] and appHed, e.g., in Ref [36]. Following the MuUer-Liu approach, Svendsen [39] also derived a generalization of Eq. (17) for a model with scalar-valued stractural parameters. 

Closer inspection of Eq. (2.34) indicates that it is possible to form also derivatives of the same kind, e.g., from the Helmholtz energy, so it is advisable to add the set of variables to the respective susceptibility. 

In this section we present those class 1 and class 11 derivatives that show how properties respond to changes in temperature. First, we consider the effects of temperature changes on two measurables— pressure and volume then we describe the effects on internal energy, enthalpy, and entropy and finally, we present the effects on Gibbs and Helmholtz energies. 

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