Thermodynamic properties Helmholtz energy

An important consideration in process calculations is the change that results in the basic thermodynamic properties, internal energy U), enthalpy H), Helmholtz... 

The Helmholtz free energy A is the second of the three derived thermodynamic properties. It is defined as... 

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... 

This simple relationship allows us to express all the thermodynamic variables in terms of our colloid concentration. The Helmholtz free energy per unit volume depends upon concentration of the colloidal particles rather than the size of the system so these are useful thermodynamic properties. If we use a bar to symbolise the extensive properties per unit volume we obtain... 

Thermodynamic properties for explosion calculations are presented for major organic chemical compounds. The thermodynamic properties include enthalpy of formation, Gibbs free energy of formation, internal energy of formation and Helmholtz free energy of formation. The major chemicals include hydrocarbon, oxygen, nitrogen, sulfur, fluorine, chlorine, bromine, iodine and other compound types. 

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. 

In thermodynamics, we focus on the most important variables needed to describe a system. Although we are interested in the size of a system (or of a phase), we usually do not concern ourselves with the shape of the system. One way in which the shape of a system does influence its thermodynamic properties is through its surface area. The surface of a phase is a different environment than its bulk region. Molecules on the surface of a material do not experience attractive interactions to other molecules in all directions and, therefore, have higher energy than molecules in the bulk of the material. Energy is increased when the surface area of a condensed system (usually a liquid) is increased at constant volume and temperature. Because the Helmholtz free energy has T and V as its natural variables, we can immediately write... 

Once the Helmholtz free energy is calculated, other thermodynamic properties of the system can be obtained using the basic relationships between various quantities. The pressure P, given by... 

Once the free energy is calculated, other thermodynamic properties can easily be determined using the basic thermodynamic relations. For example, pressure p can be obtained by differentiating the Helmholtz free energy F ... 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

In the last equations A(n) is the Helmholtz free energy of the total NVT system fully defined by the vector n = nm, n 2,... providing the molecular number in each tiny volume and iq) is the chemical potential at a given iq position, i.e., within the corresponding tiny volume. Note that the molecular number can be used as a continuous variable, given the fact that for any thermodynamic property in a macroscopic system the variation due to a single molecule is virtually equivalent to a differential. From the definition of the chemical potential and probability density in the r) space p(t, ti), we readily have... 

This derivation has been taken from the book Colloidal Dispersions by Russel, Saville, and Schowalter [30]. The thermodynamic properties of a ssrstem can be evaluated from the Helmholtz free energy, A, as a function of the volume and temperature ... 

If we take account of the thermodynamic properties of the material the energy density iPE should, more properly, be described as the density of Helmholtz free energy. 

We remarked in connection with Eq. (16.13) that the partition function Z is a state function from which all thermodynamic properties may be found once it is known as a function of T and V. Its relation to the Helmholtz energy follows from Eq. (16.19), which may be written ... 

Thus, when we know how A/RT (or In Z) is related to its canonical variables, T and V, we can evaluate all other thermodynamic properties by simple mathematical operations. The Helmholtz energy and therefore tire partition function serve as generating functions for tire otlrer tlrenrrodyiramic properties, and implicitly represeirt complete property iirfonrration. 

The values in Tables 2-23 to 2-26 were generated from the NIST REFPROP software (Lemmon, E. W, McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23 Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002). The primary source for the properties of aqueous ammonia mixtures is R. Tillner-Roth and D. G. Friend, A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture Water + Ammonia, /, Phys. Chem. Ref. Data 27 63-96(1998). 

Selected thermodynamic properties of liquid COFj have been estimated [1683]. These calculated values for the Helmholtz free energy function (-G/RT), liquid entropy (5), and heat capacity at constant volume (C,) are recorded in Table 13.11. 

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