The free energy and chemical potentials

The major goal of the theoretical developments will be a clear and practical access, on the basis of molecular information, to the chemical potential for a species of type a. The combination 

The simplest examples of chemical potentials occur when interactions between molecules are negligible. The chemical potentials are then evaluated as 

These relations will be established specifically in Chapter 3. j3 = kT, where k is the Boltzmann constant, and A is the thermal de Broglie wavelength. Q Ua = 1) = ISa is the canonical ensemble partition function of a system 

On this basis we then consider conformational and chemical equilibrium in turn (Widom, 2002, see Chapter 3)  

We emphasize that these results are for the ideal case and we will seek a natural generalization later. 

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Ion-Dipole Forces. Ion-dipole forces bring about solubihty resulting from the interaction of the dye ion with polar water molecules. The ions, in both dye and fiber, are therefore surrounded by bound water molecules that behave differently from the rest of the water molecules. If when the dye and fiber come together some of these bound water molecules are released, there is an increase in the entropy of the system. This lowers the free energy and chemical potential and thus acts as a driving force to dye absorption. 

The free-energy and chemical potentials of the solvent is slightly altered from the case of small solute molecules (Flory, 1970) ... 

Herman and Edwards [55] extended the Brochard and de Gennes approach [53] by considering in detail the stress accompanying the swelling of the polymer within the reptation model. They evaluated the contributions to the free energy and chemical potentials due to the deformation of the polymer due to swelling. The chemical potential of the solvent, pi, was obtained by taking two contributions into account. The first was the classical osmotic pressure... 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

The ideas underlying elemental structures models are to establish microstructures experimentally, to compute free energies and chemical potentials from models based on these structures, and to use the chemical potentials to construct phase diagrams. Jonsson and Wennerstrom have used this approach to predict the phase diagrams of water, hydrocarbon, and ionic surfactant mixtures [18]. In their model, they assume the surfactant resides in sheetlike structures with heads on one side and tails on the other side of the sheet. They consider five structures spheres, inverted (reversed) spheres, cylinders, inverted cylinders, and layers (lamellar). These structures are indicated in Fig. 12. Nonpolar regions (tails and oil) are cross-hatched. For these elemental structures, Jonsson and Wennerstrom include in the free energy contributions from the electrical double layer on the water... 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

To apply the preceding concepts of chemical thermodynamics to chemical reaction systems (and to understand how thermodynamic variables such as free energy vary with concentrations of species), we have to develop a formalism for the dependence of free energies and chemical potential on the number of particles in a system. We develop expressions for the change in Helmholtz and Gibbs free energies in chemical reactions based on the definition of A and G in terms of Q and Z. The quantities Q and Z are called the partition functions for the NVT and NPT systems, respectively. 

We begin by writing the chemical equilibrium constant Keq for the equilibrium between solid and liquid, Keq = [liquid]/[solid]. This quantity is determined by the difference between the free energies AF of the two forms, Keq = exp[—AF/k-fT], where kB is the Boltzmann constant and T is the absolute temperature. But AF is the difference in the chemical potentials Ap, multiplied by N. the number of particles in each system. We know that the traditional condition for equilibrium of two phases, e.g., solid and liquid, is the equality of the free energies or chemical potentials of the two forms. This is what sets the conditions for the coexistence curves required by the phase mle. 

In this question we use absolute free energies. Their relative magnitudes are more or less reasonable, but of course in real life you never get to deal with quantities like these. The exercise is useful, however, because you should get used to the idea that free energies and chemical potentials are finite, absolute quantities, even if unknown. 

If one employs the partition function Z for an ideal gas, as given by equations (28-59) and (28-87), then the Helmholtz free energy and chemical potential are calculated by combining classical and statistical thermodynamics ... 

F. Free Energy and Chemical Potential the Stumbling Block of Conventional Canonical Ensemble Calculations... 

The Relationship Between Free Energies and Chemical Potentials... 

Fig. 15.2 The molar free energy and chemical potential of Ga at each site as a function of Ga occupancy (a) in the D site of LGS (langasite) and (b) B site of LTG (langatate).

Entropies, free energies, and chemical potentials are typically needed in relative terms, as differences between distinct macroscopic states. For example, the sign of the entropy change provides information on the direction in which a change of state process will spontaneously move. Nonetheless, it is impossible to determine an actual value for the absolute entropy or free energy of a macroscopic system. There are at least two reasons. 

The alternative to direct simulation of two-phase coexistence is the calculation of free energies or chemical potentials together with solution of the themiodynamic coexistence conditions. Thus, we must solve (say) pj (P) = PjjCT ) at constant T. A reasonable approach [173. 174. 175 and 176] is to conduct constant-AT J simulations, measure p by test-particle insertion, and also to note that the simulations give the derivative 3p/3 7 =(F)/A directly. Thus, conducting... 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gibbs free energy or chemical potential of the solute in solution. This chemical potential reflects the conformational degrees of freedom of the solute and the solution conditions (temperature, pressure, and solvent composition) and provides the driving force for solute conformational transitions in solution. For a simple solute with no internal structure (i.e., no intramolecular degrees of freedom), this chemical potential can be expressed as... 

The concept of substance activity was derived by Gilbert N. Lewis in 1907 from the laws of equilibrium thermodynamics and is described in detail in the text entitled Thermodynamics and the Free Energy of Chemical Substances by Lewis and Randell (1923). In a homogeneous mixture, each component has a chemical potential (jjl), which describes how much the free energy changes per mole of substance added to the system. The chemical potential of water (pw) in a solution is given by... 

Equation D3.5.13 illustrated that the free energy of an interfacial system can be expressed in terms of the interfacial tension and chemical potential of the overall system. A simple differentiation or alternatively the reutilization of the definition of the interfacial tension used in Equation D3.5.7 at constant pressure and temperature yields ... 

Similar attempts were made by Likhtman et al. [13] and Reiss [14]. Reference 13 employed the ideal mixture expression for the entropy and Ref. 14 an expression derived previously by Reiss in his nucleation theory These authors added the interfacial free energy contribution to the entropic contribution. However, the free energy expressions of Refs. 13 and 14 do not provide a radius for which the free energy is minimum. An improved thermodynamic treatment was developed by Ruckenstein [15,16] and Overbeek [17] that included the chemical potentials in the expression of the free energy, since those potentials depend on the distribution of the surfactant and cosurfactant among the continuous, dispersed, and interfacial regions of the microemulsion. Ruckenstein and Krishnan [18] could explain, on the basis of the treatment in Refs. 15 and 16, the phase behavior of a three-component oil-water-nonionic surfactant system reported by Shinoda and Saito [19],... 

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