Chemical potential description

The McMillan-Mayer theory allows us to develop a fomialism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

The swelling pressure or osmotic deswelling data can be, therefore, described as the functions of n(w) by either of the theories [115]. This description can be then applied to determining the network parameters (see, for example, Ref. [22]). On the other hand, the swelling pressure which is directly connected with the chemical potential of water in the gel ... 

Thermodynamically the quantitative treatment of both active and passive processes requires them to be downhill or exoergic. The description of chemical potential as a function of mole fraction follows the same form as before for a neutral species (Section 8.2) ... 

Since the interplay of theory and experiment is central to nearly all the material covered in this chapter, it is appropriate to start by defining the various concepts and laws needed for a quantitative theoretical description of the thermodynamic properties of a dilute solid solution and of the various rate processes that occur when such a solution departs from equilibrium. This is the subject matter of Section II to follow. There Section 1 deals with equilibrium thermodynamics and develops expressions for the equilibrium concentrations of various hydrogen species and hydrogen-containing complexes in terms of the chemical potential of hy-... 

For historical reasons, the incompressible lattice-fluid system description is used, even if the distribution of one of the components is coupled to the distribution of vacant sites. Constant pressure SCF calculations are the same as constant chemical potential calculations for the vacant sites. These conditions are used below. 

Finally, it is important to mention that in the case of the HF method, the calculation of the chemical potential and the hardness, through energy differences, to determine I and A, leads, in general, to a worst description than in the KS approach, because the correlation energy is rather important, particularly for the description of the anions. However, the HF frontier eigenvalues provide, in general, a better description of p, and 17, through Equations 2.48 and 2.49, because they lie closer to the values of — I and —A than the LDA- or GGA-KS values, as established by Koopmans theorem. 

Even this more elaborated description of ion movements in response to gradients of chemical potential may turn out to be insufficient, in particular when uphill diffusion is active ... 

For the description of a pure quark phase inside the neutron star, as for neutrino-free baryonic matter, the equilibrium equations for the chemical potentials,... 

A description of the trapping process may be based on the chemical potentials of the hydrogen absorbed in the metal. In the general case, the chemical potential of interstitial or diffusible hydrogen may be described by the equation... 

The coefficients Lu, L2A, and L34 describe the viscous flow contributions of the transport of all three species in a total pressure gradient totai- Because a pressure gradient also imposes a chemical potential gradient on each species (eq 24), experimentally, there is always a superposition of diffusive and viscous flow e.g., for the description of the water flux in a total pressure gradient, all coefficients must be included, i.e.. 

Phase behavior of lipid mixtures is a much more difficult problem, due to nonideal mixing of lipid components. Ideal mixing implies like and unlike lipids have the same intermolecular interactions, while nonideal mixing results from differential interactions between lipid types. If the difference is too great, the two components will phase separate. While phase separation and lateral domain formation have been observed in many experiments, we lack a molecular-level physical description of the interactions between specific lipids that cause the macroscopic behavior. The chemical potential of a lipid determines phase separation, as phase coexistence implies the chemical potential of each type of lipid is equal in all phases of the system [3,4],... 

The ideal gas data in these tables were calculated directly from a statistical mechanical description of the isolated molecules. In terms of the quantities defined in these tables, jL02(T,p°)=[H°(T) H°(Tr) TS(T) [H°(0) H°(Tr). This expression is derived by noting that the chemical potential of an ideal gas is equal to the ideal gas free energy, G = H TS. 

Chemical reactions at supercritical conditions are good examples of solvation effects on rate constants. While the most compelling reason to carry out reactions at (near) supercritical conditions is the abihty to tune the solvation conditions of the medium (chemical potentials) and attenuate transport limitations by adjustment of the system pressure and/or temperature, there has been considerable speculation on explanations for the unusual behavior (occasionally referred to as anomalies) in reaction kinetics at near and supercritical conditions. True near-critical anomalies in reaction equilibrium, if any, will only appear within an extremely small neighborhood of the system s critical point, which is unattainable for all practical purposes. This is because the near-critical anomaly in the equilibrium extent of the reaction has the same near-critical behavior as the internal energy. However, it is not as clear that the kinetics of reactions should be free of anomalies in the near-critical region. Therefore, a more accurate description of solvent effect on the kinetic rate constant of reactions conducted in or near supercritical media is desirable (Chialvo et al., 1998). 

It is seen in Eqs. (11) and (12) that the diffusion coefficients are superficially comprised of two factors a frictional term as represented by f12 or f21 and a thermodynamic term 0 /01 or 0p2/0m2. However, some caution should be levelled at this description because the two terms are closely connected as seen by Eqs. (8) and (9) which describe the direct relationship between the gradient of the chemical potential and the frictional term. 

A2 from equation (5.16) or the cross second virial coefficient from equation (5.17). In turn, this knowledge of the second virial coefficients and their temperature dependence allows calculation of the values of the chemical potentials of all components of the biopolymer solution or colloidal system, as well as enthalpic and entropic contributions to those chemical potentials. On the basis of this information, a full description and prediction of the thermodynamic behaviour can be realised (see chapter 3 and the first paragraph of this chapter for the details). 

From this descriptive introduction, it follows that the coherent spinodal decomposition is a continuous transport process occurring in a supersaturated matrix. It is driven by chemical potential gradients. Strain energy and concentration gradient energy have to be adequately included in the component chemical potentials. We expect that the initial stages of decomposition are easier to treat quantitatively than the later ones. The basic result will be the (directional) build-up of periodic variations in concentration [J.W. Cahn (1959), (1961), (1968)]. 

In considering physicochemical equilibria, that is to say, if one is interested in the internal constitution of a system in equilibrium when changes of phase and chemical reactions are admitted, one introduces the constitutive coordinates this being the number of moles of the ith constituent Ct in the a th phase. The definitions of Equations (10) through (12) remain unaltered, for die nf do not enter into the description of the interaction of the system with its surroundings. Let an amount dnf of C be introduced quasi,statically into the a th phase of the system. The work done on K shall be fi dnt> The quantity fif so defined is the chemical potential of C, in die ct th phase. It is in general a function of all the coordinates of K. Then, identically. 

For a special choice of the chemical potential the grand canonical description as used here was suggested by Des Cloizeaux [Clo75], The generalization to arbitrary pp(n) is due to Schafer and Witten [SW80]. 

It has been said chemists have solutions 3 Solutions are involved in so many chemical processes1 that we must have the mathematical tools to comfortably work with them, and thermodynamics provides many of these tools. Thermodynamic properties such as the chemical potential, partial molar properties, fugacities, and activities, provide the keys to unlock the description of mixtures. 

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