Chemical potential total

The electrochemical potential is defined as the total work of bringing a species i from vacuum into a phase a and is thus experimentally defined. It.may be divided into a chemical work p , the chemical potential, and the electrostatic work ZiC0 ... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The total number of independent equations is therefore (tt — )N + r In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations. 

Abbreviations N is the total number of particles Pdim is the average density of dimers in each of the parts I and III is the average density of monomers in each of the parts II and IV p is the average density at the middle of parts II and IV this value has been used to calculate the excess chemical potential from Eq. (148). All remaining symbols are explained in the text. 

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

More simply, the chemical potentials appear as Legendre multipliers when G is minimised, associated with the constraints on total numbers of atoms, eg. (23). The results are ... 

Figure 7.87 shows a AG -concentration diagram for Fe(j,-Zn( ). It was constructed from the experimental data shown in Table 7.37. The method of construction is described elsewhere. Figure 7.87 can now be used, by applying the constraints imposed by the tangency rule, to explain why in Fig. 7.88a and b, where the chemical potentials (shown in the diagram) of zinc vapour varied between 0 and - 1 - 81 kJ molthe total interaction surface layer consisted of T, T, 6, and flayers in Fig. 7.88c at a chemical potential only slightly lower ( — 2-11 kJmol ) only T and T, layers were present whilst at -2-55 kJ mol only a F outermost layer was formed. 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

The chemical potential of each component is known when the total potential of the solution is determined as a function of the composition. 

Equation (5.21) is especially important in that it indicates that p, the chemical potential of the ith component in the mixture is the contribution (per mole) of that component to the total Gibbs free energy/... 

Gibbs change in free energy (AG) is that portion of the total energy change in a system that is available for doing work—ie, the usefiil energy, also known as the chemical potential. 

We are now ready for computing the electron chemical potential within the u> scheme. Since ours is a Htickel-like scheme, the total energy Etot is the sum of the orbital energies multiplied by the pertinent occupations, and therefore... 

In metal deposition, the primary products form adsorbates on the electrode surface rather than a supersaturated solution. Their excess chemical potential is directly related to polarization and given by nFAE. The total excess surface energy = 2 S,o,. Otherwise, all the results described above remain valid. 

Stresses caused by chemical forces, such as hydration stress, can have a considerable influence on the stability of a wellbore [364]. When the total pressure and the chemical potential of water increase, water is absorbed into the clay platelets, which results either in the platelets moving farther apart (swelling) if they are free to move or in generation of hydrational stress if swelling is constrained [1715]. Hydrational stress results in an increase in pore pressure and a subsequent reduction in effective mud support, which leads to a less stable wellbore condition. 

At the contact of two electronic conductors (metals or semiconductors— see Fig. 3.3), equilibrium is attained when the Fermi levels (and thus the electrochemical potentials of the electrons) are identical in both phases. The chemical potentials of electrons in metals and semiconductors are constant, as the number of electrons is practically constant (the charge of the phase is the result of a negligible excess of electrons or holes, which is incomparably smaller than the total number of electrons present in the phase). The values of chemical potentials of electrons in various substances are of course different and thus the Galvani potential differences between various metals and semiconductors in contact are non-zero, which follows from Eq. (3.1.6). According to Eq. (3.1.2) the electrochemical potential of an electron in... 

The quantum contribution to the excess chemical potential is roughly 10% of the total. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

Chemical potential, p, is another name for total internal energy. Convenient units for it are energy per mole. In terms of the work done on (PdV), and the entropy (S) of a gaseous or liquid substance, it may be written in differential form (Callen, 1960) ... 

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