State chemical potential

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

For a free energy of fonnation, the preferred standard state of the element should be the thennodynamically stable (lowest chemical potential) fonn of it e.g. at room temperature, graphite for carbon, the orthorhombic crystal for sulfiir. 

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 chemical potential for an ideal Bose gas has to be lower than the ground-state energy. Otherwise the occupancy (n.p of some state j would become negative. 

Figure 1.5. Chemical potential of the initial state, the transition state and the product of the Diels-Alder reaction between methyl vinyl ketone and cyclopentadiene in water as compared to 1-propanol The data are taken from r. 56.

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

Internal and External Phases. When dyeing hydrated fibers, for example, hydrophUic fibers in aqueous dyebaths, two distinct solvent phases exist, the external and the internal. The external solvent phase consists of the mobile molecules that are in the external dyebath so far away from the fiber that they are not influenced by it. The internal phase comprises the water that is within the fiber infrastmcture in a bound or static state and is an integral part of the internal stmcture in terms of defining the physical chemistry and thermodynamics of the system. Thus dye molecules have different chemical potentials when in the internal solvent phase than when in the external phase. Further, the effects of hydrogen ions (H" ) or hydroxyl ions (OH ) have a different impact. In the external phase acids or bases are completely dissociated and give an external or dyebath pH. In the internal phase these ions can interact with the fiber polymer chain and cause ionization of functional groups. This results in the pH of the internal phase being different from the external phase and the theoretical concept of internal pH (6). 

The electrochemical potential, )T, of a species is a function of the electrical state as well as temperature, pressure, and composition is the absolute activity, which can be broken down into three parts as shown. Eor an electrolyte. A, which dissociates into cations and v anions, the chemical potential of the electrolyte can be expressed by... 

Because only differences in chemical potential can be measured, the chemical or electrochemical potential of each species is broken down as in equation 3. An arbitrary secondary reference state is defined for each compound. For instance, the chemical potential of chlorine gas is expressed as... 

Chemical Potential. Equilibrium calculations are based on the equaHty of individual chemical potentials (and fiigacities) between phases in contact (10). In gas—soHd adsorption, the equiHbrium state can be defined in terms of an adsorption potential, which is an extension of the chemical potential concept to pore-filling (physisorption) onto microporous soHds (11—16). 

However, the chemical potential is given by Eq. (4-341) for gas-phase reactions and standard states as the pure ideal gases at T°, this equation becomes... 

The term 7 7 ln p/p° is clearly the chemical potential of a surface of radius r with respect to a flat surface of the same material as standard state. It follows that the difference in chemical potential between two surfaces, where... 

In a state of equilibrium the chemical potentials of water and water vapor are equal ... 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

To test the results of the chemical potential evaluation, the grand canonical ensemble Monte Carlo simulation of the bulk associating fluid has also been performed. The algorithm of this simulation was identical to that described in Ref. 172. All the calculations have been performed for states far from the liquid-gas coexistence curve [173]. 

In the case of a Lennard-Jones fluid, the knowledge of the bulk density in the nonreactive part is all that is needed to calculate the chemical potential. Actually, one can use the equation of state of Nicolas et al. [115] (or the... 

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

X is the scalar distance between the solute molecule and the center of the imaginary membrane, with the LJ parameters of the solute used as reducing parameters. The residual chemical potential for a pure fluid (which would correspond to component 2 in its pure state at the state conditions of cell A) can then, for example, be found using the expression... 

For this expression to be valid, in cell A components 1 and 2 must be identical in all respects, so it is a rather special case of an ideal mixture. They are however, allowed to interact differently with the membrane, as described above, xa is the mole fraction of the solute in cell A, while p and p are the number densities of cells A and B respectively. The method was extensively tested against both Monte Carlo and equations of state for LJ particles, and the values of the chemical potential were found to be satisfactory. The method can also be extended to mixtures [29] by making... 

For a substance in a given system the chemical potential gi has a definite value however, the standard potentials and activity coefficients have different values in these three equations. Therefore, the selection of a concentration scale in effect determines the standard state. 

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