Chemical potential of a component

An empirical formula, due to Pick, shows that, under simple cucumstances where the chemical potential of a component in a system is dehned by the equation... 

In the section on chemical equilibrium in gases we introduced a magnitude called the molecular chemical potential of a component ... 

In the present state of thermodynamics the calculation of the chemical potential of a component of a solution can be effected explicitly in two cases only ... 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

The right-hand equality in Equation (10.10), which gives the molar free energy of a pure ideal gas, is of the same form as Equation (10.15), which gives the chemical potential of a component of an ideal gas mixture, except that for the latter, partial pressure is substituted for total pressure. If the standard state of a component of the mixture is defined as one in which the partial pressure of that component is 0.1 MPa, then... 

Similarly, we obtained the chemical potential of a component of an ideal gas mixture as [Equation (10.17)] from an analysis of the van t Hoff mixing experiment, using the same integral. 

We have defined the chemical potential of a component as the partial derivative of the Gibbs free energy of the system (or, for a homogeneous system, of the phase) with respect to the number of moles of the component at constant P and T—i.e.,... 

Now, consider particle motion in a phase (solid, liquid, gas). If the phase is initially inhomogeneous, random motion of atoms tends to homogenize the phase. If several phases are present and there are exchanges between the phases, the interphase reaction or exchange tends to make the chemical potential of all exchangeable components the same in all phases and diffusion again works to homogenize each phase. Hence, at equilibrium, the chemical potential of a component is constant. 

That is, the chemical potential of a component increases, linearly, with the total pressnre of the system. (is the partial molar volume of the component.) Thus, if we consider the change in chemical potential of the vaponr and the liquid on prodncing a cnrved surface, we have the process shown in Figure 2.10. It follows that the change in chemical potential of the vapour is given by... 

At equilibrium, equality of chemical potentials of a component in two liquid phases Li and L2 leads to... 

We have seen that it is possible to control electron (and electron hole) concentrations by the chemical potential of a component of the crystalline compound within a finite range of homogeneity. This observation leads to an effect that is known as... 

Consider a material or system that is not at equilibrium. Its extensive state variables (total entropy number of moles of chemical component, i total magnetization volume etc.) will change consistent with the second law of thermodynamics (i.e., with an increase of entropy of all affected systems). At equilibrium, the values of the intensive variables are specified for instance, if a chemical component is free to move from one part of the material to another and there are no barriers to diffusion, the chemical potential, q., for each chemical component, i, must be uniform throughout the entire material.2 So one way that a material can be out of equilibrium is if there are spatial variations in the chemical potential fii(x,y,z). However, a chemical potential of a component is the amount of reversible work needed to add an infinitesimal amount of that component to a system at equilibrium. Can a chemical potential be defined when the system is not at equilibrium This cannot be done rigorously, but based on decades of development of kinetic models for processes, it is useful to extend the concept of the chemical potential to systems close to, but not at, equilibrium. 

Equation 5.23 may be used with Equation 5.22a to determine the chemical potential of water in hydrate /z, which is one of the major contributions of the model. The combination of these two equations is of vital importance to phase equilibrium calculations, since the method equates the chemical potential of a component in different phases, at constant temperature and pressure. 

If we have one degree of freedom, then we may assign values to one of the intensive variables within limits. When we use the temperature as an example, we can use Equation (5.67) to solve for one of the extensive variables in terms of the temperature and the other extensive variables. Thus, the value of the entropy is determined in principle from the known values of the temperature, the volume, and the mole numbers. If the system has two degrees of freedom, then two of Equations (5.67)-(5.69) may be used to solve for two of the extensive variables in terms of two intensive variables and the remaining extensive variables. We thus find that for each degree of freedom that the system has, we may substitute one intensive variable for one extensive variable. This means that, in addition to substituting the temperature and pressure for the entropy and volume, we may substitute the chemical potential of a component for its mole number. This is seldom necessary experimentally, because the determination of the number of moles of each component usually presents no problem. The arguments and results are identical for systems on which restrictions have been placed. 

This equation then gives the differential of the chemical potential of a component in terms of the experimentally determined variables the temperature, pressure, and mole fractions. It is this equation that is used to introduce the mole fraction into the Gibbs-Duhem equation as independent variables, rather than the chemical potentials. The problem of expressing the chemical potentials as functions of the composition variables, and consequently the determination of (dpjdx j P x, is discussed in Chapters 7 and 8. 

The methods for obtaining expressions for the chemical potential of a component that is a weak electrolyte in solution are the same as those used for strong electrolutes. For illustration we choose a binary system whose components are a weak electrolyte represented by the formula M2A and the solvent. We assume that the species are M +, MA , A2-, and M2A. We further assume that the species are in equilibrium with each other according to... 

Throughout the discussions in Sections 8.15-8.18, we have emphasized methods for obtaining expressions for the chemical potential of a component when we choose to treat the thermodynamic systems in terms of the species that may be present in solution. A complete presentation of all possible types of systems containing charged or neutral molecular entities is not possible. However, no matter how complicated the system is, the pertinent equations can always be developed by the use of the methods developed here, together with the careful definition of reference states or standard states. We should also recall at this point that it is the quantity (nk — nf) that is determined directly or indirectly from experiment. 

Two methods may be used, in general, to obtain the thermodynamic relations that yield the values of the excess chemical potentials or the values of the derivative of one intensive variable. One method, which may be called an integral method, is based on the condition that the chemical potential of a component is the same in any phase in which the component is present. The second method, which may be called a differential method, is based on the solution of the set of Gibbs-Duhem equations applicable to the particular system under study. The results obtained by the integral method must yield... 

The condition of equilibrium, in addition to the equality of the temperature and pressure of the two phases, demands that the chemical potential of a component in each phase in which it exists must be the same. The problem is to obtain expressions for the chemical potential of a component in terms of quantities that are experimentally observable. 

The measurement of osmotic pressure and the determination of the excess chemical potential of a component by means of such measurements is representative of a system in which certain restrictions are applied. In this case the system is separated into two parts by means of a diathermic, rigid membrane that is permeable to only one of the components. For the purpose of discussion we consider the case in which the pure solvent is one phase and a binary solution is the other phase. The membrane is permeable only to the solvent. When a solute is added to a solvent at constant temperature and pressure, the chemical potential of the solvent is decreased. The pure solvent would then diffuse into such a solution when the two phases are separated by the semipermeable membrane but are at the same temperature and pressure. The chemical potential of the solvent in the solution can be... 

The fundamental condition for equilibrium between phases, the equality of the chemical potential of a component in every phase in which the component is present still applies. We then substitute for the chemical potential of a component the equivalent chemical potential of the same substance in terms of species or appropriate sums of chemical potentials of the species, as determined by the methods of Section 8.15 and used in the preceding sections. Several examples are discussed in the following paragraphs. 

The second subject is the effect of the surface on the chemical potential of a component contained in a small drop. We consider a multicomponent system in which one phase is a bulk phase and the second phase is kept constant with the conditions that the interface between the two phases is contained wholly within the bulk phase and does not affect the external pressure. The differential of the Gibbs energy of a two-phase system may be written as... 

We therefore find that the chemical potential of a component within the drop is given by... 

The change of the chemical potential of a component with the field at constant temperature, pressure, and mole numbers is given by Equation (14.75). We note that the electric susceptibility is a function of the temperature, pressure, and mole numbers. It is also a function of the field, but may be taken as independent of the field except for high fields when saturation effects may occur. 

Usually statements of problems on chemical equilibrium include the initial amounts of several species, but this doesn t really indicate the number of components. The initial amounts of all species can be used to calculate the initial amounts of components. The choice of components is arbitrary because /xA or fiB could have been eliminated from the fundamental equation at chemical equilibrium, rather than fiAB. However, the number C of components is unique. Note that in equation 3.3-2 the components have the chemical potentials of species. This is an example of the theorems of Beattie and Oppenheim (1979) that (1) the chemical potential of a component of a phase is independent of the choice of components, and (2) the chemical potential of a constituent of a phase when considered to be a species is equal to its chemical potential when considered to be a component. The amount of a component in a species can be negative. 

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