Partial molar quantity, chemical potential

Partial molar quantity Chemical potential Perfect gas [symbol p ]... 

Solubility of water in chlorine. Expressing the free energy of a system in terms of partial molar quantities (chemical potentials) and comparing the result of differentiation of the whole function with an expression for the rate of change of free energy with composition leads to what is known as the Gibbs-Duhem relation [55,56]. The Gibbs-Duhem equation can be written as... 

In order to calculate the equOibrium composition of a system consisting of one or more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is important, This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions, the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions. 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

The chemical potential is a partial molar quantity defined by the last term in equation 4 ... 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

Equation (9.33) is one of the most useful relationships between partial molar quantities. When applied to the chemical potential, it becomes... 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

We have previously introduced one particular partial molar quantity. In Eq. (9) of Chapter 6, the chemical potential was given as... 

This shows that the chemical potential of a component is just its partial molar Gibbs free energy. Note that the definitions of the chemical potential in terms of other thermodynamic variables, given in Chapter 6, Eq. (8), are not partial molar quantities because pressure and temperature are not the variables held constant in these derivatives. 

In a binary solution, the Gibbs-Duhem relation [Eq. (15)] determines the variation of a partial molar property of one component in terms of the variation of the partial molar quantity of the other component. This relation is useful for obtaining chemical potentials in binary solutions when only one of the components has a measurable vapor pressure. Applying Eq. (15) to chemical potentials in a binary solution,... 

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

The Partial Molar Quantity of Energy and the Chemical Potential. 

Obviously, the chemical potential of a substance is the partial molar quantity of the principal energy functions with respect to the number of moles of the substance at constant values of their respective independent variables in the system as shown in Fig. 5.1. 

Equation (1.121) states that the chemical potential of the species i is the change in free energy with respect to the change in number of moles of the species i while the compositions of other species are held constant. Partial molar quantities also follow the same thermodynamic rules. 

The most important partial molar quantity in Physical Chemistry is the partial molar free energy designated as chemical potential and represented as... 

Partial molar volumes are of interest in part through their thermodynamic connection with other partial molar quantities such as partial molar Gibbs free energy, known also as chemical potential. An important property of chemical potential is that for any given component it is equal for all phases that are in equilibrium with each other. Gonsider a system... 

The chemical potential is the partial molar Gibbs free energy. Partial molar quantities figure importantly in the theory of solutions and are defined at constant temperature and pressure thus, the Gibbs free energy is a natural state function for their derivation. As an example, the partial molar volume is found from the Maxwell relation... 

Equation 41 shows that the chemical potential is a partial molar property. We will need other partial molar quantities (e.g., those for volume, enthalpy, and entropy) in dealing with pressure and temperature effects on energetics of reactions. 

The chemical potential of a species in a phase is a partial molar quantity (equation 41). The total Gibbs energy of a phase is G = E /x, n,. Other partial... 

In an earlier section the free energy of a phase and the free energy of a total system were discussed generally in terms of the potentials (e.g., equation 48). With the definition of the chemical potential as a function of activity in hand, we will now consider the Gibbs energy of a system. In a similar fashion, the enthalpy and entropy of a system can be computed using the partial molar quantities and the mole numbers of each phase. 

A review of chemical thermodynamics, especially as it relates to the properties of liquid solutions, has also been presented. Partial molar quantities such as the chemical potential are an important feature of the treatment of this subject. It is often the case that the activity and chemical potential of one quantity is relatively easy to determine directly by experiment, whereas that of another component is not. Under these circumstances, the change in chemical potential of one component can be related to that of another through the Gibbs-Duhem equation. This relationship and its use in estimating thermodynamic properties are extremely important in solution chemistry. 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

The partial molar quantities most commonly encountered in the thermodynamics of polymer lutions are partial molar volume Vi and partial molar Gibbs free energy Gi- The latter quantity is of special significance since it is identical to the quantity called chemical potential, pi, defined by... 

The chemical potentials are the key partial molar quantities. The pi s determine reaction and phase equilibrium. Moreover, all other partial molar properties and all thermodynamic properties of the solution can be found from the pi s if we know the chemical potentials as functions of T, P, and composition. For example, the partial derivatives of p with respect to T... 

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