Partial molar quantities determination

How are partial molar quantities determined experimentally Sidebar 6.3 illustrates the general procedure for the special case of the partial molar volumes VA, Vr of a binary solution (analogous to the graphical procedure previously employed in Section 3.6.7 for finding differential heats of solution). As indicated in Sidebar 6.3, each partial molar... 

The simulated plots calculated from Equations (5.15) and (5.16) are shown in Figure 5.5a and b, respectively. The partial molar quantities determined from the Monte Carlo simulation are in a good agreement with experimental results. It is noteworthy that a similar conclusion was drawn from results obtained when using the mean-field approximation based upon the lattice gas model in Section 5.2.3.I. 

A variety of procedures can be used to determine Z, as a function of composition.2 Care must be taken if reliable values are to be obtained, since the determination of a derivative or a slope is often difficult to do with high accuracy. A number of different techniques are employed, depending upon the accuracy of the data that is used to calculate Z, and the nature of the system. We will now consider several examples involving the determination of V,- and Cpj, since these are the properties for which absolute values for the partial molar quantity can be obtained. Only relative values of //, and can be obtained, since absolute values of H and G are not available. For H, and we determine H, — H° or — n°, where H° and are values for H, and in a reference or standard state. We will delay a discussion of these quantities until we have described standard states. 

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. 

When the value of an intensive property / can be expressed as an algebraic function of the composition, the partial molar quantities can be determined analytically. 

The subject of partial molar quantities needs to be developed and understood before considering the application of thermodynamics to actual systems. Partial molar quantities apply to any extensive property of a single-phase system such as the volume or the Gibbs energy. These properties are important in the study of the dependence of the extensive property on the composition of the phase at constant temperature and pressure e.g., what effect does changing the composition have on the Helmholtz energy In this chapter partial molar quantities are defined, the mathematical relations that exist between them are derived, and their experimental determination is discussed. 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

These equations are of value for solution thermodynamics. For example, if a partial molar quantity of one component in a binary solution has been determined, then the partial molar quantity of the other component is fixed ... 

A graphical method of determining the partial molar quantities from the data on the integral molar quantities is frequently employed. 

This method is known as the method of intercepts and is of proven value in determining partial molar quantities. 

This is called the relative integral molar free energy or the molar free energy of mixing. Hie method of tangential intercepts which we have applied for determination of partial molar quantities from the integral molar quantities can also apply to the relative quantities ... 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

The measurement of partial molar quantities will be illustrated with reference to partial molar volumes. We can measure absolute volumes of solution and, thus, can determine partial molar volumes directly from its definition ... 

In this section, we investigate the relations between the macroscopic susceptibilities and the molecular polarizabilities. Consistent microscopic interpretations of many of the non-linear susceptibilities introduced in Section 2 will be given. Molar polarizabilities will be defined in analogy to the partial molar quantities (PMQ) known from chemical thermodynamics of multicomponent systems. The molar polarizabilities can be used as a consistent and general concept to describe virtually all linear and non-linear optical experiments on molecular media. First, these quantities will be explicitly derived for a number of NLO susceptibilities. Physical effects arising from will then be discussed very briefly, followed by a survey of experimental methods to determine second-order polarizabilities. 

Partial Molar Quantities. — The thermodynamic functions, such as heat content, free energy, etc., encountered in electrochemistry have the property of depending on the temperature, pressure and volume, i.e., the state of the system, and on the amounts of the various constituents present. For a given mass, the temperature, pressure and volume are not independent variables, and so it is, in general, sufficient to express the function in terms of two of these factors, e.g., temperature and pressure. If X represents any such extensive property, i.e., one whose magnitude is determined by the state of the system and the amounts, e.g., number of moles, of the constituents, then the partial molar value of that property, for any constituent i of the system, is defined by... 

Determination of Partial Molar Quantities Ir Direct Method. —In view of the definition of the partial molar property Qi as... 

IV. General Methods.—In the methods described above for the determination of partial molar quantities, it has been tacitly assumed that the property G is one which is capable of experimental determination. Such is the case, for example, if 0 represents the volume or the heat capacity. However, if the property under consideration is the heat content then, like the free energj , it cannot be determined directly. In cases of this kind modified methods, which involve measurements of changes in the property, rather than of the property itself, can be used. It should be pointed out that the procedures are quite general and they are frequently adopted for the study of properties susceptible of direct measurement, as vrell as of those which are not. ... 

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 approach to the thermodynamics of solubilization in micellar solutions is based on the determination of a given partial molar property of the solute (volume, enthalpy, heat capacity, compressibility) as a function of the surfactant content. The simplest approach is to use the pseudophase model. The partial molar quantity, L will thus be an average value of Y in the micellar and aqueous phases, as described 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... 

Although partial molar quantities are in principle measurable from slopes or intercepts as in Figures 9.3 and 9.5, they are not actually measured in this way. In practice, apparent molar quantities are determined, and the corresponding partial molar quantities are calculated from these. It is standard practice to let component 1 refer to the... 

The principles of phase equilibrium do not apply to excess adsorption variables at high pressure where the excess adsorption passes throu a maximum. Under these conditions, the pressure is no longer a single-valued function of excess adsorption so that n cannot serve as an independent variable for the determination of partial molar quantities such as activity coefficients. Additional complications which arise at high pressure are (1) the selectivity for excess adsorption (S12 = (nf/j/i)/(n2/y2)) approaches infinity as nj — 0 and (2) the differential enthalpy of the ith component has a singularity at the pressure corresponding to maximum nf. For excess variables, the diffierential functions are undefined but the integral functions for enthalpy and entropy are smooth and well-behaved (1). 

In practice, many of the extensive functions (IJ, S, H, G), and their corresponding partial molar quantities, can only be determined up to an additive constant—absolute values of U, H, and can be obtained from simulation, but these then depend on the model of choice. Hence, their values are typically expressed with respect to a set of defined reference or standard states. These are usually taken as the pure solutions of each component at the same T and p. One can then define a series of mixing quantities such that... 

It is important to note that the above formulas represent fluctuations (8X=X - (X)) in the properties of the whole system, that is, bulk fluctuations. They are useful expressions but provide no information concerning fluctuations in the local vicinity of atoms or molecules. These latter quantities will prove to be most useful and informative. One can also derive expressions for partial molar quantities by taking appropriate first (to give the chemical potential) and second (to give partial molar volume and enthalpy) derivatives of the expressions presented in Equation 1.28. However, these do not typically lead to useful simple formulas that can be applied directly to theory or simulation. For instance, while it is straightforward to calculate the compressibility, thermal expansion, and heat capacity from simulation, the determination of chemical potentials is much more involved (especially for large molecules and high densities). 

The partial molar quantities of a binary mixture may be determined by graphical method if the appropriate quantities are known as a function of the mixture composition (Fig. 1-8). 

FIGURE 1 Determination of partial molar quantities Z2 and apparent molar quantities z from measurements of extensive thermodynamic quantities Zpj n n2) (also AZ2 from AZ see Section III.B). 

FIGURE 2 Determination of the partial molar quantities Zi and Z2 as a function of mole fraction X2 from mean molar quantities p,t(x2) (also Zi and AZ2 from AZ see Section III.B). 

Here, AmixZ is the integral effect of mixing AZ/ = Z/ — Z are the differential effects. For enthalpies and Gibbs energies, Z = H or G, only the differential effects AZ/ can be determined by experiments and not the partial molar quantities Z/ themselves, in contrast to volumes and heat capacities, where both quantities are available. 

In the following discussion we will sometimes call it just a molar quantity. Let Z he a function of T. P, ni,... then the partial molar quantity Z, of the tth component is determined as... 

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