Partial molar quantities experimental determination

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. 

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 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. 

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. 

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. ... 

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. 

The absolute values of some partial molar quantities, such as the volume, can be determined experimentally. Other absolute values, however, cannot be these include partial molar enthalpies, free enthalpies, and free entropies. However, their changes are measurable. 

Equation (4.70) is a starting point in the determination of diffusivities in liquid metal alloys, but in most real systems, experimental values are difficult to obtain to confirm theoretical expressions, and pair potentials and molecular interactions that exist in liquid alloys are not sufficiently quantified. Even semiempirical approaches do not fare well when applied to liquid alloy systems. There have been some attempts to correlate diffusivities with thermodynamic quantities such as partial molar enthalpy and free energy of solution, but their application has been limited to only a few systems. 

In Equations (10.43) and (10.44) Vf represents the partial molar volume of the component in the infinitely dilute solution, which is also the partial molar volume of the component in the standard state. The right-hand side of Equation (10.44) contains only quantities that can be determined experimentally, and thus A/j. [T, P, x] can be determined. However, just as in the previous case, the pressure is a function of the mole fraction. Therefore, if we require values of A/tf at some arbitrary constant pressure, the correction expressed in Equation (10.34) must be made with the substitution of Vf for... 

The experimental determination of T, yu and xt is actually redundant. However, the equation corresponding to Equation (10.70) involves the partial molar entropies of the components in the two phases. In many cases the values of these quantities are not known. Therefore, a test of the thermodynamic consistency of each experimental point generally cannot be made, neither can the value of one of the three quantities T,yu and xt be calculated when the other two are measured. The test for the consistency of the overall data according to Equation (10.78) can be made when the isothermal values of both A i[T0, P, x] and Anf[T0> P> X1 have been determined. 

In this expression Cp2 is the partial molar heat capacity of the solute in the infinitely dilute solution. Although the experimental significance of this quantity is not immediately obvious, it will be shown below to be capable of direct determination. Thus, from a knowledge of the variation of Zs, the partial molar heat content of the solute, with temperature, it should be possible to derive, with the aid of equation (44.46), the partid molar heat capacity of the solute Cp2 at the given composition. 

Heat Capacities of Aqueous Ions The difference between the specific heat of a dilute solution of an electrolyte and that of water can be obtained by flow microcalorimetry that requires also knowledge of the corresponding densities. Extrapolation to infinite dilution of these differences yields the standard partial molar (constant pressure) heat capacity of the electrolyte, Cp. Alternatively, the heat of solution of an electrolyte in water to form a dilute solution can be measured calori-metrically at several temperatures and exhapolation of the temperature coefficients of these heats of solution to infinite dilution yields the same quantity, but somewhat less accurately. A recent review of the experimental methods by Hakin and Bhuiyan [63] may be consulted for details. Determinations of CZ are accurate to 1 to... 

In a recent study of the a, co-amino acids, Chalikian et al. [93C] determined apparent molar quantities over the concentration range 1 - 3 mg mL Within the limits of the experimental errors, the results obtained were considered to be the same as partial molar compressibilities at infinite dilution. 

The further scheme for calculating the composition of the primary products for sodium and potassium azides, decomposing to gaseous products only, and for the silver azide melt was the same as for the nitrides (see above). The composition was determined by choosing the decomposition scheme for which the molar enthalpy would fit the experimental value. For the other azides, decomposing to the solid metal and a mixture of atomic and molecular nitrogen, the molar enthalpy for one or another product composition needs to be compared to the experimental value, taking into account a partial contribution of the heat of condensation, AcH. For this purpose, the tAcH /v quantity was subtracted from the experimental value of E. Their difference E — tAcH /v) corresponds to the molar enthalpy, of... 

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