Molar volume experimental determination

The free volume approach has been an increasingly popular method to relate polymer structure to gas transport properties. The basic premise of this technique is that a polymer with an open, poorly packed structure will have a large unoccupied free volume through which a gas can diffuse with ease. In a typical model, set forth by Lee (d2.) a specific free volume, SFV, is derived from the difference between the molar volume, Vm, (determined from the experimental density of a polymer) and the occupied volume, Vo, (calculated using a group additive method, in this case, that of Bondi) (4(1). ... 

The ideal gas law offers a simple approach to the experimental determination of the molar mass of a gas. Indeed, this approach can be applied to volatile liquids like acetone (Example 5.4). All you need to know is the mass of a sample confined to a container of fixed volume at a particular temperature and pressure. 

Vapour pressure osmometry is the second experimental technique based on colligative properties with importance for molar mass determination. The vapour pressure of the solvent above a (polymer) solution is determined by the requirement that the chemical potential of the solvent in the vapour and in the liquid phase must be identical. For ideal solutions the change of the vapour pressure p of the solvent due to the presence of the solute with molar volume V/1 is given by... 

The coordination numbers of the Ln3+ ions in water are now well established from different experimental techniques (214-221). The lighter La3+-Nd3+ ions are predominantly nine-coordinate, Pm3+ Eu3+ exist in equilibria between nine- and eight-coordinate states and the heavier Gd3+-Lu3+ are predominantly eight-coordinate. The change in coordination number is also reflected in the absolute partial molar volumes, U°bs, of several Ln3+ ions determined in aqueous solutions (222,223). 

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. 

Partial molar volumes have been determined for [Fe(phen)3] +, [Fe(bmi)3] +, and [Fe(cmi)3] + (emi = (10 ) in water and in methanol-water mixtures, and for [Fe(bipy)3] + and [Fe(Me2bsb)3] + in water. Caleulations of these partial molar volumes have been compared with the experimentally derived values. ... 

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... 

Let us now consider how these quantities are related to experimentally determined heats of adsorption. An essential factor is the condition under which the calorimetric experiment is carried out. Under constant volume conditions, AadU 1 is equal to the total heat of adsorption. In such an experiment a gas reservoir of constant volume is connected to a constant volume adsorbent reservoir (Fig. 9.3). Both are immersed in the same calorimetric cell. The total volume remains constant and there is no volume work. The heat exchanged equals the integral molar energy times the amount of gas adsorbed ... 

It is relatively easy to determine the solubility parameter of a solvent. The molar volume can be obtained from pycnometry, or a value can possibly be found in the literature. Also, since most solvents of interest have significant volatility, their heats of vaporization can be determined calorimetrically. The experimentally determined heat of vaporization can be converted into the desired energy of vaporization through a conversion term that is simply the change in pressure-volume product for the process. Specifically, AH = AE + A(pV). At constant pressure this is pAV and, to the adequate approximation that the vapor is an ideal gas, the conversion term is thus simply RT (where R is the usual gas constant). 

When examining data for Henry s law constants, it is useful to compare values with data for structurally similar compounds. For a homologous series such as the chlorobenzenes, the increase in molar volume or area associated with substitution of chlorine for hydrogen causes a decrease in both solubility and vapor pressure thus H may be fairly constant for such a series. The ideal situation is one in which reliable independent experimental data are available for P , C, and H which permit a consistency check of the three determinations. 

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. 

This equation makes it possible to determine the number of moles of the component in each phase or to use n as an independent variable rather than V, if we so choose. The molar volumes are properties of the separate phases, and consequently can be considered as functions of the temperature and pressure. However, the system is univariant, and consequently the pressure is a function of the temperature when the temperature is taken as the independent variable. The relation between the temperature and pressure may be determined experimentally or may be determined by means of the Clapeyron equation. The differential of each molar volume may then be expressed by... 

The evaluation of Equation (8.41) presents some difficulty. The molar volumes, V and V", can be determined experimentally, as can (S — S"). However, the equation contains the molar enthalpy of the double-primed phase, and the absolute value of this quantity is not known. It would then... 

For a given set of values for H, V, and n, values of n, n", and n " can be calculated if the molar enthalpies and molar volumes of the three phases can be determined. The molar volumes can be obtained experimentally, but the absolute values of the molar enthalpies are not known. In order to solve this problem, we make use of the concept of standard states. We choose one of the three phases and define the standard state to be the state of the system when all of the component exists in that phase at the temperature and pressure of the triple point. If we choose the triple-primed phase as the standard phase, we subtract nH " from each side of Equation (8.50) and obtain... 

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 reaction volume may be of interest in itself, and furthermore its determination can provide a route to the volume of activation in the reverse direction if that parameter is not experimentally accessible and when AV for the reaction in the forward direction is known. As indicated above, AV may be determined from the dependence upon pressure of the equilibrium constant. It may also be obtained under certain circumstances from the partial molar volumes of the reactants and products. Density measurements d are made on several solutions of different concentrations of the reactant(s) and the product(s). The following equation is used to obtain the apparent molar volume of each species, tp, at each molar concentration c. 

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