Partial molar functions measurement

The formation and decarboxylation kinetics of [Ir(0C02)(NH3)5]+ have been studied as a function of pressure, based on activation volumes for C02 uptake (AF = —4.0 cm3 mol-1) and decarboxylation (AF1 = + 2.5 cm3 mol-1), and partial molar volume measurements. The suggested mechanism for the formation and decarboxylation is given in Scheme 16. Bond formation during C02 uptake and bond breaking during decarboxylation were believed to be ca. 50% completed in the transition state (149) of these processes.342... 

The integration of these equations gives the partial molar functions of component B. For the integration the integration constants must be known. Because the potential difference was measured between the alloy and the pure metal, the partial molar functions are relative values referring to the pure metal as a reference state. Therefore, integration is carried out between Xg=1, x = 0 and Xg, and x = 1 Xg. 

This procedure provides a second way to obtain partial molar functions of component B. The partial molar functions of A, AZ, are known from potential measurements. One can then calculate the partial molar functions of component B from Aj, Z subtracting the partial molar function AZ according to the equation... 

A variety of transition-metal hydroxo complexes, [M(NH3)50H] [M = Co(III), Rh(III), and Ir(III)], react with CO2 to give the corresponding monodentate carbonato complexes [M(NH3)s0C02]. The formation and decarboxylation kinetics of these complexes have now been studied as a function of pressure up to 1000 bar. The volumes of activation for CO2 uptake are -10.1 0.6 (Co(III)), -4.7 0.8 (Rh(III)), and -4.0 1.0 (Ir(III)) cm mor, whereas the corresponding values for decarboxylation are +6.8 0.3, +5.2 0.3, and +2.5 0.4 cm mol , respectively. Combined with partial molar volume measurements, these values enable the construction of overall reaction volume profiles. Bond formation during CO2 uptake and bond breakage during decarboxylation are approximately 50% completed in the transition state of these processes. 

By measuring the partial molar heat of solution as a function of temperature for infinitely dilute concentrations of Cu, Ag, and Au in liquid tin, Oriani and Murphy51 have determined ACp for the liquid solutes to be 1.0, 0, and 3.0 cal/deg mole respectively. These numbers bear no relationship to the sign of the heat of solution, or to atom-size disparity, but seem to be related to the deviation from unity of the ratio of the masses of the components. 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

Some earlier thermodynamic studies on rutile reported expressions involving simple idealized quasi-chemical equilibrium constants for point defect equilibria (see, e.g., Kofstad 1972) by correlating the composition x in TiOx with a function of AGm (O2), which is the partial molar free energy of oxygen. However, the structural effects were not accounted for in these considerations. Careful measurements of AGm (O2) in the TiOjc system (Bursill and Hyde 1971) have indicated that complete equilibrium is rarely achieved in non-stoichiometric rutile. 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

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 calorimetric studies of micelle formation it is often difficult to relate the measured enthalpy changes to specified steps in the aggregation process. Instead one perferably determines the partial molar enthalpy hA of the amphiphile as a function of concentration12). The ideal case of the phase separation model predicts that hA is constant up to the CMC where it discontinuously jumps to another constant value. 

Here, AH(A-B) is the partial molar net adsorption enthalpy associated with the transformation of 1 mol of the pure metal A in its standard state into the state of zero coverage on the surface of the electrode material B, ASVjbr is the difference in the vibrational entropies in the above states, n is the number of electrons involved in the electrode process, F the Faraday constant, and Am the surface of 1 mol of A as a mono layer on the electrode metal B [70]. For the calculation of the thermodynamic functions in (12), a number of models were used in [70] and calculations were performed for Ni-, Cu-, Pd-, Ag-, Pt-, and Au-electrodes and the micro components Hg, Tl, Pb, Bi, and Po, confirming the decisive influence of the choice of the electrode material on the deposition potential. For Pd and Pt, particularly large, positive values of E5o% were calculated, larger than the standard electrode potentials tabulated for these elements. This makes these electrode materials the prime choice for practical applications. An application of the same model to the superheavy elements still needs to be done, but one can anticipate that the preference for Pd and Pt will persist. The latter are metals in which, due to the formation of the metallic bond, almost or completely filled d orbitals are broken up, such that these metals tend in an extreme way towards the formation of intermetallic compounds with sp-metals. The perspective is to make use of the Pd or Pt in form of a tape on which the tracer activities are electrodeposited and the deposition zone is subsequently stepped between pairs of Si detectors for a-spectroscopy and SF measurements. 

Calculation of A//e -quantities from the dependence of AG on temperature is less reliable than direct calorimetric measurements (Franks and Reid, 1973 Frank, 1973 Reid et al., 1969). However, disagreement between published A//-functions for apolar solutes in aqueous solutions may also stem from practical problems associated with low solubilities (Gill et al., 1975). Calorimetric data have the advantage that, as theory shows, the standard partial molar enthalpy H3 for a solute in solution is equal to the partial molar enthalpy in the infinitely dilute solution, i.e. x3 - 0. A similar identity between X3 and X3 (x3 - 0) occurs for the volumes and heat capacities but not for the chemical potentials and entropies. The design of a flow system for the measurement of the heat capacity of solutions (Picker et al., 1971) has provided valuable information on aqueous solutions. 

In this experiment the partial molar volumes of sodium chloride solutions will be calculated as a function of concentration from densities measured with a pycnometer. 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

This equation, which is one example of the Gibbs-Duhem equation, shows that changes in the partial molar volume of one component may be related to changes in the same quantity for the other component. Experimentally, it means that one only has to measure one partial molar volume as a function of composition provided one has a value of the second partial molar volume at a reference point. In order to illustrate this point, equation (1.4.8) is written in a form suitable for calculating ua from ug ... 

The main reason that the only really useful choice is to have 7 equal to 1.0 in the standard state is that values of 7// in real solutions approach 1.0 in very dilute solutions, so that many properties of ideal solutions can be estimated by measuring them as a function of concentration, and then extrapolating to zero concentration to get their value in an ideal solution, i.e. where yn = 10. These are then called standard state values of partial molar volume, or partial molar enthalpy, etc. There is no way of deriving properties of solutions with other values of 7f/. The choice of m as 1.0 does not have the same degree of necessity. Any value could be used, adding a constant factor to all attendant values of solute activities and (/u /i°) values. The very dilute or infinitely dilute solutions themselves (m very small or zero) would make rather poor standard states, because although values would be 1.0, consideration of... 

Fig. 17.12. The standard partial molar heat capacity of aqueous HCl as a function of temperature. Squares are the experimental data of Tremaine et al. (1986), and the solvation and non-solvation contributions, which add to the line fitting the data, are from the HKF model. Note how the shapes of the two contributions combine to give the inverted-U shape of the measured heat capacity.

Fig. 17.14. The standard partial molar heat capacities of aqueous HCl, Na" and Cl as a function of temperature. The experimental data for Cl are in fact those for aqueous HCl of Tremaine et al. (1986) from Figure 17.12, illustrating the fact that the heat capacity of is zero by convention (see discussion in text). The heat capacity of aqueous NaCl is a measured quantity, and that of aqueous Na is obtained from the difference of the NaCl and HCl (or Cl ) data.

There is one thorough calorimetric study of the enthalpy of reaction of thorium with hydrogen, but at one temperature only, by Picard and Kleppa [1980P1C/KLE], who measured the partial molar enthalpy of solution (H2, 700 K), as a function of the... 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

In a gas-phase system at constant volume, the reaction is monitored through the formation of products and change of total pressure. The pressure is a direct measure in a closed system. If the partial molar fractions are known, the partial pressure can also be determined as a function of the total pressure. For a gas-phase reaction of the type like ... 

The KB inversion process involves the extraction of KBIs from the available experimental data. The experimental data required for this process—derivatives of the chemical potentials, partial molar volumes, and the isothermal compressibility—are all generally obtained as derivatives of various properties of the solution. Obtaining reliable derivatives can be challenging and will depend on the quality of the source data and the fitting function. Unfortunately, the experimental data often appear without a reliable statistical analysis of the errors involved, and hence the quality of the data is difficult to determine. Matteoli and Lepori have performed a fairly rigorous analysis of a series of binary mixtures and concluded that, for systems under ambient conditions, the quality of the resulting KBIs is primarily determined by the chemical potential data, followed by the partial molar volume data, whereas errors in the compressibility data have essentially no effect on the KBI values (Matteoli and Lepori 1984). Excess chemical potentials are typically obtained from partial pressure data, either isothermal or bubble point determinations, and from osmotic pressure or even electrochemical measurements. The particle number... 

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