Thermodynamic functions partial molar

Starting from the definition 5.22 we now establish several important properties of thermodynamic potentials (partial molar quantities of thermodynamic energy functions) for an ideal system of mixture. Differentiating G-H-TS with respect to n, with Tand p constant, we have pt = ht- Tsl and furthermore [d(jWf IT) / dT pn = (1 IT) (dp, / dT) - (p, / T1) = - [(r s, + pt) / T2] = -h,l T2. From this equation we obtain Eq. 5.34 for the partial molar enthalpy hf of a constituent i in an ideal mixture ... 

These mathematical properties have a simple practical result they ensure that the same relationships can be written among all forms of a thermodynamic function, including molar, total, residual, partial molar, or residual partial molar. 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

In the case of reciprocal systems, the modelling of the solution can be simplified to some degree. The partial molar Gibbs energy of mixing of a neutral component, for example AC, is obtained by differentiation with respect to the number of AC neutral entities. In general, the partial derivative of any thermodynamic function Y for a component AaCc is given by... 

In connection with the development of the thermodynamic concept of partial molar quantities, it is desirable to be familiar with a mathematical relationship known as Euler s theorem. As this theorem is stated with reference to homogeneous functions, we will consider briefly the namre of these functions. 

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. 

Therefore, the physical meaning of the solubility curve of a surfactant is different from that of ordinary substances. Above the critical micelle concentration the thermodynamic functions, for example, the partial molar free energy, the activity, the enthalpy, remain more or less constant. For that reason, micelle formation can be considered as the formation of a new phase. Therefore, the Krafft Point depends on a complicated three phase equilibrium. 

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. 

Before discussing all these biopolymer applications, we first take this opportunity to remind the reader that, in general, any thermodynamic variable can be expressed as the sum of two functions, one of which depends only on the temperature and pressure, and another which depends on the system composition (expressed as the mole fraction xt of the /-component). Therefore, for example, the chemical potential fM of the /-component of the system at constant temperature T and pressure p (the general experimental conditions), /. e., partial molar Gibbs free energy (dG/dn TtP may be expressed as (Prigogine and Defay, 1954) ... 

Figure 11. (a) The calculated partial molar entropy of oxygen (sQJ and (b) the calculated partial molar enthalpy of oxygen (fto2) as a function of 8 for La02Sr08Fe0 55Tio4503 s. Symbols are calculated by the Gibbs-Helmholtz equation. Lines correspond to the partial molar quantities calculated by statistical thermodynamics. 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

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. 

In conclusion, the partial molar quantity in thermodynamics functions consists of its unitary term and its mixing term as shown above. 

Furthermore, in analogy to the partial molar quantities of thermodynamic functions, the partial molar chemical exergy, echem l, can be defined for a substance i in a gaseous mixture, in a liquid solution, and in a solid solution as shown in Eq. 10.35 ... 

It is possible to obtain the partial molar volume of hydrogen in a metal provided one knows the solubility of hydrogen in it, corresponding to a constant pressure or overpotential, as a function of applied stress. For the applied stress to be thermodynamically significant, it should be within the Hooke s-law region for the metal (Fig. 12.73). Proceeding from the thermodynamic relations (dp/dP)T=V and p=p° + RT n c (when c is small), one has ... 

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. 

The definition of a partial molar property, Eq. (11.2), provides the me-for calculation of partial properties from solution-property data. Implicit in definition is a second, equally important, equation that allows the calculation solution properties from knowledge of the partial properties. The derivation this second equation starts with the observation that the thermodynamic propertl of a homogeneous phase are functions of temperature, pressure, and the numb of moles of the individual species which comprise the phase. For thermodyna property M we may therefore write... 

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... 

The above equations again correlate partial molal and molar energies and enthalpies only when all intensive variables are held fixed is the partial molal and molar enthalpy the same. In most cases one may drop the term involving Vs - One may also use Eqs. (5.2.2) to access other thermodynamic functions of interest in terms of differential quantities. 

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 ... 

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