Excess thermodynamic quantities

Excess Thermodynamic Quantities The excess Gibbs energy is defined by 

Note that the independent variables chosen here are T, F, N AG is due to the charging process against the ionic atmosphere potential only, at constant volume [see Eq. (6.12.61)]. When the charging work is taken at constant F, k is also constant. However, if the charging process is carried out at P constant, the volume F changes and therefore K would be a function of A (see section 6.12.8). 

There are two limiting cases of interest. First, if all p, = 0 and py - 0, then we approach the ideal-gas limit therefore at this limit and thus C = 0. Since at this limit the dielectric constant is unity, we define 

The other limit of interest occurs when all p/ - 0, but the density of the solvent is Pw- pC, in which case the pressure in (6.12.88) must tend to the pressure of the pure solvent, hence 

Note that here k is k ( ) with the value of of the pure solvent. The first term on the rhs of (6.12.92) is the osmotic pressure of an ideal solution. The correction term is the change in the osmotic pressure due to the ionic interactions only. 

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On the other hand, we showed that the coii5>osition of surfactant in a mixed adsorbed film can be estimated thermodynamically from experimental results without introducing such a supposition (9-11). Further, the composition of a mixed micelle was calculated assuming that the micelle behaves thermodynamically like a macroscopic bulk phase whose thermodynamic quantities are given by the excess thermodynamic quantities similar to those used for the adsorbed film (i8). Therefore, we can now compare the composition of surfactant in the mixed adsorbed film with that in the mixed micelle at the critical micelle concentration (CMC). 

The surface pressure—area and surface dipole moment-area data for both components and a 1 to 1 molar ratio of all but the last mixture were obtained for at least two temperatures in the range 0° to 40 °C. From such data excess thermodynamic quantities were calculated. For reasons... 

While the thermodynamic treatment is applicable to the mixed cholesterol systems we listed, we are not reporting the excess thermodynamic quantities for these systems at this time because of difficulties in allowing for differences in the initial physical states of the pure component mono-layers. The conclusions drawn by van Deenen et al. (9) for the dimyris-toyl lecithin-cholesterol system must be regarded as invalid since they did not take this factor into account. 

In the previous sections concerning reference and standard states we have developed expressions for the thermodynamic functions in terms of the components of the solution. The equations derived and the definitions of the reference and standard states for components are the same in terms of species when reactions take place in the system so that other species, in addition to the components, are present. Experimental studies of such systems and the thermodynamic treatment of the data in terms of the components yield the values of the excess thermodynamic quantities as functions of the temperature, pressure, and composition variables. However, no information is obtained concerning the equilibrium constants for the chemical reactions, and no correlations of the observed quantities with theoretical concepts are possible. Such information can be obtained and correlations made when the thermodynamic functions are expressed in terms of the species actually present or assumed to be present. The methods that are used are discussed in Chapter 11. Here, general relations concerning the expressions for the thermodynamic functions in terms of species and certain problems concerning the reference states are discussed. 

Similarly, for other surface excess thermodynamic quantities, the corresponding molar quantities are as follows ... 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

The difference between a molar surface excess thermodynamic quantity xar r and the corresponding molar quantity x p for the gaseous adsorptive at the same equilibrium T and p is usually called the integral molar quantity of adsorption, and is denoted Aads r,r ... 

The second source of data available for multicomponent mixtures are the excess thermodynamic quantities. These are equivalent to activity coefficients that measure deviations from symmetrical ideal solutions and should be distinguished carefully from activity coefficients which measure deviations from ideal dilute solutions (see chapter 6). In a symmetrical ideal (SI) solution, the... 

The excess chemical potential of solute, or the solvation free energy , at infinite dilution is of particular interest, because it is the quantity which measures the stability of solute in solvent, and because all other excess thermodynamic quantities are derived from the free energy. The excess chemical potential, which is defined as an excess from the ideal gas, can be expressed in terms of the so called Kirkwood coupling parameter. The excess chemical potential is defined as the free energy change associated with a process in which a solute molecule is coupled into solvent [41]. The coupling procedure can be expressed by. 

Dielectric or Debye, or self diffusion Excess thermodynamic quantity Of formation (thermodynamic quantity) Of a gas Ideal gas... 

Recently, a new way of studying, analyzing, and interpreting liquid mixtures has been suggested." The traditional approach to mixtures is based on the study of the excess thermodynamic quantities such as excess free energy, excess entropy, and enthalpy volume. These quantities convey macroscopic... 

As we have noted in the introduction, the local properties provide more detailed information on the mixture than the global properties provide. This fact is obviously true for mixtures of HRs, for which all the excess thermodynamic quantities are zero, but the local quantities are not. 

Hence, a mixture of HRs always forms a SI solution (Ben-Naim 2006). Therefore, for such mixtures, all the excess thermodynamic quantities are zero. 

We have seen that systems of hard rods form SI solutions. Therefore, all the excess thermodynamic quantities are zero. We have already examined the dependence of the local properties on the ratio of diameters in Section 2.6. Therefore, in this section, we choose equal diameters for the particles = Ogg = 1, and explore the dependence of the thermodynamic properties of the mixture on the ratio of the energy parameter e. In the succeeding calculation, we choose dimensionless parameters. 

In Equations 3.75 through 3.78, E b rftlA. Note that in these equations the interfacial excess thermodynamic quantities and E depend on the choice of the dividing plane, whereas y is independent of it. F° presents the most simple integral expression for the interfacial tension y, namely ... 

The full solution of the binding MSA for dimer association was discussed elsewhere (BIMSA)[48, 39]. Imposing an exponential closure reminiscent of Bjerrum s approximation [39] for the contact pair distribution function results in simple analytic expressions for the excess thermodynamic quantities. 

Similarly, the excess thermodynamic quantities may also be related to the reversible e.m.f. Thus... 

The relationship between the immersion and surface excess thermodynamic quantities has been further discussed by Schay, Everett [2-5], and Woodbury and colleagues [29,30]. The standard enthalpy entropy Ajoisf , and free enthalpy of component 1 in component 2 at... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

Silver-copper, energy of solutions, 142 Silver-gold, excess entropy, 132, 136 excess free energy, 136 Silver-lead, alloy (AgsPb5), calculation of thermodynamic quantities, 136 Silver-zinc, alloy (Ag5Zn5), 129... 

For correlation of solubility, the correct thermodynamic quantities for correlation are the activity coefficient y, or the excess Gibbs free energy AG, as discussed by Pierotti et al. (1959) and Tsonopoulos and Prausnitz (1971). Examples of such correlations are given below. 

In order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

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. 

This relation can be inverted to yield fip(n) as function of c.p(nr) (eh Appendix A 4.1). Substituting the result into 77[/xp], we find the osmotic pressure as function of the concentration, which is the standard form of the osmotic equation of state. Also all the other thermodynamic quantities can be calculated from n[fip]. The excess free energy due to the solute, for instance, takes the form... 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

Thermodynamic properties, such as the excess energy [Eq. (4)], the pressure [Eq. (5)], and the isothermal compressibility [Eq. (7)] are calculated in a consistent manner and expressed in terms of correlation functions [g(r), or c(r)], that are themselves determined so that Eq. (17) is satisfied within 1%. It is usually believed that for the thermodynamic quantities, the values of the correlation functions B(r) and c(r), e.g.] do not matter as much inside the core. This may be true for quantities dependent on g(r), which is zero inside the core. But this is no longer true for at least one case the isothermal compressibility that depends critically on the values of c(r) inside the core, where major contribution to its value is derived. In addition, it should be stressed that the final g(r) is slightly sensitive to the consistent isothermal compressibility. 

This last relation involves previouly defined thermodynamic quantities. Note that, in the case of the HS system, fiEex/N = 0. Once again, it is easy to guess that an accurate predictive SCIET is needed first to obtain the excess chemical potential with a good degree of confidence and then to obtain accurate results on the excess entropy. In order to be calculated in a consistent manner, the excess chemical potential has to satisfy the following condition... 

The Gibbs representation provides a simple, clear-cut mode of accounting for the transfer of adsorptive associated with the adsorption phenomenon. The same representation is used to define surface excess quantities assumed to be associated with the GDS for any other thermodynamic quantity related with adsorption. In this way, surface excess energy (U°), entropy (Sa) and Helmholtz energy (Fa) are easily defined (Everett, 1972) as ... 

The enthalpy, internal energy and their excess quantities of the Lennard-Jones binary mixture have been determined using the PY approximation. The values obtained are in good agreement with the results of MC calculation. The enthalpy and isobaric heat capacity are calculated using the extended expression of the thermodynamic quantities in terms of pair correlation functions. 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

We have calculated enthalpy, internal energy, excess molar enthalpy, and excess molar internal energy based on the integral equation theory. Validity of its use has been confirmed by the comparison of our results with those of MC calculation. Then, we have calculated the differential thermodynamic quantities of the isobaric heat capacity Cp and the excess isobaric molar heat capacity, Cp. ... 

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