Excess thermodynamic function entropy

Here S is the entropy and N j is the composition of the system. It is convenient to subtract the corresponding bulk phase equations pertaining to the volumes va and vp from these relations and to introduce intensive excess thermodynamic functions. The familiar result is given by... 

In this section it was shown that the excess entropy and excess enthalpy can be determined from various temperature derivatives of the excess Gibbs energy. These and other excess thermodynamic functions can also be computed directly from derivatives of the activity coefficients. Show that in a binary mixture the following equations can be used for such calculations ... 

Figure 4. Molecular-weight dependence of excess thermodynamic functions of dextran aqueous solutions at 37°C (9) excess virial coefficient, B (O) excess enthalpy coefficient, Bh (excess entropy coefficient, Bs (16).

The ideal solutions are characterized by zero excess thermodynamic functions. In thermodynamics, ideal solutions are often characterized by their excess entropy and enthalpy. Note that to obtain these, we need to assume differentiability of (4.91) with respect to temperature. This assumption is still quite weaker than the requirement that (4.91) be valid at all T and P. 

Now let us consider the non-electrolytes. Here we have two very distinct types of behaviour. There are the so-called hydro-phobic, and the hydrophilic effects. The hydrophobic effect can be shown schematically from a consideration of the thermodynamics of hydrocarbon solutions. Usually a non-ideal solution arises because the two components either strongly attract each other or strongly repel each other the effects are shown in the enthalpy. Figure 8 shows various types of behaviour, as reflected in the excess thermodynamic functions (Rowlinson, 1969). The drawn out lines are free energies, the broken lines are enthalpies and the dotted lines are the entropy curves. A positive free energy means a positive deviation from ideal behaviour. In normal systems AG follows the AH curve fairly benzene-MeOH. In... 

The difference in thermodynamic functions between a non-ideal solution and a comparative perfect solution is called in general the thermodynamic excess function. In addition to the excess free enthalpy gE, other excess functions may also be defined such as excess entropy sE, excess enthalpy hE, excess volume vE, and excess free energy fE per mole of a non-ideal binary solution. These excess functions can be derived as partial derivatives of the excess free enthalpy gE in the following. 

Thermodynamic functions have been calculated for liquid binary Ga-Pb alloys in the composition range 10—90 atom % Pb. Enthalpies and excess entropies of mixing at 1000 K were reported. ... 

Flory and Krigbaum defined an enthalpy (Kj) parameter and an entropy of dilution ( /i) parameter such that the thermodynamic functions used to describe these long-range effects are given in terms of the excess partial molar quantities... 

This chapter deals with experimental methods for determining the thermodynamic excess functions of binary liquid mixtures of non-electrolytes. Most of it is concerned with techniques suitable for measurements in the temperature range 250 to 400 K and the pressure range 0 to 100 kPa. Techniques suitable for lower temperatures will be briefly reviewed. Techniques for measuring the molar excess Gibbs function G, the molar excess enthalpy and the molar excess volume will be discussed. The molar excess entropy can only be determined indirectly from either measurements of (7 and at a specific temperature = (If — C /T], or from the temperature dependence of G m [ S m = The molar excess functions have been defined by... 

Binary mixtures of non-aromatic fluorocarbons with hydrocarbons are characterized by large positive values of the major thermodynamic excess functions G , the excess Gibbs function, JT , the excess enthalpy, 5 , the excess entropy, and F , the excess volume. In many cases these large positive deviations from ideality result in the mixture forming two liquid phases at temperatures below rSpper. an upper critical solution temperature. Experimental values of the excess functions and of Tapper for a representative sample of such binary mixtures are given in Table 1. 

Table 5.4.3. Thermodynamic properties of the liquid mixtures used as cosolvents of PMMA. Excess Gibbs function G, and excess entropy S, of the binary mixtures at equimolecular composition (at 25°C). From Prolongo et al. (Copyright by Butterworth-Heineman Ltd., used with permission)...

Vp is the specific pore volume of the material typical values are 200-400 cm /kg for zeolites and up to 1000 cm /kg for activated carbon, n is the actual number of molecules contained in the micropores the excess adsorption n subtracts fix>m n the number of molecules which would have been present in the micropores at the bulk density in the absence of adsorption. The (oversimplified) case when absolute adsorption is described by the Langmuir equation and the gets obeys the perfect gas law p = P/RT) has been worked out in detail for the isotherms and thermodynamic functions (enthalpy, entropy, etc.) [2]. 

The data for Hultgren, Orr, Anderson, and Kelley s compilation were prepared between 1955 and 1963. Information on 65 elements and 167 alloy systems is presented, and selected values for heat capacity, entropy, enthalpy, free energy functions , and vapour pressures of phases are given in tabular form. For alloys, the preferred values of integral, partial, and excess thermodynamic properties are listed or are presented as analytical functions. Phase diagrams and graphs are also included. 

As to the unreal limiting values of enthalpies [see Eq. (18)], in the past 10-15 years it has been proven both theoretically and experimentally that the enthalpies and entropies have finite values at total monolayer coverage. For example, in adsorption from solutions a total excess coverage is always formed, and, evidently, the changes in enthalpies (entropies) can be measured exactly. The experimental data prove [9] that the enthalpy of a total monolayer coverage can never become infinite. Since the infinite or finite character of thermodynamic functions is independent of the nature of the adsorptive system (gas/solid, vapor/solid, liquid/solid) the supposition of the classical isotherm equation concerning limiting values [Eqs (17) and (18)] should be rejected. 

The thermodynamic excess functions differ from the thermodynamic functions of mixing only for quantities which involve the entropy. For example, the excess enthalpy A is identical with the enthalpy of mixing given by (1.6.6). Furthermore the excess volume v is identical with the volume of mixing given by (1.6.7). The excess entropy (in terms of activity coefficients) is given by (cf. 1.6.5)... 

It was not d in section 1.1 that we can go over from (1.37) to (1.33) only if AS does not change or if all changes in entropy are proportional to the corresponding heats of reaction. Such a relationship between the changes in entropy and energy has been repeatedly noticed, including the relations between excess thermodynamic properties, i.e. the deviation of the thermodynamic functions of real... 

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-... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

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. 

---