Excess chemical potential and entropy

The excess molar enthalpy hV is simply the heat of mixing at constant pressure related to 1 mole of solution.) And from the excess molar enthalpy and the excess chemical potential, we can obtain the excess molar entropy of the system from the following equation ... 

The evaluation of the derivatives thus requires a knowledge of the entropies and volumes of the three phases as well as a knowledge of the excess chemical potential of the first component in the primed phase. 

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. 

One of the most challenging tasks in the theory of liquids is the evaluation of the excess entropy Sex, which is representative of the number of accessible configurations to a system. It is well known that related entropic quantities play a crucial role, not only in the description of phase transitions, but also in the relation between the thermodynamic properties and dynamics. In this context, the prediction of Sex and related quantities, such as the residual multiparticle entropy in terms of correlation functions, free of any thermodynamic integration (means direct predictive evaluation), is of primary importance. In evaluating entropic properties, the key quantity to be determined is the excess chemical potential (3pex. Calculation of ppex is not straightforward and requires a special analysis. 

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... 

Once the thermodynamic properties and the excess chemical potential have been calculated in a consistent manner, the entropy of the system can be derived. As seen in Table VI, a close agreement for the previous thermodynamic properties is found against CS data at all densities, while the HNC approximation fails at higher densities. 

Now, we turn to the entropy. In Fig. 15, the excess calculated entropy is plotted in comparison with the reference data drawn from MD simulation [34] owing to Eq. (86). For the three supercritical temperatures investigated, the curves follow the reference data extremely well from low densities up to p 0.7. Beyond this limit, the accordance becomes less and less satisfactory. This is not surprising since the excess chemical potential (see Fig. 13) is itself less accurately predicted for high densities. The bump observed on the curve at T 1.35 is also not surprising as it lies around the critical point located, with different theoretical and simulation approaches, at temperature between 1.30 and 1.35 and density between 0.30 and 0.35. 

Another very common misinterpretation of experimental results is the following. Suppose we measure deviations from a DI solution in a T, P, NA, NB system. The corresponding activity coefficient is given by (6.34) the same quantity is often referred to as the excess chemical potential of the solute. One then expands the activity coefficient (or the excess chemical potential) to first order in pA and interprets the first coefficient as a measure of the extent of solute-solute interaction. Clearly, such an interpretation is valid for an osmotic system provided we understand interaction in the sense of affinity, as pointed out above. However, in the T, P, NA, NB system, the first-order coefficient depends on the difference — G°B. It is in principle possible that G be, say, positive, whereas the first-order coefficient in (6.34) can be positive, negative, or zero. This clearly invalidates the interpretation of the first-order coefficient in (6.34) in terms of solute-solute correlation. Similar expansions are common for the excess enthalpies and entropies where the first-order coefficient in the density expansion is not known explicitly. 

Equation (6-45) contains the excess chemical potential instead of the chemical potential itself because the ideal term X X 2 has been omitted. This omission was necessary because the considerations have to be restricted to a small volume of solution where the segment distribution is uniform enough so that the lattice theory can be applied. Although in this equation the quantity (xo 0 ) is derived from the enthalpy of mixing and the factor 0.5 from the entropy of mixing, it is convenient to replace this combination of terms by another with a new enthalpy parameter k and a new entropy parameter... 

Since the excess chemical potential is a fi-ee energy then % must also be a free energy, i.e. it has both an entropy and an enthalpy component. Flory defined enthalpy and entropy of dilution parameters, Ki and respectively, such that the partial molar enthalpy of dilution was... 

At the theta state, the contributions of the intra-molecular attraction and repulsion to the coil size compensate with each other. Flory first gave a thermodynamic treatment to the theta state (Flory 1953). He assumed that the solvent molar mass was Nj, and the excess chemical potential = dAF IdN] contained two parts of contributions, i.e., enthalpy and entropy... 

It is sometimes convenient to rewrite the adsorption equation in other more symmetrical forms. In a system of c components we can specify the state by the values of (c + 1) independent densities, for example, the c molar densities and the entropy density, i). Let all these densities be denoted by the symbol p, a vector of (c +1) components, and their surface excesses F. Shnilaily let e be the oonjugnte set of fields.t In the example given the first c of these would be the chemical potentials and the last would be the temperature. The adsorption equation (2.31) can then be written in more symmetric form... 

The excess chemical potential obviously results from the sum of partial excess enthalpy and entropy as expressed, in dimensionless terms, by the following well known relationship ... 

The two-moment ITM applied to hard sphere solutes predicts entropy convergence for those cases. Additionally, test particle simulation methods used to study more realistic, Lennard-Jones models of inert gas atoms in water also provide a reliable description of the temperature dependence of the solvation free-energy. This theoretical success permits a simpler understanding of entropy convergence. We argue as follows a continuous Gaussian distribution reliably approximates the two-moment information model,exhibits the entropy convergence, and produces an explicit result for the excess chemical potential ... 

Figure 2 Excess chemical potentials for methane and neon gases plotted as a function of temperature (data taken from Ref. 11). The positive slope and negative curvature of each function indicate a negative entropy and a positive heat capacity of...

A zero excess chemical potential does not imply that both the enthalpy of solution and the excess entropy of solution are zero as in the case of true ideal solutions where AH = 0 and = 0. Rather, a theta state means only... 

As already pointed out, Yu is 1 if a compound forms an ideal solution. In this rather rare case, the term RTkiyu, which we denote as partial molar excess free energy of compound i in solution t, Gpe, is 0. This means that the difference between the chemical potential of the compound in solution and its chemical potential in the reference state is only due to the different concentration of the compound i in the two states. The term R In xtf=S 1 expresses the partial molar entropy of ideal mixing (a purely statistical term) when diluting the compound from its pure liquid (xiL =1) into a solvent that consists of otherwise like molecules. 

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