Partial molar quantities, calculating from

The simulated plots calculated from Equations (5.15) and (5.16) are shown in Figure 5.5a and b, respectively. The partial molar quantities determined from the Monte Carlo simulation are in a good agreement with experimental results. It is noteworthy that a similar conclusion was drawn from results obtained when using the mean-field approximation based upon the lattice gas model in Section 5.2.3.I. 

CALCULATION OF PARTIAL MOLAR QUANTITIES AND EXCESS MOLAR QUANTITIES FROM EXPERIMENTAL DATA VOLUME AND ENTHALPY... 

The dependence of XE on mole fraction can be analysed to obtain the dependence on mole fraction of the relative partial molar quantities, Xx — X and X2 — X2, which are calculated from the intercepts on the and X2-axes of the tangent to the XE -curve. Consequently the XE -data must be precise if these partial molar values are to be meaningful. For example, one of the features of some aqueous mixtures is a minimum in the value of V2 — V2 at low values of x2. This means that the VE -curve has a point of inflexion at this mole fraction. 

The foregoing treatment permits the calculation of the partial molar heat capacities of the solute it will now be shown how the values for the solvent may be obtained from the same data. By combining equations (44.47) and (44.48) with the general relationship for partial molar quantities, viz.,... 

On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in fig. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in fig. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one defines and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be defined as... 

Although partial molar quantities are in principle measurable from slopes or intercepts as in Figures 9.3 and 9.5, they are not actually measured in this way. In practice, apparent molar quantities are determined, and the corresponding partial molar quantities are calculated from these. It is standard practice to let component 1 refer to the... 

So far we have considered only the volume as a partial molar quantity. But calculations involving solutes will require knowledge of all the thermodynamic properties of dissolved substances, such as H, S, Cp, and of course G, as well as the pressure and temperature derivatives of these. These quantities are for the most part derived from calorimetric measurements, that is, of the amount of heat released or absorbed during the dissolution process, whereas V is the result of volume or density measurements. 

It is important to note that the above formulas represent fluctuations (8X=X - (X)) in the properties of the whole system, that is, bulk fluctuations. They are useful expressions but provide no information concerning fluctuations in the local vicinity of atoms or molecules. These latter quantities will prove to be most useful and informative. One can also derive expressions for partial molar quantities by taking appropriate first (to give the chemical potential) and second (to give partial molar volume and enthalpy) derivatives of the expressions presented in Equation 1.28. However, these do not typically lead to useful simple formulas that can be applied directly to theory or simulation. For instance, while it is straightforward to calculate the compressibility, thermal expansion, and heat capacity from simulation, the determination of chemical potentials is much more involved (especially for large molecules and high densities). 

In order to calculate the equOibrium composition of a system consisting of one or more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is important, This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions, the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions. 

Calculation of partial molar quantities from experimental data... 

For more elaborate methods of calculating partial molar quantities from experimental data the reader is referred to the literature.f... 

Fig. 6.1. Graphical Method of Calculating Partial Molar Quantities from Molar Quantities.

It must be remembered that F in the above equation is expressed in cal/V/g-equivalent. The partial molar quantities can also be calculated from knowledge of the e.m.f. Thus,... 

The integral and partial quantities can often be obtained easily from experiments. Partial molar quantities are used to describe the thermodynamic behaviour of the individual components. In a binary system, the partial molar Gibbs energy Ga of component A can be calculated from the molar Gibbs energy, G, at constant temperature and pressure by the well-known relation ... 

From the very beginning the heat capacity measured by DSC was recognised to be a partial molar quantity [52]. This means that every change in the properties of the solution caused by addition of a protein molecule is ascribed to this molecule. This is manifested in the procedure by which the raw data are treated to calculate the heat capacity. The value used for the mass of the protein, mproteinj in equation 2 is that of the dry polypeptide chain and not that of a hydrated polypeptide chain. Only if the dry mass is employed in the calculations, does one obtain agreement between the van t Hoff enthalpy values, A/fy.H.) and the calorimetric enthalpies A/fcai-This is good evidence for the correctness of the choice. 

In order to evaluate each of the derivatives, such quantities as (V" — V-), (S l — Sj), and (dfi t/x t)T P need to be evaluated. The difference in the partial molar volumes of a component between the two phases presents no problem the dependence of the molar volume of a phase on the mole fraction must be known from experiment or from an equation of state for a gas phase. In order to determine the difference in the partial molar entropies, not only must the dependence of the molar entropy of a phase on the mole fraction be known, but also the difference in the molar entropy of the component in the two standard states must be known or calculable. If the two standard states are the same, there is no problem. If the two standard states are the pure component in the two phases at the temperature and pressure at which the derivative is to be evaluated, the difference can be calculated by methods similar to that discussed in Sections 10.10 and 10.12. In the case of vapor-liquid equilibria in which the reference state of a solute is taken as the infinitely dilute solution, the difference between the molar entropy of the solute in its two standard states may be determined from the temperature dependence of the Henry s law constant. Finally, the expression used for fii in evaluating (dx Jdx l)TtP must be appropriate for the particular phase of interest. This phase is dictated by the particular choice of the mole fraction variables. 

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. 

H2 — H i are the relative partial molar enthalpies of the two cosolvents which can be calculated from the dependence of HE on x2 as described on p. 281. Grunwald and Effio (1974) list some of the important quantities for four solvent mixtures. 

The partial molar volume, which is a very important quantity to probe the response of the free energy (or stability) of protein to pressure, including the so-called pressure denaturation, is not a canonical thermodynamic quantity for the (V, T) ensemble, since volume is an independent thermodynamic variable of the ensemble. The partial molar volume of protein at infinite dilution can be calculated from the Kirkwood-Buff equation [20] generalized to the site-site representation of liquid and solutions [21,22],... 

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. 

The purpose of this Appendix is to provide expressions for calculating the KBIs for binary mixtures from measurable thermodynamic quantities such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. 

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 ... 

Now these expressions clearly define the composition-free energy relationships required to ensure adherence to the general equilibrium relationship of equation (1-138) and are useful for calculations if we know quantities such as G, 7, or fi- However they are still somewhat far removed from the number that is normally of primary interest to us, which is the equilibrimn constant. To get to this point in a practically useful way, we can define a reference or standard state for each species to define the partial molar free energy at a reference mol fraction, x°, the temperature of interest, T, and a pressure of 1 atm. Following through with a little bit of thermodynamics, here is the following sequence of equations along the road to an equilibrium constant ... 

The subsequent step is the calculation of the quantities that appear in the matrix elements A-. By means of Equation 4.15 through Equation 4.18, all chemical potential derivatives are calculated. The partial molar volumes are obtained by adding to the molar volume of the pure components the excess partial molar volumes, obtained through Equation 4.21 with = V, by applying the same procedure used to calculate In y. The mixture molar volume is then obtained as = x V + X2V2 + 3 3 and Kj. from Equation 4.5. The values of are finally obtained from Equation 4.12, for which the calculations of the concentrations, the determinant, and the cofactors are all straightforward. 

We have recently carried out Monte Carlo computer simulation of dilute aqueous solutions of the monatomic cations Li, Na and K and the monatomic anions F and Cl using the KPC-HF functions for the ion-water interaction and the MCY-CI potential for the water-water interaction. The temperature of the systems was taken to be 25° and the density chosen to be commensurate with the partial molar volumes as reported by Millero. - The calculated average quantities are based on from 600- 900K configurations after equilibration of the systems. The calculated ion-water radial distribution functions are given for the dilute aqueous solutions of Li", K" ", Na" ", F and Cl" in Figures 11-15, respectively. 

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