Partial molar entropies determination

The partial molar entropy of adsorption AI2 may be determined from q j or qsi through Eq. XVII-118, and hence is obtainable either from calorimetric heats plus an adsorption isotherm or from adsorption isotherms at more than one temperature. The integral entropy of adsorption can be obtained from isotherm data at more than one temperature, through Eqs. XVII-110 and XVII-119, in which case complete isotherms are needed. Alternatively, AS2 can be obtained from the calorimetric plus a single complete adsorption isotherm, using Eq. XVII-115. This last approach has been recommended by Jura and Hill [121] as giving more accurate integral entropy values (see also Ref. 124). 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

The configurational entropy (sQ conf) can be determined by using a model proposed by Mizusaki.20,24 The configurational entropy, partial molar enthalpy, and partial molar entropy can be expressed in terms of 8, a, n, and p values as follows... 

The experimental determination of T, yu and xt is actually redundant. However, the equation corresponding to Equation (10.70) involves the partial molar entropies of the components in the two phases. In many cases the values of these quantities are not known. Therefore, a test of the thermodynamic consistency of each experimental point generally cannot be made, neither can the value of one of the three quantities T,yu and xt be calculated when the other two are measured. The test for the consistency of the overall data according to Equation (10.78) can be made when the isothermal values of both A i[T0, P, x] and Anf[T0> P> X1 have been determined. 

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. 

For a solution of specified composition, the relative partial molar entropy can be determined from the data on activities at different temperatures. 

The importance of the entropy of activation in these reactions is clear when it is realized that if the rate constant were solely a function of AH, the rate constant would increase initially as 2 increases. Instead, the fall in rate constant is determined by the decrease in AS. Thus, as the co-solvent enhances water-structure, so the partial molar entropy of the apolar initial state increases, the solute entering the structure-enhanced solvent, possibly as a guest in a preformed cavity. In any event, the change in this entropy term is probably more important than the endothermic shift in SmH, so... 

Determination of partial molar entropies for individual ions... 

Similar tables can be found for determining standard partial molar entropies for ions in a variety of solvents. These are important in determining thermodynamic transfer data (see Sections 13.23 and 13.25.5). 

But take care these will also be dependent on the convention used in the determination of the individual partial molar entropies of ions in aqueous solution. 

Another possible cycle for determining the partial molar entropy of Ni is based on the dissolution reaction ... 

If AEj and its temperature dependence are measured, the partial molar entropy of component A can be determined... 

AGk, AH, AH thus obtained represent the stoichiometric variations of the Gibbs free energy, enthalpy and entropy, respectively, on the transfer of one mole of solute between the two phases in standard state. AG is the same for the hypothetical ideal state and the real state pro wded that the activity equals unity in both. However AHJ is different in the two cases and reference should be made to the hypothetical ideal state. Because the intermolecular attractions which determine AH are identical in the hypothetical (standard) and reference states, AH refers also to the modification of partial molar enthalpy between the reference states. The same conclusion holds true for the modification of molar heat capacities. A/Sk, like AGk, does not apply to the modification of partial molar entropy between reference states but refers to the hypothetical standard state described above. 

The chemical potential i (i.e. the partial molar free energy) is the most fundamental physical quantity to describe the thermodynamic properties of polymer solutions. If p is represented as functions of absolute temperature (7), pressure (7 ) and the composition (for example, the polymer concentration for a binary mixture), one can determine not only the molecular weight, M, of the solute (the polymer), but also many other thermodynamic quantities, such as the partial molar entropy, the partial molar enthalpy and the partial molar volume of each component. 

The partial molar entropy of mixing of gold in solid Au-Cu alloys, determined at 500°C (773 K) over the complete solution range, is tabulated below. 

Equations (6) and (7) allow determination of the temperature and pressure depend-enee of py from experimental measurements of the partial molar entropy (sometimes ealled latent heat) and partial molar volume, and Y, respectively. Unfortunately, however, Eqs (6) and (7) provide no explicit general equation for py as a function of y. From the form of Eq. (2) one can see that py and y will always appear as a product in the same term in every thermodynamic energy equation. Therefore, it will never be possible to derive a Maxwell equation from which the composition dependence of py can be experimentally determined. 

The chemical potential difference —ju may be resolved into its heat and entropy components in either of two ways the partial molar heat of dilution may be measured directly by calorimetric methods and the entropy of dilution calculated from the relationship A i = (AHi —AFi)/T where AFi=/xi —/x or the temperature coefficient of the activity (hence the temperature coefficient of the chemical potential) may be determined, and from it the heat and entropy of dilution can be calculated using the standard relationships... 

P-C-T Determinations Low Pressure Studies. Absorption isotherms obtained for the reaction of hydrogen with TiMo are shown in Figure 3 for 590°-392°C. These temperatures are above the decomposition temperature of /J-TiMo (see Figure 2) consequently, decomposition of the solid solution plays no role here. These data follow Sieverts Law only in the very dilute region—to hydrogen-to-metal ratios (H/M) of about 0.02. Thereafter, deviations in the direction of decreased solubility are observed. Data in the region of Sieverts Law can be used to determine the relative partial molar enthalpy and entropy at infinite dilution (47). From Sieverts Law (Equation 1), where Ks is a tempi/2 = Ksn (1)... 

It must be realized that because of kinetic limitations, most half-cells that can be written cannot be the basis of a practical cell which will display the appropriate emf. It has however proved convenient to include such halfequations in tables of redox potentials if their emf could be evaluated in some other way. In a large number of cases electrochemical data are not used at all. Rather, partial molar heats and entropies of the species involved are determined by calorimetric methods and these are used to derive AG°for the cell reactions. ceii values can then be calculated. 

For each extensive property, there is a corresponding partial molar and partial specific property. Consider any thermodynamic extensive property, such as volume, free energy, entropy, energy content, etc., the value of which, for a homogeneous system, is completely determined by the state of the system e.g., the temperature, pressure, and the amounts of the various constituents present. Thus, denoting by Y any extensive thermodynamic property, it can be represented by... 

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