Partial molar heats

Corresponding to the integral heat and entropy of formation of the solution are the partial molar heats A//, and entropies AS, of solution of the components where... 

Along the three-phase line liquid-clathrate-gas the variation of the composition with temperature is considerable (cf. CD in Fig. 3), because when applying Eq. 27 to this equilibrium, the relatively small quantity AH = 0.16 kcal/mole has to be replaced by the much larger difference/ —//ql between the partial molar heat functions of / -hydroquinone and the liquid phase, which amounts to about —6 kcal/mole. The argon content of the solid reaches a minimum at the quadruple point. 

By measuring the partial molar heat of solution as a function of temperature for infinitely dilute concentrations of Cu, Ag, and Au in liquid tin, Oriani and Murphy51 have determined ACp for the liquid solutes to be 1.0, 0, and 3.0 cal/deg mole respectively. These numbers bear no relationship to the sign of the heat of solution, or to atom-size disparity, but seem to be related to the deviation from unity of the ratio of the masses of the components. 

It is more interesting to examine the behavior of theory with respect to solutions of moderate dilutions. The partial molar heats of solution of copper, silver, and gold in liquid tin have been measured51 at solute concentrations from 0.0005 to 0.02. A schematic... 

Fig. 3. Schematic diagram for the variation with concentration of the partial molar heat of solution of the liquid noble metals into liquid tin, taken from reference 51. The numbers are the experimental and calculated AHt for the solutes, in cal/g atom, at the two concentrations of 0 and 0.02 mole fraction. Q-C labels the curves calculated by the quasichemical theory in first order B-W labels the curves calculated by the Bragg-Williams, or zeroth-order approximation, which assumes a random...

Gibbs-Duhem equation, 16 Gold, partial molar heat on solution in tin, 133... 

Figure 5.2 Partial molar heat capacities of EFO (Cp 1) and H2SO4 (Cp 2) against 2, the mole fraction of H2SO4.

Differentiation of equation (7.65) with respect to temperature gives an equation for Ji, the relative partial molar heat capacity, given by... 

The difference Cp. -C°pi is the relative partial molar heat capacity Jt. Thus... 

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... 

If the partial molar heat capacities are substantially constant over the temperature range of interest, this equation may be solved to determine the relationship between the temperature and the fraction conversion. 

For a nonideal solution, CPi is replaced by the partial molar heat capacity, CPl, but such information may not be available. 

Information on partial molar heat capacities [1,18] is indeed very scarce, hindering the calculation of the temperature correction terms for reactions in solution. In most practical situations, we can only hope that these temperature corrections are similar to those derived for the standard state reactions. Fortunately, due to the upper limits set by the normal boiling temperatures of the solvents, the temperatures of reactions in solution are not substantially different from 298.15 K, so large ArCp(T - 298.15) corrections are uncommon. 

Selected entries from Methods in Enzymology [vol, page(s)] Aspartate transcarbamylase [assembly effects, 259, 624-625 buffer sensitivity, 259, 625 ligation effects, 259, 625 mutation effects, 259, 626] baseline estimation [effect on parameters, 240, 542-543, 548-549 importance of, 240, 540 polynomial interpolation, 240, 540-541,549, 567 proportional method for, 240, 541-542, 547-548, 567] baseline subtraction and partial molar heat capacity, 259, 151 changes in solvent accessible surface areas, 240, 519-520, 528 characterization of membrane phase transition, 250,... 

This, of course, is the difference between the heat capacity of the solution and the sum of those of the unmixed liquid elements. Using Eq. (38) and defining relative partial molar heat capacities of the components as... 

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. 

In all measurements, the heating rate was 1 °C min-1. The partial molar heat capacity of a fully extended peptide is calculated from its amino acid composition according to the method of Privalov and Makhatadze.1 37 Partial specific volume of the peptides was calculated from amino acid composition according to Makhatadze et al.,[138l with a value of 0.751 mL-mg-1. 

Figure 17.5 Derived thermodynamic properties at T — 298.15 K and p = 0.1 MPa for (2Cic-CfiHi2 + X2n-CjHi4) (a) excess molar heat capacities obtained from the excess molar enthalpies (b) relative partial molar heat capacities obtained from the excess molar heat capacities (c) change of the excess molar volume with temperature obtained from the excess molar volumes and (d) change of the excess molar enthalpies with pressure obtained from the excess molar volumes.

Cp m of -1.4 J K-1 - mol-1 is again of moderate size. Figure 17.5b summarizes the relative partial molar heat capacity Jt = (CA m,- C t m,We note that the molar heat capacity of hexane in the infinitely dilute solution is 7.4 J-K 1 - mol-1 less than the molar heat capacity of pure hexane. 

Thermal Properties Pitzer s equations for ln7 and 4> [equations (18.18) to (18.26)] can be used to obtain relative partial molar enthalpies L and L2, and relative partial molar heat capacities8 7j and J2, by taking derivatives. For... 

The relative partial molar enthalpy and relative partial molar heat capacity are obtained from8... 

In equations (18.91) and (18.92), C° 2 and V are the partial molar heat capacity and partial molar volume of the surfactant in the infinitely dilute solution (standard state values). 

A different result is obtained when we consider the partial molar enthalpy, the partial molar volume, the partial molar heat capacity, and all other higher derivatives taken with respect to the temperature or pressure. At the composition of the reference state, AH, AP, and ACp k x are all equal to zero. Then we have, from Equations (8.78)-(8.80),... 

Thus, the value of (SJ (g) — Sj(/)) at a given temperature may be determined from the slope of the curve of In xt plotted as a function of In T at the given temperature, provided that (d Afi2/dx2)T P can be evaluated from experiment or theory. Similarly, (H (g) — f j(/)) can be calculated at a given temperature from the slope of the curve of In xt plotted as a function of 1/T at the given temperature with the same provision. The values so determined are not isothermal when isothermal values are desired, then a knowledge of the partial molar heat capacity of the solvent in the liquid phase and the molar heat capacity of the component in the gas phase would be required. 

The heat capacity C is an extensive property and, for a mixture of substances i, is given as the sum of the partial molar heat capacity cp i of all the constituent substances each multiplied by the number of moles ni of i as shown in Eq. 2.16 ... 

Recalling d(dH dIf)/dT - d(0HldT)/d%, we have from Eq. 2.15 the heat of reaction at constant pressure as a function of the heat capacities, Cp, of all the reaction species. The temperature dependence of the heat of reaction at constant pressure is thus determined by the partial molar heat capacities, cp of the reaction species as shown in Eq. 2.29 ... 

This equation enables us to calculate the heat of a reaction at any temperature, provided that we know the value of the heat of the reaction at a specified temperature and that we know the partial molar heat capacities cpi of all the species taking part in the reaction cp i may be equated to the molar heat capacities of the pure species in the case of gas reactions. By integrating Eq. 2.29 with respect to temperature we obtain Eq. 2.30 for the temperature dependence of the heat of reaction ... 

We see that the average affinity A of a reaction at a temperature T can be calculated, if we know (a) the average affinity A 11 at one specified temperature T0 at the pressure p (b) the heat of reaction (AH)T(j p at T0 and (c) the partial molar heat capacities of the constituent substances as a function of temperature throughout the whole range from roto T. 

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