Partial molar heat capacity

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... 

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. 

Fig. 4 Partial heat capacity functions of poly(NVCl-co-NVIAz) synthesized at 65 °C from the feeds with initial molar ratios of comonomers equal to a 85 15 and b 90 10 (the data from [26,27,42])...

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,... 

We now show that equations analogous to Eq. (34) follow for the enthalpy and entropy of mixing, AHM and ASM, but that, in contrast to the chemical potentials, the partial molar enthalpies and entropies for the components differ from those for the species. Finally we show that the equation for the constant pressure relative heat capacity is of a slightly more complicated form than Eq. (34). Equation (34) and its analogs for and ASM are necessary for comparison of model predicted quantities with experiment. From basic thermodynamic equations we have... 

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... 

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. 

The populations of other intermediate states are very small and can be neglected. For larger more complex proteins made up of multiple subunits, and in many fibrous proteins, this conclusion cannot be supported. Complex globular proteins appear to melt cooperatively in domains in which the smaller units melt independently, and the melting in fibrous proteins is even more complex. While the molar quantities for the heat capacity are dependent upon the size of the protein, the partial specific heat capacities of many proteins are very nearly the same. 

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. 

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). 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

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),... 

We find from this discussion that, when the reference state of a component in a multicomponent system is taken to be the pure component at all temperatures and pressures of interest, the properties of the standard state of the component are also those of the pure component. When the reference state of a component in a multicomponent system is taken at some fixed concentration of the system at all temperatures and pressures of interest, the system or systems that represent the standard state of the component are different for the chemical potential, the partial molar entropy, and for the partial molar enthalpy, volume, and heat capacity. There is no real state of the system whose properties are those of the standard state of a component. In such cases it may be better to speak of the standard state of a component for each of the thermodynamic quantities. 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

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. 

Fig. 7 Temperature dependence of partial heat capacity (Cp) of two pairs of NIPAM-co-VP copolymers in water. The weight average molar masses of NIPAM-co-VP/60/5, NIPAM-co-VP/30/5, NIPAM-co-VP/60/10 and NIPAM-co-VP/30/10 are 2.9 x 106, 4.2 x 106, 5.6 x 106 and 7.9 x 106 g/mol, respectively. The polymer concentration is 10-3 g/mL. The temperature was increased with a rate of 1.5 °C/min and pressure was maintained at 180 kPa [56]...

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