Heat capacity excess

This is of course an excess heat capacity, an amount in addition to the contributions of the heat capacities ( y... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Figure 3.2. Excess heat capacity chart (reproduced from Sterbacek et al. (1979), with permission)...

Here we will report measurements on the heat capacity of two NTD Ge 34B wafers, one non-metallized (only doped and annealed) and the other metallized (by B+ implantation and Au deposition). The comparison of data obtained from the non-metallized NTD Ge wafer and from the wafer with electrical contacts revealed an excess heat capacity, which can be attributed to the implantation process with B ions [44],... 

Instead, an anomalous trend and an excess heat capacity appear at low temperature in the case of the metallized wafer. 

In Fig. 12.16, the metallized wafer, in contrast to the non-metallized one, shows a large excess contribution in the heat capacity. The excess heat capacity cannot be attributed to the term CM of eq. (12.11), since the material of the two wafers is the same, and they were produced in the same run of neutron irradiation. Instead, this excess is to be attributed to the metallization process. 

The excess heat capacity (see Section 12.6.4) also spoils the performance of the detector below about 30 mK. 

Figure 7. Excess heat capacities of various nonelectrolytes in water at 25°C 0c° ofM and PD were obtained by extrapolation of data above 0.2m to infinite dilution.

When one has a model or related data to estimate the normal or lattice heat capacity in the transition region, the enthalpy of transition for a continuous transition can be obtained as an integration of the excess heat capacity, defined as... 

Figure 16.8 The excess heat capacity, C xcess = Cp (protein + buffer) - C/,(buffer), as a function of temperature, obtained from a differential scanning calorimeter. The shaded area is intergrated to yield the enthalpy of the transition. AtransCp is the difference in the baseline heat capacity at Tm.

With additional information, including the heat capacity of the buffer solvent, the partial specific volumes (volume per gram of the solute), and the specific volume of the solvent, one can extract the partial specific heat capacity (J K 1g I) of the solute. Privalov has summarized these calculations.8 Because the solutions are studied at very low concentrations, it is assumed that the contribution to the total heat capacity from the solvent cancels out when one calculates the excess heat capacity. With only minor exceptions, the procedures used to calculate parameters associated with the transformations in nucleic acids and in proteins are the same and yield quantities that are interpreted in similar ways, although researchers in these two fields may use a different notation for the same quantity. 

The Calorimetrically Obtained van t Hoff Enthalpy In a manner analogous to that used to obtain the van t Hoff enthalpy from the fractional change in the optical absorbance, one can use the temperature dependence of the fractional enthalpy as a function of temperature to determine an effective enthalpy. We will adopt the notation to represent the total enthalpy associated with the denaturation transition. It can be obtained from an integration of the excess heat capacity, corrected for the baselines, as discussed before ... 

Thus, the derivative can be evaluated at any temperature by dividing the baseline-corrected excess heat capacity, ACP(T), by the integrated enthalpy A /f. By substituting this result into equation (16.19) one obtains... 

A. Inglese, J.-P. E. Grolier, and E. Wilhelm, Excess Volumes and Excess Heat Capacities of Oxane + Cyclohexane and 1, 4-Dioxane + Cyclohexane , Fluid Phase Equilib., 15, 287-294 (1984) and D. D. Deshpande and S. L. Oswal, Thermodynamics of Mixtures Containing p-Dioxan or Tetrahydrofuran. 1. Excess Gibbs Free Energies and Excess Volumes , J. Chem. Thermodyn., 7, 155-159 (1975). 

Fig. 5. The excess heat capacity of PrFj, NdF3, DyFj and ErF3 as calculated from the crystal field energies.

The experimental excess heat capacity thus obtained as the difference between measured Cp and Qat can then be compared to the values calculated from the crystal field levels. As an example, fig. 7 shows the good agreement of the experimental and calculated excess heat capacity of DyF3. 

Fig. 18. The excess heat capacity inLnCl3 compounds , CeCl3 o, PrCl3 A, NdCl3 A, Pma3 . Sma3, , EuCl3.

Fig. 4. Schematic representation of the partition function [Eq. (1)] for protein folding/unfolding. Each state, from the native state (i = 0) to the unfolded state (i = n) and all intermediates (i = 1 to n - 1), is assigned a AG relative to the native state from which the statistical weights are obtained. The partition function, Q, is simply the sum of the statistical weights of all the states. Other important parameters, including the population of each state [Eq. (2)], the excess enthalpy [Eq. (3)], and the excess heat capacity [Eq. (4)], are determined from the partition function as described in the text.

The thermal stability of metmyoglobin and apomyoglobin has been extensively studied under different solvent conditions (Privalov et al., 1986). In particular, it was shown that at low pH values both the heat and cold denaturation peaks are clearly visible in the calorimetric scans. Figure 10 shows the excess heat capacity function for apomyoglobin predicted by the hierarchical partition function and the thermodynamic parameters described above. In order to simulate the experimental curve obtained at pH 3.83 (Privalov et al., 1986), the protonation of five specific histidine residues on unfolding was... 

Fig. 10. Predicted excess heat capacity function versus temperature for myoglobin. The curve simulates the experimental curve obtained at pH 3.83 by Privalov et al. (1986). Under those conditions both the cold and heat denaturation curves can be studied experimentally. The predicted values are Tm,cold = 4°C Tm>heat = 58°C A// = 59 kcal mol-1 ACp = 2.45 kcal K-1 mol-1. The experimental values are Tm>coid = 3°C Tm>heat = 57.5°C AH = 53 kcal mol-1 ACP = 2.5 kcal K-1 mol-1 (Privalov et al., 1986). [Reprinted from Freire and Murphy (1991).]...

Furthermore, for excess heat capacity cE at constant pressure we obtain Eq. 8.22 ... 

This excess heat capacity cE is the difference between the mean molar heat capacity of the non-ideal binary solution, - C kat + j), and the mean molar heat capacity of... 

Figure 38. Excess heat capacities of mixing for t-butyl alcohol + water mixture (Arnaud etal., 1972).

The enthalpy, AH, and entropy, AS, of the transition were calculated from the heat capacity curve by integration of the excess heat capacity, ACp, and ACp/T, respectively. The normal heat capacities for the calculation were determined by interpolation of the heat capacities outside the transition region using the linear function... 

Hammers, W. E. De Ligny, C. L., "Determination of the Flory-Huggins X Parameter, the Heat of Dilution, and the Excess Heat Capacity of Some Alkanes in Polydimethylsiloxane by Gas Chromatography," J. Polym. Sci., Polym. Phys. Ed., 12, 2065 (1974). 

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