The Difference in Heat Capacities

From the definition of enthalpy applied to one mole of an ideal gas dH = dU d(PV) = dU RdT Introduction of Eqs 2.6.3 and 2.6.2 into this expression yields  

Consider one mole of an ideal gas that undergoes an adiabatic and reversible change from the initial condition P Vy, and Tj, to a final pressure Pj. What are the corresponding values of T2 and V-p. 

From the first law for an adiabatic process dW =-dU. Since the process is reversible, dW = PdV, which for an ideal gas becomes  

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AC is interpreted as the difference in heat capacities between the transition state and the reactants, and it may be a valuable mechanistic tool. Most reported ACp values are for reactions of neutral reactants to products, as in solvolysis reactions of neutral esters or aliphatic halides. " Because of the slight curvature seen in the Arrhenius plots, as exemplified by Fig. 6-2, the interpretation, and even the existence, of AC is a matter of debate. The subject is rather specialized, so we will not explore it deeply, but will outline methods for the estimation of ACp. 

A thermogram from a differential scanning calorimeter. The peak indicates a phase change in the sample, and the difference in base line before and after the phase transition is due to the difference in heat capacities of the two phases. 

Kirchhoff s law The relation between the standard reaction enthalpies at two temperatures in terms of the temperature difference and the difference in heat capacities (at constant pressure) of the products and reactants. 

Actually, the reaction peak in 7.2.5. is an idealized curve since the baseline is a function of the difference in heat capacities between reference to sample and reference and product. Usually, we have a different baseline, vis ... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

It can be concluded from equations 12.11 and 12.12 that the small deviation of the zero line relative to the isothermal baseline under the same scanning conditions is proportional to the heating rate and the difference in heat capacities of the two empty crucibles. This deviation can be positive (as in figure 12.4) or negative, depending on the magnitude of the intrinsic thermal asymmetry of the system under scanning conditions and the relative masses of the two crucibles. When the sample is introduced in the sample crucible,... 

Hence, the difference in heat capacities of products and reactants is given by the equation... 

According to Equation (4.5) the ideal solubility of a compound is only dependent upon the heat of fusion, the difference in heat capacity of the solid and supercooled liquid and the melting point of the compound. Since there are no properties of the solvent included in the ideal solubility equation, the solubility of a compound should be the same in all solvents. This equation overlooks all solute-solvent and solvent-solvent interactions. 

In Table IV the heats of formation of several nitroaromatic compounds in the ideal gas state are calculated from the measured heats of formation of the solid and the measured heats of sublimation. In cases where the heat of sublimation is not measured at 298°K there should be a correction for the differences in heat capacities of the solid and ideal gas. The data required to make these corrections are not available but in general it is expected that the corrections will be small and can be neglected. From the heats of formation of each compound in the ideal gas state, a value for the group Cb-N02 (ideal gas) has been derived (Table IV). A weighted-average value (CB-N02(ideal gas)) = 3.0 kcal/mole was used, and a heat of formation was estimated for each compound. In Table IV the difference between observed and estimated heat of formation in the ideal gas state is less than + 3.2 kcal/mole in all cases... 

According to Hildebrand et al. (1970), the simpliLed solubility equation (assuming the difference in heat capacity of the crystalline and the hypothetical molten forms of the solute is negligible) for a crystalline solute in a regular solution is... 

The difference in heat capacity between unfolded and folded protein is assumed, based on good experimental data, to be positive and constant [Eq. (17.3)]. 

Similar to the DTA system noted above, differential scanning calorimetry (DSC) consists of a sample cell and reference cell, where platinum sensors detect the temperature in each cell. In the case of DSC, however, each cell possesses an individual heater (Figure 18.11B) so that energy input to the individual heaters is recorded as the instrument attempts to maintain equivalent temperatures in each cell while scanning a preset temperature range. The detector thus is able to directly measure the difference in heat capacity between the sample cell and the reference cell. 

Notice that we have used the standard state heat of reaction to estimate the heat evolved at 515 K. This is only approximately true since the standard state heat of reaction should be adjusted for the difference in heat capacities between the products and the reactants at the high temperature. The only point we are trying to convey is that the available work is significantly reduced when we use steam to drive a turbine. Furthermore, because of inefficiencies of the turbine and the generator, the electrical energy generated this way ends up as a small fraction of the 43.1 kJ mol toluene that is theoretically available from the reaction. 

Although a large number of thermal analysis techniques have been developed, the most commonly applied are those of thermogravimetry (TG, the measure of thermally induced weight loss of a material as a function of applied temperature), differential thermal analysis (DTA, the difference in temperature existing between a sample and a reference as a function of temperature), and differential scanning calorimetry (DSC, the difference in heat capacity between the sample and... 

For example, the heats of fusion and solution have been reported for the polymorphs of auranofm [48], and these are summarized in Table 3. The similarity of the heats of transition deduced in 95% ethanol (2.90 kcal/mol) and dimethylformamide (2.85 kcal/mol) with the heat of transition calculated at the melting point (3.20 kcal/mol) indicates that the difference in heat capacity between the two polymorphs is relatively small. 

The unique feature of DSC is the determination of heat capacities (specific heats). As noted, the differential power (cal/sec) divided by the heating rate (°C/sec) yields the difference in heat capacities between the sample and the reference (in the baseline regions only). A change in heat capacity is seen by a shift in the baseline. A sharp increase in the baseline of the plot is typical of glass transitions in polymers. By comparing the heat capacity of the sample with the known heat capacity of the standard, the absolute heat capacity of the sample can be calculated. 

Here a. (T) is the activity of component i in a stoichiometric liquid at the liquidus temperature, T, AS (IC) is the entropy of fusion of compound IC at the melting temperature, T, and AC (IC) is the difference in heat capacity between themstoichiom6tric liquid and the solid compound. This sequence is the same as that proposed for binary III-V systems by Vieland (5.). 

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