Standard transformed heat capacity

The standard transformed heat capacity at constant pressure of a reactant is discussed later in Chapter 10 on calorimetry. The calculation of A H ° using equation 4.5-3 looks simple, but note that the standard transformed Gibbs energies of formation of all of the species are involved in the calculation. These equations were applied to the ATP series by Alberty and Goldberg (1992). 

ArCp° standard transformed heat capacity of reaction (J K 1 mol )... 

These calculations can be checked in additional ways by use of Af Gj ° = Af Hj ° - TAf Sj ° and dAf Gj °/dpH = dAf Hj 7d pH - TdAf Sj °/dpH. More partial derivatives can be taken, but taking a second derivative with respect to the same variable is not likely to be very accurate. An example of a second derivative is the standard transformed heat capacity since Af Cp ° = -Td Af G/ Another example is the binding capacity, defined by di Cera, Gill, and Wyman (4). 

It is necessary to specify zero ionic strength here because Debye-HUckel adjustments for ionic strength depend on the temperature. Heat capacities and transformed heat capacities are discussed in an Appendix to this chapter. However, since there is not very much information in the literature on heat capacities of species or transformed heat capacities of reactants, the treatments described here are based on the assumption that heat capacities of species are equal to zero. When molar heat capacities of species can be taken as zero, both standard enthalpies of formation and standard entropies of formation of species are independent of temperature. When Af H° and Af 5° are independent of temperature, standard Gibbs energies of formation of species at zero ionic strength can be calculated using... 

Rand and Kubaschewski have provided a critical assessment of all the available thermochemical data for binary compounds of uranium and a consistent set of values, including in some instances estimates, for enthalpies of formation and standard entropies, enthalpies of transformation, heat capacities, vapour pressures, and free energies of formation. 

Reiterating, heat of transformation, or H, is involved in change of form of the material while heat capacity relates to its internal change in temperature as it approaches another point of change. Both of these constants are based upon the standard of energy, or heat, of one (1.00) calorie. Heat capacity is also known as thermal capacity. 

As we mentioned, it is necessary to have information about the standard enthalpy change for a reaction as well as the standard entropies of the reactants and products to calculate the change in Gibbs function. At some temperature T, A// j can be obtained from Af/Z of each of the substances involved in the transformation. Data on the standard enthalpies of formation are tabulated in either of two ways. One method is to list Af/Z at some convenient temperature, such as 25°C, or at a series of temperatures. Tables 4.2 through 4.5 contain values of AfZ/ at 298.15 K. Values at temperatures not listed are calculated with the aid of heat capacity equations, whose coefficients are given in Table 4.8. 

When the pH is specified, the standard transformed molar heat capacity of a species is given by (Alberty, 200Id)... 

In Chapter 4 the effects of temperature on Af G ° and AfH ° and on ArG ° and ArH ° are discussed on the basis of the assumption that A,H° at zero ionic strength is independent of temperature. Therefore the effects of heat capacities of species were not treated. When a biochemical reactant contains two or more species, the standard transformed molar heat capacity of the pseudoisomer group is given by (Alberty, 1983a)... 

The second term in this equation is always positive because the weighted average of the squares of the individual standard transforme enthalpies of formation of the species is always greater that the square of the weighted average enthalpy of formation (6). This is the quantitative expression of Le Chatelier s principle for the heat capacity. 

By way of illustrations we display in Fig. 1.17.2a plot of the molar heat capacity of oxygen under standard conditions. The plot of Cp vs. In T is then used to determine the entropy of oxygen from the area under the curves. Note that the element in the solid state exists in three distinct allotropic modifications, with transition temperatures close to 23.6 and 43.8 K the melting point occurs at 54.4 K, and the boiling point is at 90.1 K. All the enthalpies of transition at the various phase transformations are accurately known. An extrapolation procedure was employed below 14 K, which in 1929 was about the lower limit that could conveniently be reached in calorimetric measurements. 

The standard entropy of a-SnSe was evaluated to be 86.93 J-K" -mol , corresponding to Af5° (SnSe, a, 298.15 K) = - 6.3 J-K -mor, in the thermodynamic optimisation and assessment of the Sn-Se system in [96FEU/MAJ]. The value originates mainly from the modelling and assumptions made about the liquid phase in the system and the recalculation to 298.15 K by the use of enthalpies of phase transformations and heat capacities. The only experimental determination of the entropy at low temperatures was made by Melekh, Stepanova, Fomina, and Semenkovich [71MEL/STE] who performed emf measurements on the galvanic cells... 

TABLE 6. Heat Capacity, Standard Entropy, Heats of Transformation, and Fusion of the Rare Earth Metals... 

Rare earth Heat capacity at 298K(J/molK) Standard entropy S ,a/molK) Heat of transformation (kj/mol) Heat of fusion... 

The situation is more difficult for the ArS values. These can be calculated according to Equation (3.2.1-12) from the standard entropies of the reactants S j, which, however, are often not available and can only be determined with major experimental effort form the heat capacities c (T) and the heats of transformation A H by the schematic integration scheme of Equation (3.1.2-13) ... 

Rare earth metal Heat capacity at 298 K (J/mol K) Standard entropy 298 (J/mol K) Irans. 1 Heat of transformation (kJ/mol) trans. 2 Heat of fusion (kJ/mol)... 

Thermodynamics. Owing to the occurring polymorphism for the trivalent R oxides and the redox instability for the mixed trivalent-tetravalent R oxides, even precisely obtained thermodynamic data may refer to uncertain compositions. In particular, this is the case for the cl to mC transformation, where the transformation temperatures are not well defined and the high-temperature heat capacities are therefore uncertain. On the other hand, the standard enthalpies and entropies themselves of these transformations are known, yet rather small (around +1.0 and +0.6 per R2O3, respectively). Owing to these problems, data involving these transitions as well as data for oxides subject to extensive redox interactions are not included in the essential thermodynamic data listed in tables 6 and 7. 

Kubaschewski, Evans, and Alcock have tabulated many data of interest to metallurgists, including enthalpies of formation and standard entropies at 298.15 K, heat capacities, enthalpies and temperatures of transformation, melting and boiling temperatures, vapour pressures, and standard free energies of reaction. Some data on binary metallic systems are also given. 

For the determination of standard Gibbs energies of reaction, a wide variety of experimental methods have been devised. These may be subdivided into e.m.f. measurements, equilibria with a gaseous phase, and distribution equilibria. From the temperature coefficients of the Gibbs energies, enthalpies and entropies of reaction can be deduced, but experience has shown that these cannot be relied upon when one or more solid phase takes part in the reaction, and errors are very difficult to assess. In such cases, it is recommended that the enthalpies of reaction are measured caloriraetrically and combined with the standard Gibbs energies to yield standard entropies of reaction. Calorimetric methods are also used to determine heat capacities, enthalpies of transformation, and enthalpies of fusion. Only for the determination of enthalpies of evaporation may... 

Adiabatic Calorimeters.—The adiabatic principle has also been applied in calorimetry for a long time and adiabatic calorimeters can be used for the determination of heat capacities and enthalpies of transformation and fusion, as well as enthalpies of reaction. The first adiabatic calorimeters were designed for room temperature. Low-temperature heat capacities for the determination of standard entropies, etc., are also measured by adiabatic methods. However, low-temperature calorimetry is the subject of Chapter 4. 

---