State heat capacity

The ideal-gas-state heat capacity Cf is a function of T but not of T. For a mixture, the heat capacity is simply the molar average X, Xi Cf. Empirical equations giving the temperature dependence of Cf are available for many pure gases, often taking the form... 

The ideal state heat capacity of ethylene is given by the equation ... 

Dr. Adamson If there is a distribution of reaction paths, then the apparent activation energy should indeed change with temperature, and the effect would appear as a heat capacity of activation. However, it does not seem possible to distinguish this situation from that of a single reaction path where the transition state heat capacity is different from that of the reactants. That is to say, the formal thermodynamics would be identical for the two cases. 

The methods used in predicting these thermodynamic properties employ (a) an equation of state, relating the pressure-volume-temperature characteristics of the fluids (b) ideal gas state heat capacities of the individual components and (c) binary interaction coefficients between the components. The development of these basic relationships is not within the scope of this paper. Technical literature sources of the thermodynamic equations and data are given in the references. 

Property values in the standard state are denoted by the degree symbol (°). For example, C°P is the standard-state heat capacity. Since the standard state for gases is the ideal-gas state, C% for gases is identical with Cj , and the data of Table 4.1 apply to the standard state for gases. All conditions for a standard state are fixed except temperature, which is always the temperature of the system. Standard-state properties are therefore functions of temperature only. 

Analytic expressions for p,° and are obtainable by use of a polynomial expansion of the standard-state heat capacity C° = d f /dT ... 

Just as the fundamental property relation of Eq. (11.50) provides complete property information from a canonical equation of state expressing G/RT as a function of T, P, and composition, so the fundamental residual-property relation, Eq. (11.51) or (11.52), provides complete residual-property information from a PVT equation of state, from PVT data, or from generalized P VT correlations. However, for complete property information, one needs in addition to PVT data the ideal-gas-state heat capacities of tile species tliat comprise tlie system. In complete analogy, thefundamentalexcess-property relation, Eq. (11.86) or (11.89), provides complete excess-property information, given an equation for G /RT as a function of its canonical variables, T, P, and composition. However, tliis formulation represents less-complete property information tlian does the residual-propertyfonmilation, because it tells us no tiling about the properties of the pure constituent chemical species. 

Standard-state heat capacity, constant pressrne... 

The most satisfactory calculation procedure for the thermodynamic properties of gases and vapors is based on ideal gas state heat capacities and residual properties. Of primary interest are the enthalpy and entropy these are given by rearrangement of the residual property definitions ... 

It should be noted that Equation 49 leads to Equation 46 so that the MRK and RK equations have precisely the same difficulties with the heat capacity. Unfortunately, the LHW equation-of-state heat capacity is no better. 

HEP/HOV] Hepler, L. G., Hovey, J. K., Standard state heat capacities of aqueous electrolytes and some related undissociated species. Can. J. Chem., 74, (1996), 639-646. Cited on page 88. 

Harris FE, Rice SA (1954) A chain model for polyelectrolytes. Int J Phys Chem 58 725-732 Heller GT, Marcus Y, Waghorne WE (2002) Enthalpies and entropies of transfer of electrolytes and ions from water to mixed aqueous organic solvents. Chem Rev 102 2773-2836 Helgeson HC, Kirkham DH, Flowers GC (1981) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures tmd temperatures IV. Calculation of activity coefficients, osmotic coefficients, and apparent moled and standard and relative partial moM properties to 600 °C and 5 kb. Am J Sci 281 1249-1516 HeplerLG, Hovey JK (1996) Standard state heat capacities ofaqueous electrolytes and some related undissociated species. Can J Chem 74 639-649... 

Here the partial differential term can be calculated from an equation of state. Heat capacity data are available at very low pressures or at very large specific volume, where all fluids are ideal gases ... 

Figure 10.12 Standard state heat capacity of aqueous NaCI as a function of temperature. Squares - Criss and Cobble (1961). Open circles - Gardner, Mitchell and Cobble (1969). Crosses - Helgeson et al. (1981). Line - Pitzer et al. (1984).

The normal state heat capacity obeys the usual Cp = yT + AT + BT law. On 100% a-La samples Ohtsuka and Satoh (1966) found T = 4.9 K, = 10.0 mJ/mole-K and dD. (0) = 142 K. From the magnitude of the discontinuity in Cp at the superconducting transition comparison can be made with the law of corresponding states for the two lanthanum phases (Finnemore et al., 1965 Finnemore and Johnson, 1966 Lounasmaa and Sundstrdm, 1%7) similarly for superconducting alloys containing lanthanum (Ohtsuka and Satoh, 1966-LaY and Bonnerot et al., 1966-LaGd). 

Recent experiments on (LaPr)Sm alloys have rekindled interest in the possibility of a Kondo effect in a Pr impurity system. The early observation by Bucher et al. (1968) of a large depression of the superconducting transition temperature Tc of LaSns by Pr impurities (relative to other R impurities) is shown in fig. 11.11. This initial observation stimulated McCallum et al. (1975a) to perform more detailed measurements of the depression of with Pr concentration, the superconducting and normal state heat capacity, and the normal state magnetic susceptibility of (l Pr)Sn3 alloys. Calculated values of the ther-... 

A standard state pressure of 1 standard atmosphere (101,325 Pa) has been changed to 1 bar (lO Pa). The standard state for a monomer in liquid form is the pure monomer at 1 bar, liquid or solid. For monomers as gas, the standard state is the pure monomer in the ideal gas state at 1 bar. C° is the standard state heat capacity. The standard heat of polymerization can be written as... 

Ernst G., Biisser J. Ideal and real gas state heat capacities Cp of C3H8, i—C4H10, C2F5CI, CH2CICF3, CF2CICFC2 and CHF2CI—J. Chem. Thermodyn., 1970, v. 2, p. 787—791. 

A reliable Ahyd// and a reliable enthalpy of formation of either the reactant or the product of hydrogenation leads to Af// of the other participant in the reaction because Af//(H2) = 0 at 298 K by definition. Computed thermochemical properties are valid for a single, isolated molecule, hence they should be compared to experimental measurements made on the sample in the ideal gas state. Heat capacities are the most difficult of the common thermochemical properties to calculate, consequently temperature corrections are questionable if the temperature range is large. The most valid comparison is between calculated Ahyd// at 298 K and measured Ahyd//(gas, 298),... 

Equation 38 defines Tf for the general case when the glass-state heat capacity Cu.p depends on temperature. 

The 0% crystalline amorphous state heat capacities in various temperature ranges, all in J/K/mol. 

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