Heat capacity functional forms

For many crystalline compounds, Cp values are experimentally known and tabulated at various T conditions. In several cases, the data are also presented in the interpolated form of a heat capacity function through Maier-Kelley-type polynomials. 

Table 5.36 Thermodynamic properties of pure pyroxene components in their various structural forms according to Saxena (1989) (1), Berman (1988) (2), and Holland and Powell (1990) (3) database. = standard state entropy of pure component at 7) = 298.15 K and Py = bar (J/mole) Hjp p = enthalpy of formation from elements at same standard state conditions. Isobaric heat capacity function Cp is... 

Figure 5.56 Maier-Kelley heat capacity functions for various structural forms of the KAlSi30g (A) and NaAlSi30g (B) feldspar end-members (solid curves), compared with T derivative of hnite differences at various T (hlled symbols) (from Helgeson et ah,...

Table 5.65 Thermodynamic properties of feldspar end-members in various structural forms according to Helgeson et al. (1978). Heat capacity function Cp= K + K Y +...

The high temperature form can easily be quenehed to room temperature. Thus, the heat capacity function for the stoichiometric composition of this phase can be derived from... 

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

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 coefficient of the 8-function reflects the pile-up of the two-level systems that would have had a value of e < S were it not for quantum effects. These fast two-level systems will contribute to the short-time value of the heat capacity in glasses. The precise distribution in Eq. (69) was only derived within perturbation theory and so is expected to provide only a crude description of the interplay of clasical and quantum effects in forming low-barrier TLS. Quantitative discrepancies from the simple perturbative distribution may be expected owing to the finite size of a tunneling mosaic cell, as mentioned earlier. 

Heat capacity data are often available in some form of polynomial as a function of temperature, for example6 ... 

Johnson et al. [143] used low-temperature adiabatic calorimetry and high-temperature drop calorimetry to obtain the heat capacity of both forms of mordenite as a function of the temperature. These results and the results of the reaction-solution calorimetric studies discussed herein, enabled the tabulation of the thermodynamic properties (C°, S°, Af H°, and Af G°) of mordenite from 0 K to 500 K and dehydrated mordenite from 0 K to 900 K. 

Table 22.1), which has been modified by appropriate substitutions to yield the desired molecule. Thus, aliphatic hydrocarbons can be built up from methane by repeated substitutions of methyl groups for hydrogen atoms. Other compounds are formed by substitution of functional groups for CHn groups. The heat capacity constants are those for a cubic polynomial in the temperature, which are similar to those discussed in Chapter 4. 

Gibbs Free Energy of a Phase at Higb P and T, Based on tbe Functional Forms of Heat Capacity, Thermal Expansion, and Compressibility... 

We have seen (section 3.2) that heat capacity at constant P can be expressed in a functional form representing its T-dependency. Adopting, for instance, the Haas-Fisher polynomial ... 

Medium-chain alcohols such as 2-butoxyethanol (BE) exist as microaggregates in water which in many respects resemble micellar systems. Mixed micelles can be formed between such alcohols and surfactants. The thermodynamics of the system BE-sodlum decanoate (Na-Dec)-water was studied through direct measurements of volumes (flow denslmetry), enthalpies and heat capacities (flow microcalorimetry). Data are reported as transfer functions. The observed trends are analyzed with a recently published chemical equilibrium model (J. Solution Chem. 13,1,1984). By adjusting the distribution constant and the thermodynamic property of the solute In the mixed micelle. It Is possible to fit nearly quantitatively the transfer of BE from water to aqueous NaDec. The model Is not as successful for the transfert of NaDec from water to aqueous BE at low BE concentrations Indicating self-association of NaDec Induced by BE. The model can be used to evaluate the thermodynamic properties of both components of the mixed micelle. 

We cannot answer the question posed by Anfin-sen s hypothesis. Does the native state have a minimum value of the Gibbs energy Nevertheless, it is observed that proteins usually behave as if folded, unfolded forms are in a true thermodynamic equilibrium, and that this equilibrium is attained rapidly. The difference AG between a folded and a denatured protein is only 21-63 kj mol-1, which shows that folded proteins are only marginally more stable than are unfolded polypeptide chains.645 The value of AG of unfolding as a function of temperature T is given by Eq. 29-13, where AH(T) and ACp are the changes in enthalpy and heat capacity upon unfolding.645 646... 

The currently preferred method for the study of gas forming reactions as function of temp is DSC. Here the specimen and the ref sample are heated at programmed heating rates (in controlled atms, if desired) while the differential energy input to the specimens is recorded. Hence the pen movement is directly proportional to the heat capacity while the area under the curve represents the enthalpy change. New equipment is now on the market which can operate up to 1200° (Ref 79) and which is therefore adequate for expl and propint studies. Limitations on the use of DSC for kinetics studies of expls will be discussed under the entry Thermochemistry in this Vol... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The plot in Fig. 3.2 of the acid dissociation constant for acetic acid was calculated using equation 3.2-21 and the values of standard thermodynamic properties tabulated by Edsall and Wyman (1958). When equation 3.2-21 is not satisfactory, empirical functions representing ArC[ as a function of temperature can be used. Clark and Glew (1966) used Taylor series expansions of the enthalpy and the heat capacity to show the form that extensions of equation 3.2-21 should take up to terms in d3ArCp/dT3. 

Reference data on total energies of forms 19-23 optimized by means of different theoretical methods in the gas phase are given in Table 2. Various energetic characteristics of tetrazoles can be successfully estimated. The vertical adiabatic ionization potentials of both neutral tautomers 20 and 21 were calculated for a- and Tt-radical cations <2000CPL(330)212>. The standard molar thermodynamic functions (enthalpies, heat capacities, and entropies) of... 

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