Heat capacity theory

By theory, heat capacity is infinite at the critical point, which has also been proved experimentally. 

V. V. Tarasov. Zhur. Fiz. Khim. 26, 1374-82 (1952). Theory heat capacity, H2SO43H2O. 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

However, the possibility that might not go to zero could not be excluded before the development of the quantum theory of the heat capacity of solids. When Debye (1912) showed that, at sufficiently low... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

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. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

CA 78, 161665 (1973) A math analysis of the theory is presented on the basis of the combustion rate, the thermal conductivity, the heat capacity, the surface temp of the proplnt grains, and other factors. Expts were made to determine the relation of the combustion rate to acceleration for various proplnts. The rate of combustion at 70 atm was compared with the initial rate. The. relation of the critical pressure of transitional laminar combustion to acceleration, and the dependence of the combustion rate of nitroglycol to the pressure at various acceleration rates were determined. Exptl observations were compared with results of theoretical calcns... 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

Moreover, the mesophase-volume fractions Oj for the same inclusion-contents were determined from the experimental values of heat-capacity jumps ACp at the respective glass transition temperatures T f by applying Lipatov s theory. Fig. 7 presents the variation of the differences Ars oi the radii of the mesophase and the inclusion (rf), versus the inclusion volume content, uf, for three different diameters of inclusions varying between df = 150 pm and df = 400 pm. 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

Section V, other quantum effects are indeed present in the theory and we will discuss how these contribute both to the deviation of the conductivity from the law and to the way the heat capacity differs from the strict linear dependence, both contributions being in the direction observed in experiment. Finally, when there is significant time dependence of cy, the kinematics of the thermal conductivity experiments are more complex and in need of attention. When the time-dependent effects are included, both phonons and two-level systems should ideally be treated by coupled kinetic equations. Such kinetic analysis, in the context of the time-dependent heat capacity, has been conducted before by other workers [102]. 

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. 

We have carrried out an analysis of the multilevel structure of the tunneling centers that goes beyond a semiclassical picture of the formation of those centers at the glass transition, which was primarily employed in this chapter. These effects exhibit themselves in a deviation of the heat capacity and conductivity from the nearly linear and quadratic laws, respectively, that are predicted by the semiclassical theory. 

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