Heat capacity approaches zero

Classical thermod)mamics using the equipartition of energy principle predicts that the lattice molar heat capacity will be given by 1/2R for each of the six degrees of freedom in a solid (three kinetic and three potential energy) for a total of 3R. As the temperature is increased, the observed heat capacity of materials approaches this value, which is known as the Dulong-Petit limit. However, at low temperatures, the observed heat capacity approaches zero as 7. ... 

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

The formation of S o in sulfur melts is a slow reaction, and it takes about 1 h at 160 °C to establish the equilibrium concentration [24, 58]. From the temperature dependence of the polymer content, from the heat capacity Cp of the melt [29] as well as from calorimetric measurements [56, 58] it was concluded that the reaction Ss Sqo is endothermic with an estimated activation energy of ca. 120 kj mor (Ss) [58]. The same value was derived from DSC measurements of liquid sulfur [58]. In this context it was observed that the sudden viscosity increase of liquid sulfur takes place at exactly 159 only if the heating rate approaches zero. If the heating rate is varied between 1.25 and 40 K min higher transition temperatures are observed as the data in Table 1 show [58]. 

Heat capacities of solids are always functions of temperature, as illustrated for several ceramics in Figure 34.1. Note the units are J g-atom internal energy of a solid is capacity is zero. As the temperature rises the heat capacity increases, which is indicative of the various mechanisms by which energy is absorbed. The heat capacity approaches... 

It is manifestly impossible to measure heat capacities down to exactly 0 K, so some kind of extrapolation is necessary. Unless were to approach zero as T approaches zero, the limiting value of C T would not be finite and the first integral in equation (A2.1.71) would be infinite. Experiments suggested that C might... 

For Cy/T to approach zero as T approaches zero, CV must go to zero at a rate at least proportional to T. Earlier, we summarized the temperature dependence of Cy on T for different substances and showed that this is true. For example, most solids follow the Debye low-temperature heat capacity equation of low T for which... 

Figures 9.17-9.19 clearly show that, as the Biot number approaches zero, the temperature becomes uniform within the solid, and the lumped capacity method may be used for calculating the unsteady-state heating of the particles, as discussed in section (2). The charts are applicable for Fourier numbers greater than about 0.2.

Note that we have drawn CP approaching zero as 0. This feature is a general phenomenon, and explained by quantum mechanics. At low temperatures, the energy available is so small that there is not enough to stimulate transitions to higher energy states, so the sample cannot take up energy, and its capacity for heat is zero. 

Thus, even at temperatures well above absolute zero, the electrons are essentially all in the lowest possible energy states. As a result, the electronic heat capacity at constant volume, which equals d tot/dr, is small at ordinary temperatures and approaches zero at low temperatures. 

Since co2 =K/m, the mean potential and kinetic energy terms are equal and the total energy of the linear oscillator is twice its mean kinetic energy. Since there are three oscillators per atom, for a monoatomic crystal U m =3RT and Cy m =3R = 2494 J K-1 mol-1. This first useful model for the heat capacity of crystals (solids), proposed by Dulong and Petit in 1819, states that the molar heat capacity has a universal value for all chemical elements independent of the atomic mass and crystal structure and furthermore independent of temperature. Dulong-Petit s law works well at high temperatures, but fails at lower temperatures where the heat capacity decreases and approaches zero at 0 K. More thorough models are thus needed for the lattice heat capacity of crystals. 

For Solid or Liquid. For either of these final states, it is necessary to have heat capacity data for the solid down to ternperamres approaching absolute zero. 

Entropy of Gaseous Cyclopropane at its Boiling Point. Heat capacities for cyclopropane have been measured down to temperatures approaching absolute zero by Ruehrwein and Powell [12]. Their calculation of the entropy of the gas at the boiling point, 240.30 K, is summarized as follows ... 

At low f thermal energy is not enough to ensure statistical occupancy of all energy levels accessible to each atom, and the heat capacity (either Cy or Cp) approaches zero as absolute zero is approached. 

Debye s investigations on the energy content of substances at low temperatures be expressible in the form u — Uq aT , where a is determinable from the heat capacity of the surface film, and the temperature coefficient of the heat of wetting should decrease rapidly as we approach the absolute zero. Furthermore it is evident that at this temperature the free and total surface energies should be identical in value, the total surface energy sinking first slowly and then rapidly as the critical temperature is reached. Confirmation likewise of the assumption Lt = 0 or that the temperature... 

Frequently, the context of a particular problem requires us to consider the limiting behaviour of a function as the value of the independent variable approaches zero. For example, consider the physical measurement of heat capacity at absolute zero. Since it is impossible to achieve absolute zero in the laboratory, a natural way to approach the problem would be to obtain measurements of the property at increasingly lower temperatures. If, as the temperature is reduced, the corresponding measurements approach some value m, then it may be assumed that the measurement of the property (in this case, heat capacity) at absolute zero is also m, so long as the specific heat function is continuous in the region of study. We say in this case that the limiting value of the heat capacity,... 

Thus vibrations with frequencies v much higher than k T / h contribute essentially nothing to the total energy or to the heat capacity dE/dT. As the temperature approaches zero, all of the vibrational modes have frequencies much higher than k T / h, and thus they cease to contribute to the heat capacity. 

Show that in order to use the third law to calculate entropies, the heat capacity must approach zero as T —> 0 at least as quickly as does T. 

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