Debye temperature entropy-based

An alternative to deriving the Debye temperature from experimental heat capacities is to derive an entropy-based Debye temperature by calculation of the s that reproduces the observed entropy for each single temperature using... 

The S(T) on the left-hand side of Eq. (39) could be an entropy 5vib,atp(7 ) determined from experiments or it could be the result of a detailed model calculation based on a certain E((o) in Eq. (27). There is a solution 0d to Eq. (39) for each temperature T. We shall call that 0d a Debye temperature. Because we represent the property of a system that has 3rL degrees of freedom with a single parameter 0d, it is obvious that we have to pay a price. In this case, the price is that 0d varies with T and also with the physical property that is modeled. (Here 0d refers to the entropy.) One may therefore introduce one Debye temperature 0 that describes the vibrational entropy, another Debye temperature 0c that describes the vibrational heat capacity, etc. The heat capacity Debye temperature 0c would be the solution to... 

Here 0O is the characteristic temperature at volume V0. An average value for the volume dependence of the standard entropy at 298 K for around 60 oxides based on the Einstein model is 1.1 0.1 J K-1 cm-3 [15]. A corresponding analysis using the Debye model gives approximately the same numeric value. 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

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... 

The heat capacity and entropy of TiBr Ccr) have been measured over the temperature range 51 to 800 K by King et al. (2). Heat capacities above 800 K are estimated from graphical extrapolation. The value of S"(298.15 K) is derived from these data, based on S (51 K) - 8.60 cal K mol. The value of S (51 K) is estimated from a Debye-Einsteln extrapolation of the measured heat capacities, the equation being C - D(70.0/T) + E(120/T) + 2E(306/T). It is assumed that all electronic entropy is... 

There remains the experimental problem of determining S(riowest). the entropy at the lowest attainable temperature. If very low temperatures (within a few kelvins of T = 0) can be reached and reliable measurements of Cp made, it is possible to use an extrapolation based on the observation that many non-metallic substances have a heat capacity that obeys the Debye T -law ... 

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