Contributions to the Data on Theoretical Metallurgy: [Part] 11. Entropies of Inorganic Substances: Revision (1948) of Data and Methods of Calculation Page: 6
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6 CONTRIBUTIONS TO DATA ON THEORETICAL METALLURGY
infinite and the integral evaluated as 7r4/15. At low temperatures,
E= 5h37m3 , (12)
C, = T3 a T3,
where a is substituted for the more-complicated constant multiplier.
Many substances have been studied to low enough temperatures to
demonstrate that the T3-law and general Debye equation usually are
adequate means of extrapolating heat-capacity data for the evalua-
tion of entropies. Consequently, for substances whose heat capacities
have been determined to low enough temperatures, these relationships
are employed in evaluating the first integral of equation (7) and the
integral of equation (6). The procedure is to consider these inte-
grals in two parts, the first of which accounts for the temperature
range covered by the measured heat capacities and is evaluated me-
chanically, as are the other integrals of equation (7). The second part
is entirely extrapolation. Tables of the heat capacity at constant
volume and the entropy for values of 0DI/T for the Debye function
may be found in several places (208, 309, 374, 375), so that the labor
involved in the extrapolation is slight. When the T3-law is obeyed
below a temperature T, the entropy at T is simply C,/3 for
T C,d T oT2dT aT3 C,
ST= T 3 3 (14)
The fact that the heat capacities of some substances, such as lead
and mercury, have been shown to deviate in various ways from Debye
behavior need not be considered here. The fluctuations that have
been observed up to the present do not produce errors in the extra-
polations that are of much importance, relatively speaking, in the
entropies at 298.160 K. However, they are of great theoretical interest.
The above remarks do not apply to the departure from Debye behavior
caused by splitting a multiple lowest-energy state, such as was
observed for Gd2(SO4)3.8H20 (to be discussed later).
Unfortunately, the Debye function alone is of little aid in the extra-
polation of the entropies of many substances as they are not monatomic
solids and their heat-capacity curves have been extended experiment-
ally down only to some point in the range 500 to 1000, which, except
for some metals, is rarely low enough to give coincidence with a
Debye function. For such substances, it is necessary to employ a
more-complicated and less-sure method of extrapolation based upon
a simplified version of the theory of Born and von Karman (49, 50, 51).
It is assumed here that, at low enough temperatures, any substance
will obey the Debye law well enough to permit a satisfactory entropy
extrapolation. At higher temperatures the Debye equation no longer
will suffice, and the specific heat will rise above the Debye function
because C,-C, becomes appreciable, and, in the case of polyatomic
crystals, because degrees of freedom come into play that are not
accounted for by the Debye function.
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Kelley, K. K. Contributions to the Data on Theoretical Metallurgy: [Part] 11. Entropies of Inorganic Substances: Revision (1948) of Data and Methods of Calculation, report, 1950; Washington D.C.. (https://digital.library.unt.edu/ark:/67531/metadc12637/m1/10/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.