Contributions to the Data on Theoretical Metallurgy: [Part] 11. Entropies of Inorganic Substances: Revision (1948) of Data and Methods of Calculation Page: 7
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CALCULATING ENTROPIES FROM EXPERIMENTAL DATA
The assumption that the atoms in a crystal behave as harmonic
oscillators led Einstein (141) to the expression
Nke kT )2 (15)
for the contribution of a vibrational degree of freedom. The symbols
in this equation have the same meaning as in that of Debye. Tables
of Einstein functions for different values of also are available
(208, 309, 374, 375). Born and von Karman (49, 50, 51) have con-
sidered the vibrations in a crystal lattice, and Born (48) has summed
up by stating that the heat content of a p-atomic crystal, which con-
sists of n elementary parallelepipeds, may, to a close approximation,
be considered as composed of two portions; the first part is given by
the sum of 3 Debye energy-functions of characteristic temperatures
OD1, 0D2 and 0D3, which are related closely to the elastic properties,
and the second part consists of the sum of 3 (p-1) Einstein energy-
functions of 0E's, which may be found from a study of the infrared
dispersion of the crystal. In terms of heat capacity this may be
C= 3D(OD) +31) E(i (16)
T=1 i=3 T
where D () and E )represent, respectively, Debye and Einstein
specific-heat functions. This equation gives the heat capacity per
chemical formula mass of the n unit parallelepipeds. The factors
"Y3" and "3" occur because crystals generally have different frequencies
of vibration in different directions.
Equation (16) must be modified before it is readily and generally
adaptable to the extrapolation of heat-capacity curves. As the extra-
polated part of the entropy usually is only a relatively small portion
of the total at 298.160 K., the assumptions to be made should not be
objectionable. First, let p mean the number of atoms in a molecule of
the substance, if that is known, or else the number of atoms in the
simplest chemical formula that may be written to represent the com-
position; also instead of having 3 Debye functions and 3(p--1) Ein-
stein functions, take 1 Debye function and (p-1) Einstein functions
and remove the factor %^ from equation (16). The result is
C ,= D () 1E Ei). (17)
Each Debye and Einstein function now may be considered as being
equivalent to one-third the sum of the three it is replacing. In other
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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/m1/11/: accessed April 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.