Contributions to the Data on Theoretical Metallurgy: [Part] 11. Entropies of Inorganic Substances: Revision (1948) of Data and Methods of Calculation Page: 5
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CALCULATING ENTROPIES FROM EXPERIMENTAL DATA 5
The integrals of equation (7), except the first, are evaluated readily
by plotting C, against In T or C against T and mechanically com-
puting the area bounded by the resulting curve, the In T or T axis,
and the two ordinates corresponding to the limits of integration. The
first integral of equation (7) and that of equation (6) need further
consideration, as an extrapolation of the heat-capacity curve, between
the lowest temperature reached in the measurements and the absolute
zero, is required for their evaluation.
As a crystal is cooled to low temperatures the difference between
the heat capacities at constant pressure and constant volume grad-
ually diminishes and usually is nearly negligible for inorganic crystals
around 500 K. Debye (123), assuming a monatomic solid to be an
isotropic, elastic medium, deduces for the energy relationship
kT3 h \3 d (hv \
E=9NkT(kT f TkT \kT/ \kT
9 \hvm J0 ehP/kT (8)
Differentiating with respect to the temperature at constant volume,
there is obtained
C,~=9N k 4 (kT) hVm/kT (s d e,'-I .
In these equations,
N= Avogadro's number,
k= gas constant per molecule,
h= Planck's constant,
v= frequency of vibration, and
v,= limiting frequency of vibration.
By substituting k --
S T\ T D
C=9 k 4 OD 0 ehvlkT- 1 eD/T- (10)
0D is a characteristic constant for each substance and has the dimen-
sions of temperature. Equation (10) expresses the heat capacity at
constant volume as a function of this characteristic constant divided
by the temperature. When T is large, so that hv/kT is small, eh^lkT
may be approximated as 1+ T' and equation (8) approaches
E= N k T, (11)
and C, approaches 3 N k=3R, where R is the gas constant per mole.
The latter is the rule of Dulong and Petit (132, 330), which holds
approximately for many elements near room temperature. When T
is small, the upper limit of integration in equation (8) may be taken as
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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/m1/9/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.