MCEN 5024. Fall 2003.
Thermal Energy of Solids.
Introduction.
Having briefly discussed atomic structure, bonding and the classification of
solids, we now consider their thermal properties.
The thermally induced vibrations of the atoms that comprise a crystal are
responsible for such physical properties as heat capacity, temperature
coefficient of thermal expansion and thermal conductivity.
The most important ways in which a solid may absorb thermal energy are:
•
•
•
Stimulation of atomic vibration
Stimulation of electronic motion or excitation
Stimulation of molecular rotation
Of these, atomic vibration is common to all solids and is the most important
of the three with respect to the behavior of solids.
Internal Energy and Heat Capacity.
The total energy of a solid consists of two parts, the thermal energy and any
other energy which might exist at 0 K; the sum of these components is
termed the internal energy, U.
The internal energy is a well-defined bulk property, which is dependent upon
temperature.
Of equal or perhaps greater importance, is the temperature derivative of U,
the heat capacity at constant volume (Cv):
in which Cv has units of cal / mole / K
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A precise calculation of Cv for a solid is extremely difficult because it requires
knowledge of every oscillating atom in the solid.
However, a successful result can be obtained by the use of extensive
approximations about the behavior of the oscillating atoms.
Classical Theory: Approach of Dulong and Petit
Dulong and Petit showed that the heat capacities of many substances were
related to their atomic weights, i.e. the product of their specific heats and
atomic weights being approximately constant at -- 6 cal / mole / K.
In fact, this is only an approximation so that for many elements the heat
capacity lies between 5-7 cal /mol / K at 0oC with an average value of 6.2.
The heat capacities are not constant but increase by approximately
0.04%/oC for temperatures above 0oC.
Curves have a common sigmoidal shape where in the low temperature
range Cv varies as T3.
(a)
The heat capacities of some elements. (From F. K. Richtmyer, E. H.
Kennard, and T. Lauritsen, Introduction to Modern Physics, 5th ed., p. 410,
McGraw-Hill, New York, 1955.)
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The "universal" curve (from F. Seitz, Modern Theory of Solids, p. 109,
McGraw-Hill, New York, 1940).
Classical approach.
The classical theory assumed that all the internal energy of a solid could be
considered to reside in the ion cores of a solid.
A solid was thought of as being composed of an assembly of non-interacting
ion cores that behaved as simple harmonic oscillators, vibrating about an
equilibrium position, in thermal equilibrium at a given temperature.
This approach neglects any contribution of the valence electron to the
internal energy of the solid.
The thermal equilibrium of the ion cores is treated, as in the case of ideal
gases, as though the energy distribution is continuous and makes use of
Maxwell-Boltzmann statistics.
The ions were considered to have three degrees of freedom that correspond
to their energies of translation parallel to the three Cartesian coordinates.
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MCEN 5024. Fall 2003.
Therefore
_
3
3
6
Ui = KE + PE = − kT + − kT = − kT (cal / ion)
2
2
2
_
6
UI = NAUi = − NA kT (cal / mole)
2
_ 6
UI = − RT
2
Letting R ≈ 2 (cal / mole) / K and taking the definition of heat capacity as dU /
dT:
Cv ≈ 6 (cal / mole) / K
This result is in agreement with the findings of Dulong and Petit but fails to
show a temperature dependence for the heat capacity.
This arises because the solid is treated as an assembly of independent
oscillators
The very close proximities of the ions in the three dimensional arrays that
constitute crystalline solids cause this assumption to give a great
oversimplification.
In fact, the oscillations of a given ion affect those of its neighbors.
These in turn influence their neighbors and so on.
In addition, if the internal energy of a solid resides primarily in the ions, their
amplitudes of oscillation must be expected to vary with temperature.
Thus, a more realistic description of heat capacity must take these factors in
account.
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MCEN 5024. Fall 2003.
Einstein model.
Einstein studied the classical prediction and observed that significant
deviations were found at low temperatures.
Einstein utilized a quantum mechanics approach and incorporated Plank's
hypothesis of discrete vibrational frequencies.
He assumed that the internal energy of a solid was associated only with the
ion core, i.e., the energy of the electrons was not taken into account.
Hence, the solid was treated as an assembly of independent simple
harmonic oscillators in thermal equilibrium whereby each oscillated with the
same frequency, v.
The energy of a single quantized oscillator, E = hv, is called a phonon.
The average energy of such an oscillator is:
where h is Plank’s constant.
This expression is quite different from that given by classical mechanics: E =
kT.
Now, if each ion of the solid:
• independently oscillated at the same frequency, vo, and
• has three translational degrees of freedom,
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MCEN 5024. Fall 2003.
then the internal energy of a mole of the solid is given by:
where NA is Avogadro's number.
Therefore,
Now multiplying the numerator and denominator by k.
At very high temperatures kT becomes much larger than hvo, and so,
where the term [exp (-hvo / kT)] can be expanded using:
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MCEN 5024. Fall 2003.
e-x = 1 – x + … and keeping only the lowest–order terms in hvo / kT.
This gives Cv = 3R in accord with classical calculations.
At low temperatures Cv decreases and approaches zero in an exponential
fashion so that the model accounts qualitatively for the lowering of the heat
capacity at low temperatures.
Einstein's model is a simplification because it requires all atoms in the solid
to oscillate with the same, single frequency whereas they actually vibrate
with a range of frequencies.
At very high temperatures kT becomes much larger than hvo, and so,
where the term [exp (-hvo / kT)] can be expanded using:
e-x = 1 – x + … and keeping only the lowest–order terms in hvo / kT.
This gives Cv = 3R in accord with classical calculations.
At low temperatures Cv decreases and approaches zero in an exponential
fashion so that the model accounts qualitatively for the lowering of the heat
capacity at low temperatures.
Einstein's model is a simplification because it requires all atoms in the solid
to oscillate with the same, single frequency whereas they actually vibrate
with a range of frequencies.
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Debye Theory.
In the Debye approach, the solid is treated as an isotropic, homogeneous
medium instead of an ensemble of oscillating particles.
In effect, the oscillations are averaged over the frequencies present.
As a consequence of this approach, a temperature, the Debye temperature,
θD, is defined by equating the classical and quantum expressions for energy:
kθD = hvc or
θD = hvc/k
where vc is the maximum frequency.
This effectively defines θD as the maximum temperature at which quantum
mechanics must be used. At higher temperatures, classical mechanics can
be utilized.
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MCEN 5024. Fall 2003.
Comparison of theories.
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A more common way of presenting heat capacity results is:
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MCEN 5024. Fall 2003.
At 0 K, the heat capacity is zero.
At temperatures above 0 K it climbs rapidly and is proportional to T3 in this
region.
At high temperatures it reaches a nearly constant value of approximately 6
cal/mole/K.
The constant θD has a different value for each solid. When T = θD, Cv reaches
approximately 96% of its final value.
Representative values of θD are given in the following table of Debye
Temperatures θD in Kelvin for a range of different materials.
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