Temperature reflects the average
randomized kinetic energy of particles in matter. Heat is the transfer of
thermal energy across a system boundary into the body or from the body to the
environment. Translation, rotation, and a combination of the two types of
energy (kinetic and potential) in vibration of atoms represent the degrees of
freedom of motion which classically contribute to the heat capacity of matter,
but loosely bound electrons may also participate
At the first decade of the twentieth
century thermodynamics and solid states physics obeyed the classical expression
for the molar specific heat capacity of a crystal known as the “Dulong–Petit law”. It was a “chemical
law” proposed in 1819 by French physicists Pierre Louis Dulong and Alexis
Thérèse Petit. Experimentally the two scientists had found that the heat
capacity per weight (the mass-specific heat capacity) for a number of
substances became close to a constant value, after it had been multiplied by number representing the presumed
relative atomic weight of the substance. These atomic weights had shortly
before been suggested by Dalton. Dulong and Petit found that the heat capacity
of a mole of many solid substances is about 3R (where R is the
modern constant called the universal gas constant). The value of 3R is
about 25 joules per Kelvin, and Dulong and Petit essentially found that this
was the heat capacity of crystals, per mole of atoms they contained.
The “Debye model” is a method developed by Peter
Debye in 1912 for estimating the phonon contribution to the specific heat (heat
capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons
in a box. The Debye model is a solid-state equivalent of Planck's law of black
body radiation, where electromagnetic radiation is treated as a “gas of photons”
in a box. Thus, as the solid is heated up, it should be a reasonable first
approximation to take all the atoms to be jiggling about independently, and the
“Equipartition of Energy” as seen by classical physics, would assure us that at
temperature T each atom would have on average energy 3kT, k
being Boltzmann’s constant.
The “Dulong–Petit”
law offers fairly good prediction for the specific heat capacity of many solids
with relatively simple crystal structure at high temperatures. This is because
in the classical theory the heat capacity of solids approaches a maximum of 3R
per mole of atoms, due to the fact that full vibrational-mode degrees of
freedom amount to 3 degrees of freedom per atom each corresponding to a
quadratic kinetic energy term and a quadratic potential energy term. By the Equipartition
theorem, the average of each quadratic term is 1⁄2kT,
or 1⁄2RT per mole.
Multiplied by 3 degrees of freedom and the two terms per degree of freedom,
this amounts to 3R per mole heat capacity.
However, the
Dulong–Petit law fails at room temperatures for light atoms bonded strongly to
each other, such as in metallic beryllium, and in carbon as diamond, for
example. The problem starts when it predicts higher heat capacities than are actually
found, with the difference due to higher-energy vibrational modes not being
populated at room temperatures in these substances.
In the very
low (cryogenic) temperature region, where the quantum mechanical nature of
energy storage in all solids manifests itself with larger and larger effect,
the law fails for all substances.
The modern day theory states that the
heat capacity of solids is due to lattice vibrations in the solid. It was first
derived from this assumption by Albert Einstein, in 1907. The “Einstein solid”
model thus gave for the first time a reason why the Dulong–Petit law should be
stated in terms of the classical heat capacities for gases. For quantum
mechanical reasons, at any given temperature, some of these degrees of freedom
may be unavailable, or only partially available in terms of capacity for
storing thermal energy. In such cases, the specific heat capacity is a fraction
of the maximum. As the temperature approaches absolute zero, the specific heat
capacity of a system also approaches zero, due to loss of available degrees of
freedom. Einstein realized that exactly the same considerations must apply to
mechanical oscillators, such as atoms in a solid. He assumed each atom to
be an independent simple harmonic oscillator, and, just as in the case of black
body radiation, the oscillators can only absorb energies in “quanta”.
Consequently, at low enough temperatures there is rarely sufficient energy in
the ambient thermal excitations to excite the oscillators, and they freeze out.
Later on some improvements were
introduced and the basic set of oscillators was taken to be standing sound wave
oscillations in the solid rather than individual atoms (even more like black
body radiation in a cavity) but the main conclusion was not affected. In
the more modern picture of sound waves in a solid, the “elementary” sound wave,
analogous to the photon, is called the phonon, and has energy hf,
where h is again Planck’s constant, and f is the sound frequency.
Oscillations of molecules can usually be analyzed fairly accurately as simple
harmonic oscillations, in particular the diatomic molecule.
References:
1.
Albert
Einstein; Wikipedia; https://en.wikipedia.org/wiki/Albert_Einstein (accessed on: 6/21/13);
2.
Fowler,
M.; The Simple Harmonic Oscillator; University of Virginia; Access: http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/SimpleHarmonicOscillator.htm (accessed on:
6/21/13);
3.
Einstein
solid; Wikipedia; Access: http://en.wikipedia.org/wiki/Einstein_solid (accessed on:
6/21/13);
4.
The
heat capacity of a solid; Access: http://ruelle.phys.unsw.edu.au/~gary/PHYS3020_files/SM3_6.pdf (accessed on: 6/21/13);
5.
Ilustration
www.wikipedia.org/wiki/Debye_modlel (accessed on:
6/21/13);
