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The heat capacity of a substance is related to the question of how much energy does it take to raise the temperature of that substance by one unit. That would depend upon how much of the substance is being considered so the answer should be in terms of the amount of energy per standardized unit of the substance. The standardized unit could be a unit of mass but the standardized unit that makes comparison between different substances easiest is a mole; i.e., the amount containing Avogadro's number (6.025×10^{23}) of molecules (or atoms as single unit molecules).
The heat capacity of a substance is defined in the reverse direction from what was referred to above. The heat capacity per unit substance, C, is the increase in internal energy of a substance U per unit increase in temperature T:
If the substance is a gas then it is important to specify whether the gas is being held at constant volume or constant pressure. For solids the difference is negligible.
A good deal of insight may be obtained from a very simple model of a solid. Consider the solid to be a three dimensional lattice of atoms in which the atoms are held near equilibrium positions by forces. If the force on an atom is proportional to its deviation from its equilibrium position then it is called a harmonic oscillator. For zero deviation the force is zero so the force for small deviations is proportional to the deviation. Thus any such solid can be considered to be composed of harmonic oscillators. In a cubic lattice the atoms can oscillate in three directions.
The average energy E of a harmonic oscillator in one dimension is kT, where k is Boltzmann's constant. In three dimensions the average energy is 3kT. If there are N atoms in the lattice then the internal energy is U=N(3kT). Let A be Avogadro's number (6.025×10^{23}). Then dividing and multiplying the equation of U by A gives
The ratio (N/A) is the number of moles of the substance, n, and Ak is denoted as R. Thus
The heat capacity per unit mole of a substance at constant pressure is then defined as
The value for C_{p} of 3R is about 6 calories per degree Kelvin. This is known as the Dulong and Petit value. It is a good approximation for the measured values for solids at room temperatures (300°K). At low temperatures the Dulong and Petit value is not a good approximation. Below is shown the heat capacity of metallic silver as a function of temperature.
The shape of the curve for T near zero is of interest. It appears to be proportional to a power of T, say T² or T³.
According to Planck's law for the distribution of energy for an ensemble (collection) of harmonic oscillators the average energy E is given by
where h is Planck's constant h divided by 2*pi; and ω is the characteristic frequency (circular) of the
oscillators. This frequency is in the nature of a parameter for the substance and has to be determined empirically.
For temperature such that kT is much larger than hω the denominator becomes approximately
hω/kT so E is approximately kT.
In general however
With a little rearrangement this can be put into the form
Now if the numerator and denominator are divided by [exp(hω/kT)]² the result is
The attempt to obtain the limit of this expression as T→0 produces the ambiguous result of ∞/∞. The application
of l'Hospital's Rule two times finally produces the result that the limit of C_{p} is zero as T→0.
As mentioned above the empirical heat capacity curve for silver seems to be proportional to T² or T³ for small values of T. The graph of the quantum mechanical heat capacity function derived above indicates that for values of T near zero the heat capacity function is zero and flat.
To investigate the behavior of the heat capacity function for small T it simplify matters let hω/k,
which has the dimensions of temperature,
be denoted as θ. This parameter is somethimes called the Einstein temperature because it was Einstein who first formulated
this line of analysis. (Einstein was far more adept at realizing the implications of the quantization of energy found by Planck than
Planck himself.)
The heat capacity function is then:
Matters can be made even simpler by replacing θ/T with z. Then
This means that ln(C_{p})→−∞ as z→+∞; i.e., C_{p}→0 as T→0.
(To be continued.)
The term [1 − exp(−hω/kT)]² approaches the limit of 1 as T→0 so let us ignore that
term. Thus
When T goes to zero the above expression goes to ∞/∞. By l'Hospital's Rule we should consider the limit of the derivatives of the numerator and denominator. Thus
Thus l'Hospital's rule must be applied again, which gives
Thus C_{p} → 0 as T→0.
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