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The Heat Capacities of Solids

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 (6.025×1023) 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:

C = (∂U/∂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 Classical Approach to Heat Capacity

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 (6.025×1023). Then dividing and multiplying the equation of U by A gives

U = (N/A)(3AkT)

The ratio (N/A) is the number of moles of the substance, n, and Ak is denoted as R. Thus

U = n(3RT)
UM = (U/n) = 3RT

The heat capacity per unit mole of a substance at constant pressure is then defined as

Cp = (∂UM/∂T)
and thus from the above
Cp = 3R

The value for Cp 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.

A Quantum Mechanical Derivation of Heat Capacity

According to Planck's law for the distribution of energy for an ensemble (collection) of harmonic oscillators the average energy E is given by

E = hω/[exp(hω/kT) − 1]

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

U = NE = N3hω/[exp(hω/kT) − 1]
UM = 3Ahω/[exp(hω/kT) − 1]
and hence
Cp = −3Ahω/[exp(hω/kT) − 1]²[exp(hω/kT)](−hω/kT²)

With a little rearrangement this can be put into the form

Cp = 3R(hω/kT)²exp(hω/kT)/[exp(hω/kT) − 1]²

Now if the numerator and denominator are divided by [exp(hω/kT)]² the result is

Cp = 3R(hω/kT)²exp(−hω/kT)/[1 − exp(−hω/kT)]²

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 Cp is zero as T→0.

The Heat Capacity Function for Low Temperatures

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:

Cp = 3R(θ/T)²exp(−θ/T)/[1 − exp(−θ/T)]²

Matters can be made even simpler by replacing θ/T with z. Then

Cp = 3Rz²exp(−z)/[1 − exp(−z)]²
ln(Cp) = ln(3R) + 2ln(z) − z − 2ln[1 − exp(−z)]

This means that ln(Cp)→−∞ as z→+∞; i.e., Cp→0 as T→0.

(To be continued.)

Limit Using l'Hospital's Rule

Cp = 3R(hω/kT)²exp(−hω/kT)/[1 − exp(−hω/kT)]²

The term [1 − exp(−hω/kT)]² approaches the limit of 1 as T→0 so let us ignore that term. Thus

Cp = 3R(hω/kT)²/exp(hω/kT)
and to simplify matters let hω/k,
which has the dimensions of temperature,
be denoted as θ. Then
Cp = 3R(θ/T)²/exp(θ/T)

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

3R(2(θ²)(−1/T³)/(exp(θ/T)(−θ/T²) = 6Rθ(1/T)/exp(θ/T)
which, as T→0, goes to ∞/∞

Thus l'Hospital's rule must be applied again, which gives

6Rθ(−1/T²)/(exp(θ/T)(θ/T²) = 6R/exp(θ/T)
which has a limit of zero as T→0.

Thus Cp → 0 as T→0.

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