William M. White - Geochemistry

(2.92)

where h is Planck's constant and ν 0is the vibrational frequency of the ground state. Higher quantum levels have higher frequencies (and hence higher energies) that are multiples of this ground state:

(2.93)

where n is the quantum number (an integer ≥ 0).

Now consider a monatomic solid, such as diamond, composed of N identical atoms arranged in a crystal lattice. For each vibration of each atom, we may write an atomic partition function, . Since vibrational motion is the only form of energy available to atoms in a lattice, the atomic partition function may be written as:

(2.94)

We can rewrite eqn. 2.94as:

(2.95)

The summation term can be expressed as a geometric series, 1 + x + x 2+ x 3+..., where . Such a series is equal to 1/(1 − x ) if x < 1. Thus, eqn. 2.95may be rewritten in a simpler form as:

(2.96)

At high temperature, , and we may approximate in the denominator of eqn. 2.96by , so that at high temperature:

(2.97)

Using this relationship, and those between constant volume heat capacity and energy and between energy and the partition function, it is possible to show that:

(2.98)

This is called the Dulong-Petit limit, and it holds only where the temperature is high enough that the approximation holds. For a solid consisting of N different kinds of atoms, the predicted heat capacity is 3NR. Observations bear out these predictions. For example, at 25°C the observed heat capacity for NaCl, for which N is 2, is 49.7 J/K, whereas the predicted value is 49.9 J/K. Substances whose heat capacity agrees with that predicted in this manner are said to be fully activated . The temperature at which this occurs, called the characteristic or Einstein temperature , varies considerably from substance to substance (for reasons explained below). For most metals, it is in the range of 100−600 K. For diamond, however, the Einstein temperature is in excess of 2000 K.

Now consider the case where the temperature is very low. In this case, and the denominator of eqn. 2.96; therefore, tends to 1, so that eqn. 2.96reduces to:

(2.99)

The differential with respect to temperature of ln is then simply:

(2.100)

If we insert this into eqn. 2.90and differentiate U with respect to temperature, we find that the predicted heat capacity at T = 0 is 0! In actuality, only a perfectly crystalline solid would have 0 heat capacity near absolute zero. Real solids have a small but finite heat capacity.

On a less mathematical level, the heat capacities of solids at low temperature are small because the spacings between the first few vibrational energy levels are large. As a result, energy transitions are large and therefore improbable. Thus, at low temperature, relatively little energy will go into vibrational motions.

We can also see from eqn. 2.93that the gaps between energy levels depend on the fundamental frequency, ν 0. The larger the gap in vibrational frequency, the less likely will be the transition to higher energy states. The ground state frequency, in turn, depends on bond strength. Strong bonds have higher vibrational frequencies and, as a result, energy is less readily stored in atomic vibrations. In general, covalent bonds will be stronger than ionic ones, which, in turn, are stronger than metallic bonds. Thus, diamond, which has strong covalent bonds, has a low heat capacity until it is fully activated, and full activation occurs at very high temperatures. The bonds in quartz and alumina (Al 2O 3) are also largely covalent, and these substances also have low heat capacities until fully activated. Metals, on the other hand, tend to have weaker bonds and high heat capacities.

Heat capacities are more difficult to predict at intermediate temperatures and require some knowledge of the vibrational frequencies. One simple assumption, used by Einstein, *is that all vibrations have the same frequency. The Einstein model provides reasonable predictions of C vat intermediate and high temperatures but does not work well at low temperatures. A somewhat more sophisticated assumption was used by Debye, †who assumed a range of frequencies up to a maximum value, ν D, now called the Debye frequency , and then integrated the frequency spectrum. The procedure is too complex for us to treat here. At low temperature, the Debye theory predicts:

(2.101)

where and is called the Debye temperature .

Figure 2.10shows an example of the variation in heat capacity. Consistent with predictions made in the discussion above, heat capacity becomes essentially constant at T = h ν/k and approaches 0 at T = 0. Together, the Debye and Einstein models give a reasonable approximation of heat capacity over a large range of temperature, particularly for simple solids.