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 (Al2O3) 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 Cv at 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, νD, 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
Figure 2.10 shows an example of the variation in heat capacity. Consistent with predictions made in the discussion above, heat capacity becomes essentially constant at T = hν/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.
Nevertheless, geochemists generally use empirically determined heat capacities. Constant pressure heat capacities are easier to determine, and therefore more generally available and used. For minerals, which are relatively incompressible, the difference between Cv and Cp is small and can often be neglected. Empirical heat capacity data is generally in the form of the coefficients of polynomial expressions of temperature. The Maier-Kelley formulation is:
(2.102)
where a, b, and c are the empirically determined coefficients. The Haas–Fisher formulation (Hass and Fisher, 1976) is:
(2.103)
with a, b, c, f, and g as empirically determined constants. The Hass–Fisher formulation is more accurate and more widely used in geochemistry and heat capacity data are commonly tabulated this way (e.g., Helgenson, et al., 1978; Berman, 1988; Holland and Powell, 1998). We shall use the Maier−Kelly formulation because it is simpler, and we do not want to become more bogged down in mathematics than necessary.
Figure 2.10 Vibrational contribution to heat capacity as a function of kT/hν.
Since these formulae and their associated constants are purely empirical (i.e., neither the equations nor constants have a theoretical basis), they should not be extrapolated beyond the calibrated range.
2.8.5 Relationship of entropy to other state variables
We can now use heat capacity to define the temperature dependency of entropy:
(2.104)
(2.105)
The dependencies on pressure and volume (at constant temperature) are:
(2.106)
(2.107)
2.8.6 Additive nature of silicate heat capacities
For many oxides and silicates, heat capacities are approximately additive at room temperature. Thus, for example, the heat capacity of enstatite, MgSiO3,, may be approximated by adding the heat capacities of its oxide components, quartz (SiO2) and periclase (MgO). In other words, since:
then
Substituting values:
The observed value for the heat capacity of enstatite at 300 K is 82.09 J/mol-K, which differs from our estimate by only 0.1%. For most silicates and oxides, this approach will yield estimates of heat capacities that are within 5% of the observed values. However, this is not true at low temperature. The same calculation for Cp-En carried out using heat capacities at 50 K differs from the observed value by 20%.
The explanation for the additive nature of oxide and silicate heat capacities has to do with the nature of bonding and atomic vibrations. The vibrations that are not fully activated at room temperature are largely dependent on the nature of the individual cation–oxygen bonds and not on the atomic arrangement in complex solids.
2.9 THE THIRD LAW AND ABSOLUTE ENTROPY
2.9.1 Statement of the third law
The entropies of substances tend toward zero as absolute zero temperature is approached, or as Lewis and Randall expressed it:
If the entropy of each element in some crystalline state may be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy, but at absolute zero, the entropy may become zero, and does so become in the case of perfectly crystalline substances.
2.9.2 Absolute entropy
We recall that entropy is proportional to the number of possible arrangements of a system: S = klnΩ. At absolute zero, a perfectly crystalline substance has only one possible arrangement, namely the ground state. Hence
The implication of this seemingly trivial statement is that we can determine the absolute entropy of substances. We can write the complete differential for S
