Geochemistry. William M. White

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      (2.88)

      It is also easy to show that , so the internal energy of the system is:

(2.89)

      For 1 mole of substance, n is equal to the Avogadro number, N_A. Since R = N_Ak, eqn. 2.89, when expressed on a molar basis, becomes:

(2.90)

      We should not be surprised to find that entropy is also related to Q. This relationship, the derivation of which is left to you (Problem 13), is:

      (2.91)

      Since the partition function is a sum over all possible states, it might appear that computing it would be a formidable, if not impossible, task. As we shall see, however, the partition function can very often be approximated to a high degree of accuracy by quite simple functions. The partition function and Boltzmann distribution will prove useful to us in subsequent chapters in discussing several geologically important phenomena such as diffusion and the distribution of stable isotopes between phases, as well as in understanding heat capacities, discussed below.

       2.8.4.3 Energy distribution in solids

      According to quantum theory, all modes of motion are quantized. Consider, for example, vibrations of atoms in a hydrogen molecule. Even at absolute zero temperature, the atoms will vibrate at a ground state frequency. The energy associated with this vibration will be:

      (2.92)

      where h is Planck's constant and ν₀ is 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. 2.94 as:

(2.95)

      The summation term can be expressed as a geometric series, 1 + x + x² + x³ +..., where . Such a series is equal to 1/(1 − x) if x < 1. Thus, eqn. 2.95 may be rewritten in a simpler form as:

(2.96)

      At high temperature, , and we may approximate in the denominator of eqn. 2.96 by , 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. 2.96 reduces to:

      (2.99)

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

      (2.100)

      If we insert this into eqn. 2.90 and 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. 2.93 that the gaps between energy levels depend on the fundamental frequency, ν₀. The larger the gap in vibrational frequency, the less likely will be the transition to higher energy states.

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