Читать книгу Geochemistry - William M. White - Страница 62

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 in terms of T and P as:

(2.108)

Substituting eqns. 2.105 and 2.106 into this, we have:

(2.109)

The coefficient of thermal expansion is 0 at absolute zero; the choice of 1 atm for the heat capacity integration is a matter of convenience because C_P measurements are conventionally made at 1 atm.

Actually, the absolute entropies of real substances tend not to be zero at absolute zero, which is to say they are not “perfectly crystalline” in the third law sense. A residual entropy, S₀, which reflects such things as mixing of two or more kinds of atoms (elements or even isotopes of the same element) at crystallographically equivalent sites, must also be considered. This configurational entropy is important for some geologically important substances such as feldspars and amphiboles. Configurational entropy can be calculated as:

(2.110)

where m_j is the total number of atoms in the j^th crystallographic site (in atoms per formula unit) and X_i,j is the mole fraction of the i^th atom (element) in the j^th site (see Example 2.3). We will return to this equation when we consider multicomponent systems.