Isotopic Constraints on Earth System Processes. Группа авторов

the subject of heated debate and will not be considered here. The range of magnesium and silicon isotopic composition CAIs is more easily understood, in no small part by comparison to the results of laboratory evaporation experiments described in the next section.

Figure 1.16 The panel on the left shows a false color image of a Type B CAI from the Allende meteorite shown on the right. Type B CAIs are characterized by having large euhedral to sub‐euhedral crystals of melilite shown in blue. The pink grains are anorthite, the light blue grains are pyroxene, and the small red “dots” are spinel. Most of the more or less spherical inclusions in Allende are chondrules while the less abundant, typically larger and more irregular, inclusions are most likely CAIs like the one indicated by the arrow.

      1.6.1. The Hertz‐Knudsen Evaporation Equation

      The standard formulation for the net flux J_i of an element i between a condensed phase and a surrounding gas is the Hertz‐Knudsen equation:

(1.9)

J_i is the flux of i in moles per unit area per unit time (positive indicating net evaporation), n is the number of atoms of i in the gas molecule containing i, γ_iv is the evaporation coefficient, γ _ic is the condensation coefficient, P_i,sat is the saturation vapor pressure of the gas molecule containing i if equilibrium with the condensed phase were realized, P_i is the pressure of the gas molecule containing i at the surface of the condensed phase, M_i is the molar mass of the gas molecule containing i, R is the gas constant, and T is the absolute temperature. This version of the Hertz‐Knudsen equation applies when there is only one gas molecule containing the element i. In the case of a CAI‐like liquid the dominant gas species of the most volatile elements magnesium and silicon are Mg and SiO (not Si). Since these gas species contain only one atom of the element of interest, n is equal to one in equation 1.9. The total pressure in that part of the solar nebula where the CAIs formed was about 10^–3 bars or less (mostly hydrogen with negligible amounts of Mg and SiO) and therefore P_i<<P_i,sat. Equation 1.9 when n=1 and P_i<<P_i,sat becomes

(1.10)

The isotopic composition of the evaporation flux is given by the ratio of the flux of isotopes i and j calculated using equation 1.10 with the further simplification that because of the thermodynamically ideal behavior of isotopes P_i,sat/P_j,sat = N_i/ N_j where N_i/ N_j is the atom ratio of the isotopes in the evaporating material. The equation for the isotopic composition of the evaporation flux is then

(1.11)

      The quantity , which we will call α _ij, is the kinetic isotope fractionation of the evaporation flux relative to the isotopic ratio of the evaporating material. Prior to having results from vacuum evaporation experiments it was often assumed that , (i.e., the inverse square root of the mass “law”) but as we will see, this is not correct.

      1.6.2. Rayleigh Fractionation

The kinetic isotope fractionation factor α _ij is the key parameter for calculating the isotopic evolution of vacuum evaporation residues as a function of the amount of the parent element evaporated when diffusion in the residue was sufficiently fast to continuously homogenize the residue. To arrive at this formulation as a form of Rayleigh fractionation consider the mass conservation equation of a uniformly distributed isotope i in a volume V bounded by a surface of area A, which is dN_i /dt =−J_i A, where N_i is the total atoms of isotope i in V. A similar conservation equation applies for isotope j. Taking the ratio of the two conservation equations and using equation 1.11 for the ratio of flux gives

      (1.12)

      which when rearranged becomes

(1.13)

Integrating equation 1.13 from an initial state with isotopic abundances N_{i, o} and N_{j, o} to a later state when the abundances have become N_i and N_j gives

      (1.14)