Answers · Sharp, Principles of Stable Isotope Geochemistry, 1st ed. (2007) · ← Problems
Not printed in the 1st ed. (review question).
Not printed in the 1st ed.
Not printed in the 1st ed.
It is generally considered that the CCAM line has a slope of unity and passes through δ17O = δ18O = −1000‰ (pure 16O) and δ17O = δ18O = 0‰ (VSMOW). In fact, the line passes through the carbonaceous chondrite reference standard (CCRS) with a δ18O value of ~9‰ (vs VSMOW) and a δ17O value of ~5‰ (vs VSMOW). The slope of the CCAM line on the VSMOW scale is therefore 1005/1009 = 0.996. If the CCAM line intersected the TFL at +50, it would have a δ17O value of 0.52 × 50 = 26‰ and a δ18O value of +50‰. The CCAM slope (on the VSMOW scale) would then be 1026/1050 = 0.977.
A plot of δ17O = 0.52·δ18O is trivial; it is a straight line with a slope of 0.52. For the second part, we need to rearrange two equations. First, the definition of δ18O is rearranged to
which plots as a curve falling increasingly below the straight 0.52 line at high δ18O. [The printed figure (Fig. 13.q.5) is not reproduced; at δ18O = 1000‰, the power-law form gives δ17O = 434‰ rather than 520‰.]
Worked grids for the constructed problems M1–M6. All values below follow from the editable parameter defaults, which are representative placeholders; they rescale automatically when the parameters are set from the preferred references.
The measured solar wind sits well below the slope-1 line, and correcting along the mass-dependent slope places the Sun far down the line, strongly 16O-rich in both δ17O and δ18O. Of the printed Problem 3’s alternatives, the very light Sun survives.
[With the default parameters the Sun lands near δ18O ≈ −97‰, δ17O ≈ −101‰, and the Earth–Sun line has a slope near 1.]
Each system’s exponents follow from the masses alone. Equilibrium and kinetic laws differ by about 0.01 to 0.015 in λ, so a measured array off both slopes requires either another process or an inherited component.
Mixing is a line in ε–ε space under the equal-concentration assumption. Matching ε54Cr = 0 with the default endmembers requires f(CC) ≈ 0.24, but that mixture predicts ε50Ti ≈ +0.25 rather than 0, so either the endmembers or the equal-concentration simplification must give.
The regression returns the isochron slope exactly; converting with 26Mg/24Mg gives (26Al/27Al)₀ = 2.60·10−5, one half-life below canonical, so the inclusion is 0.72 Myr younger than the canonical reference.
α = 0.97530, i.e. 10³lnα ≈ −25‰. The Rayleigh residue reaches +58‰ by 90% K loss, so a bulk planetary δ41K only a few tenths of a per mil heavy limits free-evaporation loss to about a percent, which is the standard argument against simple kinetic evaporation as the cause of planetary K depletion.
Because the grain is extremely 17O-rich while slightly 18O-poor, the mixing trajectory has a steep negative slope in δ17O vs δ18O (−6.7 with the defaults), unlike any mass-dependent array (≈0.52) and unlike the slope-1 line, which is the inheritance fingerprint.
Transcribed from Z. Sharp, Principles of Stable Isotope Geochemistry, 1st ed. (Pearson Prentice Hall, 2007), end-of-chapter problem set (published source). Reference-scale names are written VSMOW, VPDB and VCDT where the original used SMOW, PDB and CDT, with the original numerical constants unchanged; historical usages (e.g. Craig's definition of SMOW) are kept as printed.
The source PDF's text layer scrambles mathematics, so equations and some values were reconstructed during transcription. Bracketed notes and anything tagged reconstructed or check vs print are editorial (constructed), not part of the original, and the warn-tagged items should be verified against the print copy.