The Embarrassing Details about Geochemical Mass Balance Models for Calculating Mantle Composition (they are trivially simple, but they do give some interesting results if you ask the right questions)

The Physical Principal – Conservation of Mass Bulk Silicate Earth (the hypothetical homogeneous composition of the solid silicate Earth after core formation, but before any other differentiation process) [a] (  g/g) Mass BSE Mass of “a” = Mass BSE x [a] BSE Bulk Silicate Earth (the hypothetical homogeneous composition of the solid silicate Earth after core formation, but before any other differentiation process) [a] (  g/g) Mass BSE Mass of “a” = Mass BSE x [a] BSE Melt or Crust Residue or Depleted Mantle Mass of element “a” = Mass DM x [a] DM + Mass C x [a] C + Mass (BSE-DM-C) x [a] BSE

The Physical Principal – Conservation of Mass Bulk Earth (the hypothetical average homogeneous composition of the Earth) [a] (  g/g) Mass BE Mass of “a” = Mass BE x [a] BE Bulk Earth (the hypothetical average homogeneous composition of the Earth) [a] (  g/g) Mass BE Mass of “a” = Mass BE x [a] BE Mass of “a” = Mass A x [a] A + Mass CC x [a] CC + Mass OC x [a] OC + Mass DM x [a] DM + Mass Core x [a] Core + Mass BSE x [a] BSE + Mass Moon x [a] Moon + etc., etc. etc. Reservoir A (atmosphere) Reservoir CC (Cont. Crust) Reservoir OC (Oceanic Crust) Reservoir DM (Depleted Mantle) Reservoir BSE Reservoir Core Reservoir Moon

The Necessary Inputs Mass and composition of (n-1) reservoirs If interested only in composition, then all that matters is the mass balance, not the rates of exchange (unless you are interested in how a given reservoir changed in composition with time) If interested in the radiogenic isotope compositions (e.g. 87 Sr/ 86 Sr), then how and when the various reservoirs formed is critical

Some Examples of The Questions You Can Ask If the BSE differentiates into continental crust and depleted mantle, given the mass and composition of the continental crust, what portion of the mantle can be as depleted as the source of MORB? How does this change with different estimates of the BSE composition? Do you get the same answer when you use a highly incompatible element (e.g. Ba) and a moderately incompatible element (e.g. Sm)? If the continental crust has the average composition given in the program, could it be 5 times its current size at the same composition? If not, why not? How much Nb must be in the core to explain the non-chondritic Nb/U ratio of the mantle? How does the composition of the EER vary as a function of its size? Can the Moon be the EER? What would its composition be? The mass of the Moon is approximately twice the mass of the continental crust. The mass of the atmosphere is 5 x g of which 0.93% is argon, and most of that argon is 40 Ar produced by the decay of 40 K over Earth history. 40 K decays to 40 Ar and 40 Ca with a decay constant of 5.54 x yr -1, with 11 % of the decay going to 40 Ar. The BSE has 240 ppm of K, only 0.012% currently of which is 40 K. If all of the 40 Ar in the atmosphere came from 40 K decay, how much 40 Ar remains in the mantle? If all the radiogenic 40 Ar is in the atmosphere, what is the K content of the BSE?