Assume that depleted upper mantle and un-depleted lower mantle have the followin
ID: 535377 • Letter: A
Question
Assume that depleted upper mantle and un-depleted lower mantle have the following Nd and He concentrations and isotopic compositions. Using the binary mixing relationships for two isotope ratios, plot the mixing curve between upper mantle and lower mantle in a ^4 He/^3 He vs. epsilon_Nd space (For a proper spread of your calculated data points, use 10% increments between 1-90% of upper mantle, 1% increments between 91 to 99%, and 0.1% between 99.1% to 100%). A Hawaiian basalt sample has ^4 He/^3 He = 29600 and epsilon_Nd = +9.2. Based on your mixing calculation and your mixing curve in ^4 He/^3 He vs. epsilon_Nd space, how much lower mantle component (percentage) is in this sample? Which isotope ratio (Nd or He) gives the clearest signature of the possible presence of a lower mantle component in Hawaii?Explanation / Answer
Binary mixtures: When two components (A and B) of different chemical composition mix in varying proportions, the chemical compositions of the resulting mixtures (M) vary systematically depending on the relative abundances of the end-members. We can therefore define a mixing parameter that describes the proportions of end-members:
fA = WA/(WA + WB)
fB = 1 - fA
Where WA and WB are the weights or volumes of the components A and B in a given mixture. If fA = 0, then the mixture contains only the end-member “B”. Alternatively, if fB = 0, then the mixture contains only the end-member “A”.
The concentration of any conservative element (X) in a binary mixture of A and B depends on the concentration of that element in components A and B and on the abundances of components A and B in the mixture. Therefore, the concentration of element X in a mixture (M) of components A and B is calculated as a weighted average:
(X)M == (X)AfA + (X)B(1-fA)
where the parentheses mean concentration.
The resultant theoretical mixing line can be used to model the change in the He isotopic signature of groundwater as it acquires He from aquifer materials along a flow path. The model can then be compared to actual observed ratios and concentrations to ascertain if the model describes what is actually happening in the aquifer. Equation 1 describes the variation in the 4He/3He ratio to be expected in a mixture generated by combining various proportions of He from two different end-members with different 4He/3He ratios :
(4He/3He) mix = a/ [He] mix + b ---------(1)
The slope of the mixing line is calculated from
a ={ [He]A[He]B{(4He/3He)B-(4He/3He)A}} / { [He]A - [He]B }
and the y-intercept from
b ={ [He]A(4He/3He)A - [He]B(4He/3He)B } / { [He]A - [He]B }
These equations require input of the 4He/3He ratios and He concentrations ([He]) of the end-members. The second end-member (B) is the aquifer material for which we used the average 4He/3He ratios and strontium concentrations.