G 445/545
Geochemistry
PROBLEM SET 2




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Problems 1-10 to be completed by all students:
 

1. Silver (Ag1+) is a common trace element in galena (PbS), where it replaces Pb2+.  (a) Schematically describe the coupled substitution that is needed for this case, with Pb2+ on one side of the equation, Ag1+ on the other, and R = a generic cation of appropriate valence (make sure to show the valence of the R cation).  (b) Identify the ion of a chaclophile element that is suited to enter with Ag1+ in the coupled substitution you wrote for part (a).
 

2. Write an expression relating composition (thermodynamic mole fraction) to activity for the diopside (CaMgSi2O6) component in pyroxene, assuming ideal mixing on all cation sites and a standard state of pure diopside at the pressure and temperature of interest. (Use the activity-composition mixing model described in class.)
 

3.  An analysis of pyroxene is given in Table 1.   (a) For this pyroxene, calculate the atomic formula unit proportion of cations normalized to 6 oxygens.  Keep totals for Fe3+ and Fe2+ cations (Fe2O3 and FeO oxides) separate.  Assume cations are distributed among sites according to the site preference and filling sequence given in a class handout.  Express answers to the number of significant digits implied by the analysis.  Traditionally, in a formula expression the proportion of a cation is written as a subscript to the right of the cation. (b)  Using your answers from part (a) and from Q2,  calculate the activity of the diopside component for the pyroxene shown in Table 1.

        Table 1
wt% pyroxene
SiO2 52.70
TiO2 0.34
Al2O3 1.84
Fe2O3 2.12
FeO 5.42
MnO 0.16
MgO 15.15
CaO 21.58
Na2O 0.49
K2O 0.01
99.96

 

4.  For metamorphic reaction [1]:

[1]    2Mg2SiO4      + 3 H2O       =      Mg3Si2O5(OH)4       +  Mg(OH)2
           forsterite          fluid                      serpentine                   brucite

(a)   Sketch a P-T diagram, showing the slope of the reaction boundary and making sure to label stability fields.  (b)   Provide justification for drawing the diagram in the manner you have done.
 

5.  Consider the mineral gypsum (CaSO4 * 2H2O).  (a) Using the data given in Table 3-2 of your book, calculate the value of KSP for this mineral at 25 oC.   (This will differ somewhat from the KSP value given in Table 4-2.)  (b) Demonstrate whether seawater is oversaturated, saturated, or undersaturated compared to pure gypsum at 25 oC, using your answer from part (a) and the data for seawater given in Table 3-4 of your Text.  Aprroximate the activity of H2O in seawater as 1.0.
 

6. Three-isotope oxygen data for whole-rock chondrites form trends that differ from one class of chondrite to another and which can be explained by mixing of isotopically distinct components.  Regression lines through the data for carbonaceous and ordinary chondrites yield the following equations:

carbonaceous chondrites:     del17O = -4.20 + 0.944 x del18O      (CCAM line)
ordinary chondrites:              del17O = -1.35 + 1.0 x del18O           (OCM line)

where CCAM = carbonaceous chondrite anhydrous materials, and OCM = ordinary chondrite materials.  Assuming that these apparent mixing lines represent the addition of the same component to both carbonaceous and ordinary chondrites, what is the O-isotope composition of the common component?
 

7.   Consider equilibrium and fractional crystallization of a melt under plutonic conditions that has composition Z in Fig. 1 (see below). The phase diagram in Fig. 1 shows the liquidus diagram for the system An-Di-Fo, where An = anorthite, Di = diopside, Fo = forsterite, Lq = liquid, Sp = Mg-Al-spinel (by mass). (a) Using Fig. 1, show how the composition of the residual melt evolves using an arrow to indicate the direction of compositional changes. (b) Does your answer in part (a) differ between equilibrium and fractional crystallization? (c) How many degrees of freedom are present at point E, assuming some liquid remains? (d) Describe the rock types that would be produced for fractional and (e) equilibrium crystallization. For parts (d) and (e), use the key in Table 2:

          Table 2.
rock contains these minerals plutonic rock analogue
Fo alone dunite
An alone anorthosite
Di alone clinopyroxenite
Di + An gabbro
Fo + Di wehrlite
An + Fo troctolite
An + Di + Fo olivine gabbo

link to GIF version of Fig.1

link to TIF version of Fig. 1
 

8.  Write a balanced chemical weathering reaction showing the breakdown of enstatite (Mg2Si2O6) into aqueous Mg ions, assuming an acid titration process.

9.  The diameter of a nearly circular lagoon in a Pacific atoll is 3.5 km and its average depth is 500 m. Water flows into the inlet at a rate of 108 m3/year. What is the residence time of the water in the lagoon?
 

10. You have collected rocks from two metamorphic terranes that appear to have experienced two different burial and exhumation histories, according to the literature survey you have performed.  Rock A was collected from a region of regional metamorphic terrane far from a plate boundary that experienced gradual burial and temperature increase followed by gradual uplift, whereas rock B was collected from a subduction zone wedge that experienced rapid burial of near-surface sediments, followed at a much later date by rapid exhumation during mountain building.  Your task is to constrain the metamorphic history of the two rocks by evaluating the reactions they have experienced. Discuss whether one would expect prograde or retrograde metamorphic mineral assemblages for these two rocks and why.  
 
 

Additional problems 11-15 to be completed by G545 students:

11. Assuming fractional crystallization of melt z in Fig. 1 (Q7), use the lever rule to estimate (a) the mass proportions of phases that will crystallize in each crystallization step (i.e. for each rock type), and (b) how the mass fraction of liquid varies during each crystallization step.  HINT: For part (b), draw tie lines between key liquid compositions through the bulk composition.

12.  It was shown in class that for rivers and groundwater, pH is correlated with the logarithm of activity of bicarbonate ion (HCO31-), and that the exact correlation also depends on the value of fCO2 (= pCO2 for an ideal gas). Derive an expression for pH as a function of aHCO3 and fCO2, assuming that the following reactions involving carbonic acid (H2CO3) dissolved in water control water chemistry equilibria:

[2]         H2CO3      =       H2O      +     CO2                  which at 25 oC has K2 = 10-1.47
       carbonic acid             fluid               gas

[3]         H2CO3      =        H 1+     +     HCO3 1-            which at 25 oC has K3 = 10-6.35
       carbonic acid              aq.                aq.

HINT: your expression should be in the form of y = mx + b, where y = pH, x = log aHCO3, and b includes the values for K2 and K3.  Your first step should be to write out the expressions for the two equilibrium constants.
 

13. Figure 2 (see below) shows chemical data for various groups of iron meteorites, which formed magmatically. Groups IIIAB and IIAB are considered to have formed by fractional crystallization of metallic liquids in a core setting, whereas group IAB may have formed by partial melting in a parent body that had not yet formed a core.

Link to Fig. 2

(a) Based on the slopes in Fig. 2, estimate values of the solid metal/liquid metal partition coefficients for DAu, DW and DIr for the IIIAB and IIAB groups, taking two or three measurements for each line at different points to determine the slope (multiple measurements are needed because it is hard to accurately estimate slopes on log-log plots). Assume that DNi = 0.9. To determine slopes on log-log plots for two x and y coordinate pairs:

slope = (log y2 - log y1) / (log x2 - log x1) = (log y2/y1) / (log x2/x1)

(b) How do the values you determine compare between the two groups?
 

14. Model variation of Ni and Ir concentration for the IIIAB iron meteorites, assuming that these meteorites formed by fractional crystallization, that DNi = 0.9, DIr = 4, and that the starting melt composition is given by 7.6 wt% Ni and 5.0 ppm Ir, with the latter close to the solar ratio of these elements. How do your model results compare to the observations (see Fig. 2), and in light of your answers to Q13, what might account for any discrepancies?
 

15.  You have been tasked to design an apparatus to condense metal alloy from a vapor phase in an sealed experimental chamber.  As preparation you need to do a thermodynamic analysis to constrain the condensation temperature as a function of alloy composition and gas pressure.  The relevant reaction is given by [4]:

[4]            Fe (solid) = Fe (gas)

(a) For reaction [4], derive an expression for equilibrium constant in terms of activities and the standard free energy change of the reaction, showing all superscripts and subscripts, and assuming that the gas behaves ideally.  (b) Rearrange the expression in part (a) to solve for the condensation temperature in terms of the standard enthalpy change of reaction, standard entropy change of reaction, partial pressure of Fe gas, and activity of Fe in metal alloy.  (c) What will be the condensation temperature of a pure Fe metal alloy corresponding to a pressure of ideal Fe gas of 10-4 bar, assuming for reaction [4] an enthalpy change of 416.3 kJ, and an entropy change of 0.155 kJ/K under the relevant conditions?     
 

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