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Problems 1-11 to be completed by all students:
 

1. What is the (a) weight % and (b) mol% and atom % concentration of Cr in the mineral chromite (FeCr₂O₄)?
 

2. Table 1 shows the partial analysis of a deep sea sediment composed of quartz, clay, and carbonates. Calculate the abundance of each mineral phase in the sample.

        Table 1

wt%	sediment	quartz	clay	carbonate
SiO2	45.7	100	51.4	0
Al2O3	13.2	0	26.4	0
CaO	16.5	0	0	55

3. At a given locality at the bottom of the ocean, the rate of sedimentation is 10 mm/1000 years. The density of the sediment is 2.0 g/cm³, and its phosphorus content is 0.65 wt%. What is the advection flux of P in kg/m² per million years?
 

4. Partial melting of a peridotite in Earth's mantle produces basaltic liquids, which you will model assuming equilibrium (batch) partial melting. (a) Calculate the bulk partition coefficient for Ni, La, and Yb assuming the mineral/melt partition coefficients given in Table 2, if the residuum contains 70 wt% olivine (ol), 20 wt% orthopyroxene (opx), and 10 wt% clinopyroxene (cpx). (b) If the peridotite source contains 2500 ppm Ni, 0.5 ppm La, and 0.4 ppm Yb, calculate the concentrations of Ni, La, Yb and the La/Yb ratio in melts produced for melt fractions of 0.002, 0.01, 0.02 and 0.1.

        Table 2

element i	Ni	La	Yb
Dol/liq	10	0	0
Dopx/liq	1	0	0.05
Dcpx/liq	1	0.01	0.3

5. Table 3 lists Sm-Nd isotope data for Apollo 17 basalt 75075. Plot the results in a ¹⁴³Nd/¹⁴⁴Nd vs ¹⁴⁷Sm/¹⁴⁴Nd isochron diagram, draw the isochron, and determine the age of this basalt. Express your answer to 3 significant digits. Decay constant for ¹⁴⁷Sm = 0.654 x 10^-11 a^-1.

      Table 3.  Sm-Nd isotope data for lunar basalt.

	147Sm/144Nd	143Nd/144Nd
plagioclase	0.1942	0.51300
ilmenite	0.2416	0.51417
whole-rock	0.2566	0.51454
pyroxene	0.2930	0.51542

6. For water that contains del¹⁸O = -10 per mil, calculate the value of del¹⁸O of calcite precipitated in equilibrium at 5^oC. Use the data given in Fig 2-13 of your Text, and assume that 1000 x ln (alpha) = delta, where alpha and delta (capital delta) are the greek symbols given in class and your Text.
 

7. For an extinct radionuclide, the slope of a correlation line in an isochron diagram for D*/D vs P/D was said to equal (P*/P), where P* = the amount of short-lived parent isotope, P = the amount of stable parent isotope, D* = the amount of daughter isotope produced by radioactive decay, and D = the amount of daughter isotope not involved in decay. It was also said that this slope was related to age. (a) For a generic short-lived decay scheme, write an expression that relates the (P*/P) ratio to the time elapsed (t), assuming that the amount of radiogenic parent initially is given by P*_o. (b) Use your expression from part a to calculate the time elapsed between the formation of two objects in meteorites in which the decay of now-extinct ²⁶Al appears to have occurred, one of which has (²⁶Al*/²⁷Al) = 5 x 10^-5 (object A), the other of which has (²⁶Al*/²⁷Al) = 0.3 x 10^-5 (object B). Round your answer to 2 significant digits. (Objects A and B correspond to most "CAIs" and some "chondrules", respectively, which are objects found in chondritic meteorites that appear to have pre-dated planetary formation.) HINT: Assume that object A gives the initial P*/P ratio. Decay constant for ²⁶Al = 9.80 x 10^-7 a^-1.
 

8. Consider reaction [1]:

CaCO₃       =     CaCO₃                    [1]
aragonite          calcite

(a) Calculate the delta_r G for reaction [1] using the data in Table 3-2 of your Text, assuming pure phases and standard state conditions. (b) At 25 ^oC and 1 atm, in which direction will reaction [1] proceed and which carbonate is more stable? (c) Consider a situation in which pure aragonite and calcite are in equilibrium. What is delta_r G for reaction [1] in this case?
 

9. Calculate delta_r G for reaction [1] at a temperature of 25 ^oC and a pressure of 5 Kb (~5000 atm) assuming pure carbonate phases, and that delta_r V = constant with pressure. The molar volumes of aragonite and calcite are V^o_arag = 34.150 cm³/mol and V^o_calcite = 36.934 cm³/mol. 1 cm³ = 0.0239 cal/bar. At T = 25 ^oC and P = 5 Kb, in which direction will reaction [1] /proceed, and which carbonate is more stable?
 

10. A metamorphic rock is observed to contain plagioclase, sillimanite, quartz, and garnet. These minerals are observed to have various degrees of purity. Write a balanced reaction [2] for equilibrium between the endmember anorthite component (CaAl₂Si₂O₈) in plagioclase, the Al₂SiO₅ component in sillimanite, the SiO₂ component in quartz, and the grossular (Ca₃Al₂Si₃O₁₂) component in garnet. That is, write a balanced reaction between these endmember mineral compositions.
 

11.  (a) How many degrees of freedom are represented by reaction [2]? (b) What kind of equilibrium does this represent?
 

Additional problems 12-18 to be completed by G545 students:

12. The del ¹⁸O value of modern seawater is 0 ‰ while the average value of the polar icecap is -45 ‰. The ice cap holds 2 wt% of the oceanic water. (a) Calculate the del ¹⁸O value of an ice-free ocean. (b) Estimates suggest that during the last glacial maximum ~20,000 years ago, the amount of ice on Earth was roughly twice that of present. Assuming this ice had del ¹⁸O = -45 ‰, calculate the del ¹⁸O value expected for seawater during the last glacial maximum.
 

13. Graphically illustrate siderophile, chalcophile, and lithophile tendencies for a variety of elements using the data given in the class handout given in Lecture 2 (Table IV-8A; Li, 2000). Equate siderophile tendency with the concentration ratio = [abundance in metal]/[abundance in bulk meteorite], chalcophile tendency with the concentration ratio = [abundance in troilite (or FeS)]/[abundance in bulk meteorite], and lithophile tendency with the concentration ratio = [average abundance in silicate, oxide, and phosphate]/[abundance in bulk meteorite] = [mean abundance in olivine + hypersthene + plagioclase + chromite + phosphate + diopside]/[abundance in bulk meteorite]. Plot different elements along the x-axis, and concentration ratio along the y-axis. Group elements with primarily siderophile, chalcophile, and lithophile tendencies together. Your graph should have three plots on it ("metal/bulk" etc.), each labeled. For the y-axis, use a logarithmic scale and make sure to label it. Consider the following elements: Co, Cr, Cu, Ga, In, Mn, Ni, Sc, Se, Sr, Ti, V, Y, Zn, Zr, rare earth elements (La to Yb), P, Hf, Th, and U. Assume that the concentration of an element in a phase is zero if data for it are not given in the table; assume the concentration is the upper limit when this is given.

Based on these data, which elements are "siderophile", "chalcophile", and "lithophile"?
 

14. Calculate the value dP/dT (in bar/^oC) for the equilibrium expressed by reaction [1] (Q8), assuming delta_r V = constant and delta_r C_p = 0. Use the conversion factor 1 cm³ = 0.1 J/bar = 0.0239 cal/bar.
 

15. Construct a P-T diagram (y-axis = pressure, x-axis = temperature) for the equilibrium represented by reaction [1]. Label the stability fields for aragonite and calcite.
 

16. Assuming equilibrium for reaction [2] (Q10), write an expression which shows how the equilibrium constant is related to pressure and temperature of equilibrium. Use this expression to solve for pressure as a function of temperature and equilibrium constant, assuming delta_r V = constant and delta_r C_p = 0. HINT: First show how equilibrium constant is related to G, then expand G in terms of its dependence on H and S and other variables.
 

17. Using your expression from Q16, calculate for reaction [2] the equilibrium pressure (in bars) corresponding to a temperature of 900 K. Use the data given in Table 4, and assume the following for activities: quartz and sillimanite can be approximated as pure phases, the activity of anorthite in plagioclase = 0.2, and the activity of grossular in garnet = 0.1.

       Table 4.  Thermodynamic data for 1 bar and 298 K.

	So (J/mol-K)	delta Hfo (kJ/mol)	Vo (cm3/mol)
anorthite	199.3	-4234.0	100.79
sillimanite	95.4	-2586.1	49.9
quartz	41.46	-910.7	22.688
grossular	260.1	-6640.0	125.33

18. A roughly spherical garnet crystal 1 cm in diameter contains a uniform concentration of 1 ppm Nd in its core, and a rim of thickness L = 0.1 microns in which the Nd concentration is depleted as a result of diffusive loss during metamorphism occurring at 800 ^oC. Assuming that the concentration decreases linearly in the depleted zone, between zero at the crystal surface and the uniform core value, that the density of the garnet is 3.5 g/cm³ , and that the diffusion parameters for Nd in garnet are given by Do = 10^-9.2 cm²/s and Q = 300 KJ/mol: (a) calculate the value of the diffusion coefficient at 800 ^oC, and (b) calculate the total flux of Nd in kg/s out of the garnet, assuming a  perfectly spherical crystal that is large compared to the size of the boundary layer (that is, neglect the thickness of the boundary layer).  HINT: Note that for the flux in part b, I am asking for an answer in terms of kg/s.
 

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