Problem Set #1

 

1. Absolute Dating.  Before we can use the concepts we saw in class to determine the age of rock, we need to do a little more analysis.  In class, we saw that the number of Parents, N at any time t is

                                                                                                                          (1)

Where N0 is the initial number of parents, and l is the decay constant, a property of the parent that determines the rate of decay.  Since the number of parents that have decayed is N0-N and each parent that decays becomes a daughter, the number of daughters produced by decay is

                                                                                                  (2)

Now divide by the number of parents

                                                                                                   (3)

The total number of daughters is the number produced by radioactive decay, D, plus the number of daughters originally present at t=0, D0.  So the total number of daughters is

                                                                                                            (4)

For example, we will consider the decay in which Rb87 is the parent and Sr87 is the daughter.  For technical reasons, it is difficult to measure the absolute abundances of parents and daughters in a rock or mineral.  It is much easier to measure relative abundances.  This can be accomplished by dividing each side of our equation by the abundance of a non-radiogenic isotope, in our case, Sr86.

                                                                                     (5)

Now, to our problem.  Six samples of granodiorite from a pluton in British Columbia have strontium and rubidium isotopic compositions as follows

                                               

87Sr/86Sr          87Rb/86Sr

                                               

0.7117             3.65

0.7095             1.80

0.7092             1.48

0.7083             0.82

0.7083             0.66

0.7082             0.74

                                               

 

The decay constant l=1.42 x 10-11 years.

 

a) Find the age of the intrusion assuming that all samples had the same initial strontium isotopic ratio.

 

b) Find the initial strontium isotopic ratio.

 

c) The strontium isotopic ratio of oceanic basalts erupting today is approximately 0.704.  From this and your answer to part b, what can you conclude regarding the origin of the magma?

 

2. Exercise.  Reproduce the first four rows of the periodic table.  Include elemental symbols, names, and atomic numbers. I will ask you to do this on the midterm.

 

3. Exercise.  Write down the electronic configuration of all the elements in the first four rows of the periodic table.  You should be able to reason your way through this.  Use Table 3-7 in your book to check your results.

 

4. Ruby is the mineral corundum with small amounts (few percent) of Cr3+ substituted for Al3+.  The 3d electronic levels of the Cr3+ ion are split by its octahedral coordination environment (crystal field). Two excited states lie 2.23 eV and 3.04 eV above the ground state. 

 

a) Determine the frequency and wavelength of the photons that would induce transtions to these excited states. 

 

b) In what portion of the visible spectrum do these wavelengths lie? 

 

c) Discuss the origin of the color of ruby in the context of your findings.

 

5. The carbon dioxide molecule has three vibrational modes referred to, respectively, as symmetric stretch (1388 cm-1), asymmetric stretch (2349 cm-1), and bend (667 cm-1) (frequencies in parantheses).

a) In what portion of the electromagnetic spectrum do these frequencies lie?

 

b) Determine which of these modes are infrared active.

 

c) How are these vibrational modes related to the greenhouse effect?

 

d) Are oxygen or nitrogen greenhouse gases?  Why or why not?

 

6. The wavefunction of the 1s orbital in hydrogen-like atoms

                                              

Where Z is the number of protons in the nucleus, r is the distance from the nucleus, and a0 is the Bohr radius, a0=0.5292 .  This wavefunction is exact for atoms or ions that contain a single electron (called Rydberg atoms).  Single electron ions unfortunately are not very interesting geologically, and anything more complicated cannot be solved analytically.  We will therefore stretch the meaning of this wavefunction by assuming that it also accurately represents ions with two electrons in the 1s shell.  This assumption will be adequate  for the question we want to address:

 

How is the wavefunction related to the concept of ionic radius? The relationship is best explored with the radial distribution function, P(r).  Physically, this function represents the probability of finding an electron a certain distance r from the nucleus, P(r).  This is the quantity that is plotted in your book in figure 3.11.  The probability is related to the square of the wavefunction , so the radial distribution function is

                                                         

a) Plot the radial distribution function vs. r for Z=3 (i.e. Li+) and Z=4 (i.e. Be2+).

 

b) How does the value of Z affect the shape of the charge density?  Explain physically the origin of this effect.

 

c) Use your graph to estimate the radius of Li+ and Be2+.  What criteria did you use to estimate the radius (there is no right answer!).

 

d) Compare your estimates to the Shannon and Prewitt ionic radii for octahedral (6-fold) coordination (Table 4.8 in your text).  Do they agree?  Why or why not?