GEOL 1404 HISTORICAL GEOLOGY EXTRA CREDIT (12 pts) RADIOMETRIC AGE DATING
HAND IN WORKSHEETS AND GRAPHS ONLY!!
You are NOT to work together on this…
DUE:
This Exercise is modified from Chapter 8 – Rock Units and Time-Rock Units in Levin & Smith’s Laboratory Studies in Earth History. See “Absolute Age Determination” for diagrams of decay curves and isochron illustrations.
Absolute dating by the use of radioactive decay provides a natural “clock” that starts when certain types of rocks are formed. Radioactive decay is the spontaneous nuclear disintegration of isotopes of certain chemical elements into their stable daughter isotopes. As disintegration occurs, energy is released and can be detected by a Geiger counter or similar device.
THE DECAY CONSTANT
The rate of for isotopes is stated in terms of half-life: the amount of time it takes for a given amount of a radioactive parent isotope to be converted to its non-radioactive daughter. By comparing the amount of a radioactive parent isotope in a particular rock with its daughter product, the geologist can determine how much time has elapsed since the rock formed.
The half-life gives an impression of the rate of change of a parent into its daughter, but to utilize the decay characteristics in a calculation of age, the decay rate () must be determined.
THE DECAY EQUATION
Although the decay rate is unique for every parent-daughter pair, every radioactive decay follows the same decay curve known as the exponential decay curve. It is expressed by the following equation:
Where:
t = number of years that have elapsed
P = number of parent atoms measured in the rock today
P0 = number of parent atoms at time t (when the rock formed)
exp = natural log of e (don’t worry about this for now)
= decay constant for the parent atom
The problem with this approach is that it is not possible to determine P0 without first knowing t. But, we can determine the number of daughter atoms present today. And if we assume that every daughter atom in the rock was created by the decay of a parent atom then the sum of the current number of parent atoms and daughter atoms will represent the number of parent atoms at time t, or P0 = P + D. The equation below reflects this relationship:
Where:
D = number of daughter atoms measured today
P = number of parent atoms measured today
But we have to other problems. First, we cannot assume that every daughter in the rock was created by the decay of a parent in the rock. It is very possible that some daughter isotopes were incorporated in the rock during formation and were not formed by decay after the rock formed. Second, it is very difficult to measure the actual number of daughter (and parent) atoms directly. It is much easier to measure the ratio of the parent isotope to an isotope of the daughter element but one NOT created through radioactive decay of the parent.
We can address these uncertainties by deriving a more complicated equation:
Where:
K = isotope of the daughter element that is not created by radioactive decay
Although this equation looks ugly, it is actually in the slope-intercept format of y = mx +b. In this case:
· y = D/K (Daughter ratio measured by analyzing the rock)
· x = P/K (Parent ratio measured by analyzing the rock)
· m= the slope of a line when x is plotted vs. y; in the equation, it is (expλt-1)
· b = the y-intercept when x = 0 (in this equation, it is D0/K)
THE ISOCHRON
To determine the radiometric age of a rock, multiple samples would be collected and the appropriate ratios of parent and daughter would be determined. These ratios would then be plotted versus each other. This will produce a line known as an isochron. The slope of the isochron line (expλt-1) reflects the age of the rock; the steeper the isochron, the older the rock. Also, the y-intercept of the isochron will provide the original ratio of daughter isotope at time = 0 (D0/K). Plotting the data from multiple samples provides another piece of information. If when plotted, the parent and daughter ratios from the rock do not form a straight line, then it can be assumed that some parent or daughter escaped since the time the rock was formed. Therefore, it is important to plot the data to verify that the samples appear to be unaltered.
We can plot the data on a graph, but we want to figure out the age of the rock (t). Once the slope of the isochron is determined, we can use the slope to calculate the age of the rock using (expλt-1). We can solve this expression for t:
So by determining the slope of the isochron and the decay constant of the parent isotope, we can determine the age of the rock. The natural log function is represented on a scientific calculator as “ln”.
EXERCISE – RADIOMETRICALLY AGE DATING
You have been studying the relationship between two rock units in the Piedmont physiographic province of North Carolina. The stratigraphic relationship between the intrusive igneous complex known as the Salisbury Pluton and the surrounding Whitewater Greywacke is poorly exposed, but initial field data suggests that the pluton is older than the greywacke, suggesting that the contact between the units is nonconformity. However, it is possible that the pluton intruded into the greywacke, making their relationship an intrusive contact. The goal of this exercise is to determine the radiometric ages of each unit, and thus confirming the stratigraphic relationship between the units.
Seven samples of the clay-rich graywacke and 13 samples from the pluton have been collected for mass spectrometer analysis to determine the ratios of parent 87Rb/86Sr and daughter 87Sr/86Sr for the purposes of determining the absolute age of the units. The results of the analyses are in the table on the worksheet.
The range of the ratios simply reflects that when the rocks formed, different minerals in the rock incorporated different amounts of rubidium. Samples with higher 87Rb/86Sr ratios suggest that the minerals in that sample had a greater amount of rubidium in them at the time of their formation. For the greywacke, it is believed that during diagenesis, the radiometric “clock” was re-set, meaning that the clay minerals “locked” in an amount of rubidium that would allow the rock to be dated radiometrically.
PART I: SALSBURY PLUTON
TASK 1– CALCULATE THE 87Rb DECAY CONSTANT
The half-life of 87Rb is 4.88 x 1010 years or 48.8 billion years. From this value and using Equation 1, calculate the decay constant λ and record the value in the table on the worksheet.
TASK 2– PLOT THE WHITEWATER GREYWACKE DATA
On the Whitewater Graywacke Graph, plot the 87Rb/86Sr vs. 87Sr/86Sr. The trend of the 7 points should approximate a straight line.
TASK 3– DRAW THE ISOCHRON FOR THE WHITEWATER GRAYWACKE DATA
Draw a single straight line through the data points that follows the trend of the data – this is the isochron for the data. This should be a single line that crosses the entire graph intersecting the y-axis. Do NOT “connect the dots.”
TASK 4 – DETERMINE THE SLOPE OF THE ISOCHRON
Remember, the slope of the isochron reflects the age of the rock. Slope can be determined by:
To determine your slope you must select x and y values from the isochron. Pick two widely-separated points on your isochron line to determine the age; it is best to find to places where your isochron crosses a grid line intersection – this makes it easy to determine a value of x and y. DO NOT SIMPLY CHOOSE 87Rb/86Sr and 87Sr/86Sr VALUES FROM THE TABLE!!!! Label the two points used for determining the slope of the isochron on the graph.
(a) Determine the slope of the isochron and record the value in the table on the worksheet.
(b) Add 1 to the slope and then take the natural log of this sum and record it in the worksheet. It should be recorded without scientific notation to four (4) decimal places.
TASK 5 – CALCULATE THE AGE OF THE SALSBURY PLUTON
Calculate the age of the Salsbury Pluton using your slope, the decay constant of 87Rb and Equation 3. Record the value of “ln(slope)” on the worksheet.
Record this age value in scientific notation to two decimal places in the table on the worksheet. Please review the Math Practice posted on Blackboard if you need to brush up on scientific notation.
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PART II: WHITEWATER GRAYWACKE – A DIFFERENT APPROACH…
Another option for calculating ages of rocks using parent-daughter ratios is to re-arrange Equation 2 to solve for t:
Re-arranged to solve for t:
This equation allows for the age of each sample that was part of the isochron to be determined. The average of these ages would provide an overall age of the rock unit.
Equation (5) looks worse that it is. We know all the terms except one, and that one is easily obtained.
D0/K can be easily determined. This value represents the y-intercept of the isochron line. Once this intercept is determined, an age of each sample can be calculated.
Equation (5) includes a natural log of the following:
It is a good practice to figure out the value of the above expression separately; then take the natural log of it by using the “ln” function button on the calculator.
TASK 1: PLOT THE WHITEWATER GRAYWACKE DATA
Follow the same procedures as Task 2 in Part I.
TASK 2: DRAW THE ISOCHRON FOR THE WHITEWATER DATA
Again, follow the same procedures as Task 3 in Part I.
TASK 3– DETERMINE THE ORIGINAL RATIO OF 87Sr/87Sr AT THE TIME OF DIAGENESIS
Record the value of the isochron’s y-intercept in the appropriate location on Table 2. This value should be recorded without scientific notation to 3 decimal places. Remember, this y-intercept represents D0/K, or the original amount of 87Sr at the time of diagenesis of the graywacke.
TASK 4 – CALCULATE THE AGE OF EACH WHITEWATER GRAYWACKE SAMPLE
Calculate the age of each sample in Table 1 using Equation (5). It should be accomplished in steps to avoid errors.
(a) For each P/K and D/K pair in Table 2 on the worksheet calculate the value of Equation 6 below. Record this value on the worksheet. This is an intermediate step for calculating the age in Task 4(b) below. The value should be in non-scientific notation format to four (4) decimal places.
(b) Calculate the age and record your age in the last column in the table on the worksheet. To do this, you will use Equation (5) below. NOTE, the calculations and age determination for Sample A-2 have already been done for you as an example.
(5)
i. Take the natural log (“ln” function on the scientific calculator) of the 6 remaining values from 4(a).
ii. Then divide each of these by the decay constant (λ), which was determined in Part I, Task 1 and is listed in Table 1.
iii. Record each of the 6 remaining ages in scientific notation to two decimal places. Please review the Math Practice posted on Blackboard if you need to brush up on scientific notation.
TASK 5 – CALCULATE THE AVERAGE AGE OF THE WHITEWATER GRAYWACKE SAMPLES
Determine the average of the 7 ages from Task 4 and record the value in Table 2. This is the age of the Graywacke.
GEOL 1404 EXTRA CREDIT NAME: ____________________________
RADIOMETRIC AGE DATING WORKSHEET
DUE:
PART I: SALSBURY PLUTON
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87Rb half-life = 4.88 x 1010 yr |
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Task 1: 87Rb decay constant (λ) |
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Table 1 – Salsbury Pluton Spectrometer Results |
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Sample No. |
87Rb/86Sr |
87Sr/86Sr |
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R5945A |
8.068 |
0.6562 |
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R5946 |
1.199 |
0.5496 |
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R5945 |
9.839 |
0.7034 |
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R5945B |
7.700 |
0.6587 |
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R5948B |
4.698 |
0.6143 |
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R5949A |
8.262 |
0.6607 |
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R5943A |
2.225 |
0.5659 |
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R5939A |
2.106 |
0.5672 |
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R5942B |
5.152 |
0.6110 |
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R5947A |
4.806 |
0.6120 |
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R5948A |
4.990 |
0.6122 |
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R5946A |
6.080 |
0.6306 |
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R5949B |
6.912 |
0.6379 |
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Task 4(a): isochron slope: |
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Task 4(b): ln (isochron slope+1): |
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Task 5: Age of rock |
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PART II: WHITEWATER GRAYWACKE
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TABLE 2 – WHITEWATER GRAYWACKE MASS SPECTROMETER RESULTS |
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TASK 3: Isochron y-intercept |
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Sample No. |
87Rb/86Sr
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87Sr/86Sr
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TASK 4(a)
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TASK 4(b) Calculated age (yrs ago) |
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A-2 |
86.20 |
1.2569 |
1.0060 |
4.21 X 108 yr ago |
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C-2 |
7.22 |
0.7841 |
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D-3 |
22.30 |
0.8758 |
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F-3 |
50.20 |
1.0478 |
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G-6 |
0.76 |
0.7467 |
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I-1 |
1.28 |
0.7489 |
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I-2 |
3.40 |
0.7601 |
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TASK 5 - Average Age = |
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QUESTIONS:
Determine the geologic Eon, Era and Period (if the rock is of Paleozoic or younger) of the pluton.
Determine the geologic Eon, Era and Period (if the unit is of Paleozoic age or younger) of the graywacke.
Based on the age relationship between the Whitewater Greywacke and Salisbury Pluton, is the contact between them a nonconformity or an intrusive contact? Explain/defend your answer.
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WHITEWATER GRAYWACKE
86.2 7.22 22.3 50.2 0.76 1.28 3.4 1.25685 0.78414000000000006 0.87580499999999994 1.047795 0.74665499999999996 0.74885999999999997 0.7600949999999999787Rb/86Sr
87Sr/86Sr
SALSBURY PLUTON
changed rb/sr and sr/sr 8.0676000000000005 1.1988000000000001 9.8388000000000009 7.7004000000000001 4.6979999999999995 8.2620000000000005 2.2248000000000001 2.1059999999999999 5.1516000000000002 4.8060000000000009 4.9896000000000003 6.0804 6.9120000000000008 0.65617499999999995 0.54959999999999998 0.70342499999999997 0.658725 0.61432500000000001 0.66067500000000001 0.56587499999999991 0.56722499999999998 0.61102499999999993 0.61199999999999999 0.61214999999999997 0.63060000000000005 0.6378749999999999787Rb/86Sr
87Sr/86Sr
