Homework 1 Key (SVG)

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  1. Understanding geologic time. As we did briefly in class, consider your lifetime to correspond to the age of the earth. Calculate the following:
    1. How old were you when the dinosaurs arrived (227 Ma)?
      You need to first understand the analogy were making here. Here is a diagram showing the idea of the correspondence between your life and the life of the earth:

      Then you need to understand when on the earth's timeline the dinosaurs arrived:

      Note that the approximately correct placement of the time in earth's history shows that it is late in the earth's timeline. This shows that the answer for the question must be late in your lifetime (so if your calculation says that you were three years old, then you've done something wrong). In any numerical problem, it is wise to first estimate the answer before you write any numbers down.

      Now that we know about what the answer should be (maybe 18 or 19 years old?), we can start to calculate. First we need to figure out what fraction of the earth's lifetime has elapsed by the time the dinosaurs arrived. In order to do this, we need to know how many years have elapsed by that time. This is shown by the green line below:

      It should, I hope, be clear that this time is:

      4500 million years - 227 million years = 4273 million years

      So, by the time the dinosaurs arrived at 227 Ma, about 4273 million years had elapsed since the earth was formed. Now, how does this compare (as a fraction or a percentage) to the whole length of the earth's history? We can figure this out by:

      4273 million years / 4500 million years = 0.95 = 95%

      So by the time the dinosaurs arrived at 227 Ma, 95 % of the earth's current lifetime had elapsed. So now, we just figure out how long 95% of our lifetime is (this assumes you are 20 years old):

      20 years x 95% = 20 years x 0.95 = 19 years

      So you were 19 years old when the dinosaurs appeared. Wow, that wasn't very long ago. That means that most of the earth's history happened before the dinosaurs even appeared!
    2. How old were you when the dinosaurs mostly perished (65 Ma)?
      By a similar reasoning, this is calculated as follows:

      4500 million years - 65 million years = 4435 million years

      4435 million years / 4500 million years = 0.986 = 98.6%

      20 years x 98.6% = 20 years x 0.986 = 19.7 years (= 19 years, 8 months, about 12 days)

      So you were 19 years old when the dinosaurs perished. Wow (assuming you are exactly 20 years old), that only a few months ago. It's weird that it seems like such a long time ago that the dinosaurs perished, but it's really just like a few months out of 20 years!
    3. How old were you when the oldest known rocks were formed (4.03 Ga)?
      By a similar reasoning, this is calculated as follows:

      4500 million years - 4030 million years = 470 million years

      470 million years / 4500 million years = 0.104 = 10.4%

      20 years x 10.4% = 20 years x 0.104 = 2.09 years (= 2 years, 1 month, about 2 days)

      So the oldest known rocks were formed really early in earth history, but definitely not at the beginning. This is like the earliest baby pictures of you being when you were already 2 years old!
    4. How old were you when the first living things appeared on earth (as far as we know, about 3.5 Ga)?
      By a similar reasoning, this is calculated as follows:

      4500 million years - 3500 million years = 1000 million years

      1000 million years / 4500 million years = 0.222 = 22.2%

      20 years x 22.2% = 20 years x 0.222 = 2.09 years (= 4 years, 5 months, about 10 days)

      So life on earth appeared fairly early in earth history, but there was quite a lot of history that went by before that time!
    5. Choose two important dates in your life, and calculate how long ago they correspond to in the earth's history, both in number of years before the present, and in the divisions of geologic time (Fig. 8.15).
      Let's say one of your important dates was when you had your first kiss. You were eleven and a half (aw, that's so cute!). To figure out what this corresponds to in earth history, you go through a very similar process. Again, the first step is to estimate the answer, so you can tell if your calculation is close or not. The idea is like this:

      First, what percentage of your life had elapsed by the time of your first kiss?

      11.5 years / 20 years = 0.575 = 57.5%

      Now how long is that percentage of earth history?

      4500 million years x 57.5% = 4500 million years x 0.575 = 2588 million years

      So, by this analogy, by the time of your first kiss, 2588 million years had elapsed since the formation of the earth. However, earth times are not measured since the formation of the earth, they are measure in years before the present. So here's our picture now:

      This shows that the date in earth's history which corresponds to you being 11.5 years old is not 2588, but instead is:

      4500 million years - 2588 million years = 1912 million years ago = 1912 Ma

      This is in the Proterozoic part of the Precambrian Eon, before the first animals with hard shells appeared on earth.

      (I'm only going to do one of these, any other important date would be done in a similar fashion)
    6. Hunt around online -- searches for "historical geology" might be fruitful, or searches for the name of the time period (from Fig 8.15) -- and find out what the state of the earth was at the point in its history that corresponds to the date of your 15th birthday. Try to find the following:
  2. Radiometric Dating
    1. Feldspars commonly incorporate Potassium (K), and a portion of this K is the isotope K40. They commonly incorporate no Argon (Ar) to speak of when they initially crystallize. Let's imagine you want to find out the age of a volcanic dike containing feldspar phenocrysts. You analyze the feldspars, and find that the feldspar currently contains 250 atoms of Ar40 for each 750 atoms of K40. How old is the dike? (You will need data from Table 8.1 in your text)
      So, lets think of a bit of feldspar that contains 1000 atoms. Of these, 250 are Ar (I'm going to leave off the isotope numbers), and 750 are K. Here is the picture now:

      (250 Argon atoms, 750 Potassium atoms)

      We know that the feldspar originally incorporated only Potassium (K). So, when the feldspar crystallized, the picture was like this:

      (0 Argon atoms, 1000 Potassium atoms)

      The key idea here is a half-life, the amount of time required for half of the parent (K) to decay to the daughter (Ar). Realize what a half-life is: it's an amount of time. After one half-life has elapsed, the picture looks like this:

      (500 Argon atoms, 500 Potassium atoms)

      So, if at time zero, there were no Argon atoms, and after one half-life, there are 500 Argon atoms, then it's clear (I hope) that it requires 1/2 of one half-life to produce 250 Argon atoms.

      Another way (less intuitive, but perhaps more useful) to get this is to use Figure 8.13 in your book. This is a graph that relates the fraction of parent element remaining (i.e., still parent and not decayed to become daughter) to the amount of time that has elapsed, in terms of half-lives. [You can zoom in on the SVG figure at right to enlarge it.]

      How much of the original parent Potassium is left in this case (as a fraction)?

      Well, we now have 750 atoms of Potassium and we begain with 1000, so the fraction of the original now remaining is:

      750/1000 = 3/4 = 75%

      If you find this point on the vertical axis (in green at right), then the intersection with the red curve tells you the time (in numbers of half-lives) at which this fraction of parent remains. This time is shown in purple. Note that it is actually a little less than half, because the reasoning we used above is not, in a strict sense, correct. Using the graph is better.

      So, the age of the dike is about 0.4 x 1300 million years = 520 million years old.
    2. A different dike has feldspars with 750 atoms of Ar40 for each 250 atoms of K40. How old is the dike?
      Similarly to the above problem, lets think of a bit of feldspar that contains 1000 atoms. In this case, 750 are Ar and 250 are K. Now you cannot just reverse everything. Argon does not decay to Potassium. If you don't have a mental picture of the process, you will be lost. Here is the picture now:

      (750 Argon atoms, 250 Potassium atoms)

      How much of the original parent Potassium is left in this case (as a fraction)?

      Well, we now have 250 atoms of Potassium and we begain with 1000, so the fraction of the original now remaining is:

      250/1000 = 1/4 = 25%

      Referring to the graph at right, you can see that this fraction corresponds to 2 half-lives.

      So, the age of the dike is 2 x 1300 million years = 2600 million years old.
    3. A granitic intrusive rock frequently includes zircons, which incorporate Uranium. If you find a zircon that has 450 atoms of Lead-206 (Pb206) for each 550 atoms of Uranium-238 (U238), then how old is that zircon, assuming it had no lead when it crystallized?
      • What would be notable about a rock containing this zircon?
        Similarly to the above problem, lets think of a bit of zircon that contains 1000 atoms. In this case, 450 are Lead (daughter) and 550 are Uranium (parent). How much of the original parent Uranium is left in this case (as a fraction)?

        Well, we now have 550 atoms of Uranium and we begain with 1000, so the fraction of the original now remaining is:

        550/1000 = 55% (a little more than half)

        Referring to the graph at right, you can see that this fraction corresponds to a little less than 1 half-life. Let's say it's 0.9 half-lives. Table 8.1 tells us that one half-life for Uranium-238 is 4.5 billion years (=4500 million years).

        So, the age of the granite is 0.9 x 4500 million years = 4050 million years old. This would be notable, because it would be the oldest rock ever found - you would get your picture in the paper if you found such a rock.
    4. A different granite has zircons which you analyze, and the data say that it has 650 atoms of Lead-206 (Pb206) for each 350 atoms of Uranium-238 (U238). Why would you be forced to question the correctness of this analysis?
      In this case, 650 are Lead (daughter) and 350 are Uranium (parent). How much of the original parent Uranium is left in this case (as a fraction)?

      Well, we now have 350 atoms of Uranium and we begain with 1000, so the fraction of the original now remaining is:

      350/1000 = 35% (a little more than one third)

      Referring to the graph at right, you can see that this fraction corresponds to somewhere between one and two half-lives. Let's say it's 1.5 half-lives.

      So, the apparent age of the granite is 1.5 x 4500 million years = 6750 million years old. We would have to disbelieve this analysis, because it's older than the age of the earth! There's no way we could have granitic intrusive rocks before the earth was even formed!