Issues in Earth Science

“Eww, There’s Some Geology in my Fiction!”

Issue 18, May 2024

Teacher Resources

Suggestions for Activities and Discussions to accompany Readings of

*Umbrella Luck *** by Meridel Newton**

Lesson Objective: These exercises engage students in graphical and mathematical thinking as well as in understanding the nature of phases of matter and mass balance.

Science and Engineering Practices (SEP) addressed: Analyzing and Interpreting Data; Using Mathematics

Crosscutting Concepts (CC) addressed: Energy and Matter: Flows, Cycles, and Conservation

1) Introduction to the idea of minerals and different forms of matter—polymorphs

2) Graphing polymorphs—using graphs that show phases of solid matter—graphite and diamond

3) Using polymorphs to figure out the geological setting for the formation of ancient rocks

4) Consideration of why the carbon gems in the story are so valuable, much more valuable than diamonds! Why might they be so valuable? What can we figure out from the story and what can we speculate?

The world-building backstory for Umbrella Luck by Meridel Newton includes wide-spread presence of vining plants that extract carbon from the air, thus cleaning the air and removing a primary greenhouse gas. We aren’t really told where these vines came from—were they a product of bioengineering intended to save the world from climate change? Did they develop naturally as the composition of Earth’s atmosphere became ever more carbon dioxide rich? Whatever their origin, they now extract carbon from the atmosphere and produce black carbon spheres that have great value (in addition to their air-cleaning value), greater value than the known forms of carbon, graphite and diamond.

What is this mysterious carbon that has such great value? What properties does it possess that give it such value? These questions are not addressed in the story, which is focused on the lives and actions of Sara and Kai. But we are going to think about these questions as the foundation for an excursion into mineralogical and geological thinking (and some Science-Fiction thinking too!).

Geological
minerals are naturally occurring materials that have a definite chemical composition
(not changing composition in any arbitrary way) and a crystal structure (the
atoms are arranged in regular, repeating arrays). Sometimes it is possible to
have minerals with the same chemical composition but different crystal
structures. Such materials are called *polymorphs* (meaning many forms).
Graphite and diamond are polymorphs, having the same chemical composition (pure
carbon) but different crystal structures.

The different crystal structures cause the polymorphs to have different properties. For example, graphite is one of the softest minerals, so soft that it easily scrapes off to leave a black line behind when we use the graphite to make pencils. Diamond is the hardest mineral, so hard that it can’t be scratched with a steel knife and is used to cut or polish other hard materials or rocks.

The different properties also mean that the polymorphic minerals will form under different geological conditions. In general, the more tightly packed (denser) polymorph will form at higher pressure (for any given fixed temperature). This makes sense. If we squeeze something (higher pressure), we expect it to compress. Likewise, in general, the polymorph with the less-ordered, more random atom arrangement will be favored at higher temperature (for any particular, fixed pressure). This also makes sense from our everyday experience. As temperature increases, solid materials (in which the atoms are tightly bound by chemical bonds and so tend to be more ordered) generally convert to liquids and gases (in which the atoms are moving faster and are less tied to each other and so are more random).

A graph that
shows the conditions of temperature and pressure under which different
polymorphs naturally form or occur is called a *phase diagram*. *Phase*
is the chemical name of a material with distinct boundaries and chemical
properties. You may be familiar with the phases of water: solid, liquid, and
gas. Polymorphs are also phases.

You can practice interpreting this kind of graph using the diagram below. This graph shows the conditions of formation (which can be determined by experiments in a laboratory) for the imaginary polymorphs A and B. In this graph, the regions of temperature and pressure where A or B exist are marked. The line represents those conditions where A converts to B or B converts to A (we could think of this as the equilibrium line were A and B are in equilibrium with each other).

1) For any particular temperature, which phase, A or B, forms at higher pressure? (teacher note: A, since A always occurs at a higher P for any particular T)

2) For any particular pressure, which phase, A or B, forms at higher temperature? (teacher note: B, since B always occurs at higher T for any particular P)

3) Given the relationships seen in the graph above, and remembering the connections among density, randomness of the crystal structure, pressure and temperature, which of A and B is the most dense? Which has the most ordered crystal structure? (teacher note: most dense = A making it more preferred at higher P for any particular temperature; A has most ordered structure, since B always occurs at higher temperature for any particular , and high temperature favors the phase with the higher entropy, more random crystal structure)

Try a different graph, below, for two different imaginary polymorphs C and D.

1) For any particular temperature, which phase, C or D, forms at higher pressure? (teacher note: C)

2) For any particular pressure, which phase, C or D, forms at higher temperature? (teacher note: C, notice that the change in slope of the equilibrium line on the graph means that there is a difference in the relative properties of C and D vs A and B above in that C occurs at both higher T and higher P whereas A occurred at higher pressure but not higher temperature.)

3) Seeing the relationships between C and D from the graph, and remembering the relationships among density, randomness of the crystal structure, pressure and temperature, which of C and D is the most dense? Which has the most ordered crystal structure? (teacher note: C most dense, D most ordered)

Below is the phase diagram for carbon (we will call this the currently-known phase diagram since whatever phase occurs in “Umbrella Luck” is not shown!). This phase diagram is public domain from the English Wikipedia-- https://en.m.wikipedia.org/wiki/File:Carbon_basic_phase_diagram.png.

**Based on
this diagram:**

For any particular temperature (at which both phases can exist), which of graphite or diamond is favored at higher pressure? (teacher note: diamond, as is known from pop culture)

For any particular pressure (at which both phases can exist), which of graphite or diamond is favored at higher temperature? (teacher note: graphite)

Which is more dense, diamond or graphite? (teacher note: diamond)

Which has more ordered atoms, diamond or graphite? (teacher note: diamond)

Which has more ordered atoms, graphite or liquid carbon? (teacher note, the solid, graphite, occurs at lower temperature consistent with lower entropy and thus ‘greater order’).

Here’s a tricky one: which is more dense, liquid carbon or graphite? (teacher note: Notice that it depends on pressure! At lower pressure, the slope of the equilibrium line is positive, and graphite is on the higher-pressure side, however, at higher pressure the slope curves around to be negative, and liquid is on the higher-pressure side! This means that the properties of the liquid and/or graphite are changing as pressure changes. As pressure increases, the liquid carbon compresses more than the solid graphite, and eventually becomes more dense than the solid graphite!)

Where would you expect diamond to form (near the Earth’s surface or deep in the Earth’s interior)? Explain why you think this. (hint: 1 atmosphere pressure equals 0.0001GPa on this graph, below the bottom of the graph. The lowest value on the graph, 0.001GPa, is the pressure at about 1 km depth in the ocean, or perhaps 300-400 meters deep in rock). (teacher note: Diamond is expected to form at high pressure, which occurs deep in the Earth’s interior, pressure being caused by the weight of overlying rock. Kimberlites, one of the most common host rocks for diamonds, originate deeper than most other volcanic rocks on Earth.)

Below is the
simplified conceptual phase diagram for polymorphs of the composition Al_{2}SiO_{5}
which, depending on temperature and pressure, can occur as the polymorphic
minerals Kyanite, Silliminite, or Andalucite. Each of the three minerals has
the same composition but different crystal structures.

Based on this graph, put the three polymorphs in order of increasing density. (teacher note: pick a temperature where all three exist and see which occurs at the higher pressure. andalucite, silliminite, kyanite)

Put the three polymorphs in order of decreasing ordering of the atoms. (teacher note: pick a pressure where all three exist and see which occurs at higher temperatures. Kyanite, Andalucite, Silliminite)

Now, suppose that you found three metamorphic rocks at the surface of the Earth (exposed by years of erosion). You want to know which geological setting each of the rocks formed in before they came to be at Earth’s surface.

Rock 1 contains Andalucite

Rock 2 contains Kyanite

Rock 3 contains Silliminite

Associate each of these rocks with one of the geological settings illustrated below (in real life, there can be some overlap, but choose the conceptually ‘most fitting’ answer). This image shows a cross-sectional view of the Earth, showing a subducting plate boundary. The conditions at each of points A, B, and C can be inferred from the image below as the following (remembering that pressure and temperature tend to increase with depth):

A: High Pressure beneath a mountain range, but relatively cooler than surrounding rock because the subducting lithosphere has not yet fully heated up

B: High Pressure and high Temperature both occur at depth beneath a mountain range

C: Low pressure due to shallow depth, but high temperature due to proximity of molten rock

Teacher Note:

Rock 1 contains Andalucite (formed at location C, with moderately high T but low P )

Rock 2 contains Kyanite (formed at location A, with high pressure but a temperature lower than typically expected for that depth)

Rock 3 contains Silliminite (formed at location B, with both high pressure and high temperature due to depth in the Earth)

So, why are
the carbon gems in the story *Umbrella Luck* so valuable? What property
might give them such value? The story doesn’t really tell us, but did you
notice anything unusual about the gems when you read the story that might hint
at the reason for their value?

1) The
following four excerpts from the story *Umbrella Luck *hint at a truly
astonishing property:

*She grips
and pulls, her eyes narrowing with the effort. At last she feels the give, and
she pulls her treasure free. It is a stone—or it looks like a stone. Black and
shiny, round as a drop of water, she holds it up to the sun to examine it more
closely.*

*“Twenty
thousand cubic meters, I bet,” she mutters to herself. “At least. Maybe as many
as thirty.”*

*Such a
small stone to represent so much. Thirty thousand cubic meters of rank,
unbreathable air, compressed down to the palm of her hand. A wonder, if not a
natural one.*

. . .

*She’d let
herself relax, let her guard drop as she’d tried to collect the carbon gem,*

*. . .*

*She
swallows. “I thought…it only needed to be once. I only needed one cubic
kilometer.”*

*“Let me
see.”*

*He holds
out his hand, and, with only a moment of hesitation, she passes over her
bounty. The gems click and tumble as he runs them through his fingers.*

*“This is
half that, at most. You have a long way to go.”*

*. . .*

*“Wait.” He
catches at her hand. There’s a rustle, and then he’s pressing something into
her palm.*

*“Take these.”*

*She can
feel the carbon gems, dozens of them, filling the little pouch. It’s not just
the other half of what she needs, it’s far more.*

Before reading on, can you think of some hidden property of the gems, based on the text above, that might explain why they are so valuable? Write down your thoughts here before continuing:

Based on the story, what can we infer about these carbon gems? One thing might be that the vines take carbon out of the atmosphere, presumably as carbon dioxide, the way Earth plants do. That explains why the vines themselves are valuable--taking carbon out of the air to ‘clean’ it, and perhaps limit climate change resulting from the carbon. Although this explains why the plants are valuable, it doesn’t explain the value of the gems.

Somehow, the
vines compress pure carbon into valuable gems that Sara and Kai collect. This
gem-forming process takes a lot of raw air. The text says one stone represents
30,000 cubic meters of air. It also says that Sara needs carbon gems formed
from about *1 cubic kilometer* of air, (and the pouch Kai gives her
contains even more than she needs).

Herein lies the clue to at least one mysterious property implied by the story. How much would the carbon in 1 cubic kilometer of air weigh? How much should those pebble-sized gems weigh? Try to figure it out!

1. What are
some things we need to know to figure out the weight of carbon in 1 km^{3}
of air?

*You might
want to give students time to think about this question and brainstorm possible
ideas and questions. Listen to what your students have to say. In moving
forward, you’ll have to decide how much guidance to give or not give your
students. Questions students might come up with: *How can we know the weight of a gas?
How much carbon is in air? Is carbon present in the air as just carbon, or as
carbon dioxide? What is air made of? How do we know what atoms or molecules
‘weigh’? Once I look up all this information, what calculation method do I use
to figure out the mass of Carbon in air? Your students might come up with all
of these on their own, but most of them will likely need further guidance to
figure out the full calculation. Even so, it’s important to give them the
chance to try to figure it out on their own.

2. Here is a step-by-step primer on how to do the calculation (presuming your students have looked up some of the needed values online, and assuming that the air in the story has somewhere close to the same carbon composition as Earth’s air).

Before we
can calculate weight of carbon in air, we need to know how much CO_{2}
is in a given volume of air. A University of Chicago physicist (referenced here) helps us through the logic of using
atomic weights. This calculation is fairly obtuse and skips over some of the
important logical steps, so we have modified a bit for classroom use. *You
may need to give your students a little background on the periodic table and
atomic mass and weight*.

·
The current
concentration of CO_{2} in our air is 410 parts per million by volume
(ppmv) which for an ideal gas is the same as parts per million by mole. We can
visualize that out of every 1 million molecules of air, 410 of them are CO_{2}.

· To get the mass of carbon dioxide in air, we need to consider not just the concentration by volume, but the concentration by mass, which means we need to consider the molecular weights of carbon dioxide and other air molecules.

·
The average molecular
weight of air is about 29 (nitrogen, N_{2}, is about 28 and 0_{2}
is about 32, and these are the two main components of our air)

·
The molecular
weight of CO_{2} is about 44 (C has an atomic number of 12, oxygen an
atomic number of 16, so 1 carbon plus two oxygen atoms = 12+16+16).

·
So, to convert to
a mass fraction (the fraction of the air that is CO_{2}, by mass) we multiply
410 by 44 to get the mass of the 410 carbon dioxide molecules, and multiply
999590 (the rest of the air molecules) by 29 to get the mass of everything
else.

·
This gives us
18040 for the mass of CO_{2} and 28988110 for the mass of everything
else (if we thought of moles of molecules rather than molecules we could put
these values into units of grams)

·
So, the mass
percentage of CO_{2} = mass of CO_{2} divided by the total mass
times 1 million (to convert to ppm) = (18040 / 29006150 ) x 1000000 = 622ppm

·
Note: The
calculation in the reference above, skips some of these steps and calculates by
scaling the number of CO_{2} molecules by the relative proportion of
the mass of CO_{2} to the mass of air, 44 to 29, however this hides
some of the underlying math and does not let students see the meaning of
concentration in the process of the calculation. However, doing the math this
way yields the same result: 44/29 times 410 gives us 622 parts per million mass
units. In other words, the fraction of the air’s mass that is made up of CO_{2}
is .000622, or .0622%.

· So about 0.0622% of the mass of air is made up of carbon dioxide.

· We know that the density of dry air (the ratio of the amount of mass of a given volume) is about 1.2 kilograms per cubic meter. Note that since the amount of water vapor is variable, we’re just thinking about dry air.

·
The CO_{2}
component of the density would be 1.2 kg/m^{3} x 0.0622% = 0.00075
kg/m3. This means that within a cubic meter of air, there is 0.75 grams of CO_{2}.
*You may want to demonstrate the size of a cubic meter with meter sticks and wight
out the amount of 0.75 grams of some solid to help students get a sense of
scale.*

3. So, the amount
of carbon dioxide in the atmosphere is 0.75 grams per cubic meter or 0.75g/m^{3}.
**If Sara had a packet of gems that had concentrated carbon from 1 cubic
kilometer of air, how much would the carbon gems weigh?** *Give students a
chance to wrestle with this problem on their own with a partner or small group.
Give students a chance to explain their reasoning to the whole class.*

4. The most
straightforward way to do this calculation for most students will probably be with
the following reasoning: If one cubic meter of air holds 0.75g of carbon
dioxide, then how much carbon dioxide will be in one cubic kilometer? I need
to know how many cubic meters are in one cubic kilometer! Since a cubic
kilometer is 1000 meters on each of the three sides of a cube, the number of
meters will be 1000m x 1000m x 1000m = 1,000,000,000m^{3}. 0.75 g/m^{3}
x 1,000,000,000m^{3} = 750,000,000g (noticing that 1/m^{3} x m^{3}
= 1 and so the m^{3} cancels, giving us only grams, the mass units
that we are interested in. – this kind of dimensional analysis can help
students understand the meaning of numbers in a scientific way, not just a
mathematical way.)

5. If your students have experience with algebra, it might be useful to spend some time helping them explore this idea of dimensional analysis more thoroughly. Unit (or dimensional) analysis is a mathematical technique that keeps track of both numbers and units. Units are what give numbers meaning in science, and unit analysis is a technique that helps students know whether to ‘multiply or divide’ and gives them a real-world understanding of numbers. The heart of the technique is using conversion factors that, when stated as a fraction, have a value of 1. For example, 1 kilometer = 1000 meters. So 1 km/1000 m = 1 and 1000 m/1 km = 1.

ALTERNATIVE
METHOD (more complicated by allowing more dimensional analysis): The figure below
shows a different way to calculate how much carbon dioxide is in a cubic
kilometer of air, maximizing the use of conversion factors to multiply and
divide both numbers and units (this is the method used in the physicist
reference cited above). We start with the concentration of carbon dioxide in
air (0.75 g/m^{3}) which is kind of like a density of carbon dioxide in
the air and so is labeled with a “D,” and end with the weight of carbon dioxide
from 1 cubic km of air. We have to convert from m^{3} to km^{3},
from g to kg, and make other conversions as well. *To avoid spending too
many class periods on this, you might choose to work through this unit (or
dimension) analysis step by step with your students.*

*The basic
calculation is that D, the concentration of carbon dioxide in air, in units of
g/m ^{3}, must equal the mass of carbon dioxide in 1 m^{3}
divided by 1 cubic meter of air which also must equal the mass of carbon
dioxide in 1km^{3} of air divided by 1km^{3}. From this
starting point, a series of mathematical and algebraic operations will get to
both the correct mass (for example, in units of kg which can be converted to
pounds), and the correct units (for example, kg or pounds). If the calculation
is done correctly, other units, like m^{3}, km^{3}, and kg will
cancel (if it is done incorrectly, you might end up with weird units like kg^{2}
or km^{6}, or pound-kg).*

*0.75 g/m ^{3}
= x /km^{3}*

*Or x =
0.75 g-km ^{3}/m^{3}*

*There are
1000000000 m ^{3} in each km^{3}, or 1000000000m^{3}/km^{3}.
This value equals one (we are ratioing two different numbers of the same value
but different units), so we can multiply the expression above by this value
without changing it as follows:*

*x= 0.75
g-km ^{3}/m^{3} x 1,000,000,000m^{3}/km^{3}*

*m ^{3}
and km^{3} both cancel and the numbers multiply out to give us*

*x =
750,000,000g*

*to
convert to kg, you can recognize that one kilogram equals 1000 grams, or 1 =
1kg/1000g*

*We can
multiply by this number (which equals one) giving us*

*x =
750,000,000g x 1kg/1000g = 750,000kg.*

*We can
convert to pounds by recognizing that on pound equals 2.2pounds or 1 = 2.2lb/kg*

*Multiplying
again by this value (which equals 1) gives us*

*x =
750,000kg x 2.2lb/kg = 1,650,000lb*

*Numbers
without units often don’t tell us much in science. In science, only certain
kinds of numbers are ‘dimensionless,’ and even then we need to understand the
setting and application of the dimensionless number for it to have meaning.*

The amount of carbon dioxide in one cubic kilometer of air from the calculation illustrated above is 1.65 million pounds!

6. We aren’t
quite done yet. The gems are only made of carbon, *not carbon dioxide*.
How many pounds of __carbon__ are there in 1 km^{3} of air? *Give
your students a chance to think about how to figure this out. Give time for
them to share their ideas with the class. *We can use the ratio of atomic
weights of the atoms within a CO_{2} molecule. The atomic mass of carbon
is 12 and the atomic mass of oxygen is 16. So, the molecular mass of a CO_{2}
molecule =44. The fraction of carbon dioxide that is carbon is therefore
12/44, so we can get to the mass of carbon by considering that there are
1,650,000 lbs CO_{2} used to make the gems, and there are 12 grams of
carbon for every 44 grams of CO_{2}. So, there are 1,650,000 pounds CO_{2}
x 12 grams C/44 grams CO_{2} = 450,000 pounds C in the gems.

At this point, you could do a few more unit analysis type questions (Go deeper into the examples already given above, or a calculation of the weight of Sara’s 20,000-30,000 cubic meter gem, for example) OR you can proceed to considering the meaning of the mysterious high mass of the carbon gems, a consideration that is discussed below.

*A note
about teaching science as a practice: Taking multiple class periods to go
through one relatively small calculation, especially if you do the unit
analysis with your students, seems hard to justify with the ‘old’ mindset of
‘covering all the material.’ However, taking the time to actually work through
problems is what mathematical thinking is all about. Yes, it takes a lot of
time, which is probably why it is often skipped over in classroom learning,
which, in turn, is why so many students never really ‘get’ how math is used in
science.*

7. So the
bag of carbon gems that Kai gives Sara, combined with her already collected
gems, is *more than* 450,000 pounds! Should either of them be able to
lift the bag of gems with no apparent effort? It would seem that the young
people shouldn’t be able to carry little pouches weighing nearly ˝ million
pounds!

What in the world does this mean? Here’s the beauty of fiction – perhaps there is a mysterious property, not yet discovered, that allows all that carbon mass to not weigh very much. Perhaps this mysterious property is what makes the carbon gems so valuable.

In science, we make observations, sometimes unexpected ones, and then we try to understand the implications of the observation. As an imaginative exercise, what underlying reality might explain the observation of the mysteriously underweight gems?

Encourage your students come up with possibilities, letting them know that they are not looking for a real-world explanation, but a SF type explanation—maybe also addressing why the gems are so valuable, more valuable than diamonds. Below are some possible explanations that we came up with—you might offer one or two to your students as ‘seed’ ideas:

·
Maybe much of the
mass of the crystal is converted to energy that is held bound within the gem.
If that energy can be tapped into, it might explain their high value. How much
energy? You might calculate it from Einstein’s famous relationship: E=mc^{2}.
Compare your calculated value to the amount of energy produced by a windmill on
a typical day. Or to a 20-gallon tank of gasoline.

· Alternatively, maybe the carbon gem develops some kind of anti-gravity characteristics. Antigravity is one of the holy grails of SF, which, like FTL (faster than light) speed is contrary to known science. Could it be so dense that it repels other matter? Truly dense black holes don’t appear to do this, so if the carbon gems do, what might that mean?

·
Maybe some of the
matter is not present in our universe, but exists *outside our universe*
in an alternate dimension. Maybe what makes the gems so valuable in the story
is their ability to form a portal to another dimension!

· Students might also speculate that the protagonists are just very strong—maybe all the people are stronger in this story, or maybe the object makes people much stronger. Holding it gives great strength. This might be more fantasy than SF.

· Maybe Earth’s gravity is greatly decreased (although this doesn’t obviously explain the high value of the gems and there would be many other implications of this)—If students suggest this possibility, you might prompt them to consider other implications. Are the events in the story consistent with very low gravity, a millionth of Earth gravity? How high could a person jump? What would happen when you tried to walk? What is escape velocity where someone would jump completely off Earth? What would be the effect on the human body? Etc

· Students might also speculate that the plants have decreased the carbon in the air to such a point that the mass of carbon in a cubic kilometer is much lower than it would be today. The problem is, without that carbon in the air, earth would get too cold to live on, all liquid water would freeze. “According to scientists, the average temperature of Earth would drop from 14˚C (57˚F) to as low as –18˚C (–0.4˚F), without the greenhouse effect.”-- https://education.nationalgeographic.org/resource/greenhouse-effect-our-planet/

· Students might speculate that only the air in the vicinity of the city is depleted in carbon. However, winds and weather systems rather quickly mix batches of air on Earth, so this doesn’t quite work either without other implications for the world. For example, are the cities domed and thus isolated from the rest of the world’s atmosphere?

8. Another peculiar aspect of the gems that students might pick up on is that if Sara finds a hand-sized gem that represents 30,000 cubic meters of air, then a cubic kilometer of air would require some 33,333 gems of comparable size. There is nothing in the story to suggest that she and Kai are exchanging such huge numbers of objects. To accommodate this observation, one might speculate that, in the same way that the mass of the gems is much less than the mass of the carbon they are made from, so to does the volume of the gems not represent in any simple way the amount of carbon in them. This could lead to all kinds of additional science-fiction interpretations.

_____________________________________________________________________

The Teacher Resources for *Umbrella Luck* are
written by Russ and Mary Colson, authors of Learning to Read the Earth and Sky.

Return to Umbrella Luck by Meridel Newton

Return to “Eww, There’s Some Geology in My Fiction.”

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