# Why is the Earth Still Hot Inside?

## Summary

In this activity the teacher brings to students' attention that the Earth is still very hot inside. Deep mines are noticeably hotter the deeper you go (up to 30º per km of depth). Scientific investigations have determined that the upper mantle has a temperature of 1200ºC - 1400ºC and that the solid inner core has a temperature of up to 7000ºC. The Earth is also very old: currently estimates put its formation at 4.53 to 4.47 billion years ago. So if it was hot at its formation and all this time has passed, why is it still hot inside now? Part of the answer is that radioactive isotopes in the core and mantle continually produce heat in the interior of the Earth. But the primary reason is that heat loss from a body is less efficient when the surface area to volume ratio is low.

To investigate this problem students will be given a set of materials to look at cooling rates. They will heat glass spheres (marbles) in a hot water bath at 100ºC and transfer them to cold water in a styrofoam cup. They will then record the temperature of the water in the cup as a function of time. In this way they will generate both a total change in temperature and a total time during which the glass transfers heat to the water. Through discussion and discovery students will be led to repeat the experiment with glass spheres of a different size but the same total mass (the number of spheres will change, but the mass will be held constant). In this way students can be led to the idea that the ratio of surface to volume has important consequences in the transfer of heat.

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## Learning Goals

Students will learn about heat and temperature.

Students will learn about rates of heat transfer as a function of the size of the object: doubling the radius of a sphere increases its surface area by a factor of 4 but increases its volume by a factor of 8. Therefore it transfers heat at a rate ½ the rate of the smaller sphere. The rate of heat transfer is directly proportional to surface area but inversely proportional to volume.

Students will gain proficiency in using lab equipment to make repeatable measurements.

## Context for Use

This activity is best utilized with students in 11th grade chemistry or 12th grade physics. Students would benefit from prior exposure to thermochemistry and/or basic concepts of heat capacity and measurement of heat using calorimetry. This is an inquiry lab experience that may be completed in as little as two hours or may take four hours or longer. The length of time is up to the instructor's discretion and should depend on whether students are making productive discoveries. Students should work in groups of 2 to 4. The complexity of the activity can be modified for more advanced students to include calculations of heat content at different temperatures so that a plot of heat loss rate vs. area and vs. volume may be plotted to directly establish the nature of the proportions.

## Background

The oldest structures in the solar system are small spheres called calcium-aluminum inclusions (CAIs). These have been found in certain classes of meteorites that represent material that has remained unchanged since the formation of the solar system. CAIs have been dated to 4.567 billion years ago and the Earth is thought to have completed its formation between 30 and 90 million years later. This puts the formation of the Earth at about 4.53 to 4.47 billion years ago. The process of forming rocky planets is a violent one in which the planet grows larger through collisions with other bodies. Some of these collisions are extremely energetic and some are so large that the entire surface of the planet becomes a magma ocean. The Earth's Moon is thought to have formed as the result of such a collision.

The heat within the Earth can only be lost to space by radiation (mostly in the infrared part of the electromagnetic spectrum). The process of planet formation creates an enormous amount of heat through two mechanisms. First, the collisions that lead to the growth of a planet are very energetic and the largest of these can release more energy than any of the nuclear bombs in military arsenals. This is a powerful source of heat. Second, as the planet forms the denser materials sink toward the center of the planet. This releases gravitational potential energy as heat. In addition to these ancient heat sources, heat continues to be produced within the Earth by the radioactive decay of uranium, thorium, and potassium-40, though this source has diminished in size since the formation of the Earth. So why has this heat not all been lost to space in the past 4.5 billion years?

The Earth is still so hot inside in large part because its surface is too small, relative to its volume, for this heat to escape. The amount of heat energy is proportional to the mass (or the volume, since mass is proportional to volume). The rate of heat loss is proportional to the surface area. Small objects can radiate heat away quickly because the surface area is large relative to the volume compared to larger objects. For example, smaller planets such as Mercury and Mars are thought not to contain a liquid outer core, as does the Earth. Small moons of Jupiter and the other gas giant planets are normally expected to have completely solidified. Recent findings indicate that Europa (an icy moon of Jupiter) and Enceladus (a smaller icy moon of Saturn) both show signs of current melting and the possible extistence of sub-surface liquid water. These bodies demand a different explanation of the interior heating evidenced by recent observations by the Galileo and Cassini spacecraft. They are much smaller than the Earth and so should not have much internal heat. Scientists are currently unsure what leads to the internal heating of these small bodies (perhaps tidal interactions play a role) but for the Earth it is certain that the small surface-to-volume ratio plays an important part in preventing heat loss from the Earth.

When a sphere is scaled up by doubling its radius the surface area is scaled up by a factor of 4 and the volume is scaled up by a factor of 8. This can be understood perhaps most simply by examining the formulas for calculating surface area and volume. Surface area is proportional to the square of the radius (SA = 4πr2) and volume is proportional to the cube of the radius (V = 4/3πr3). Generally, if a body is scaled up by a length factor of k then the area scales up by k2 and the volume scales up by k3. The area/volume ratio scales down by a factor of 1/k (k2/k3) as an object is scaled up.

It is for this reason that polar bears are larger than black bears. Polar bears (8 – 10 ft) stand twice as tall as black bears (4 – 6 ft). As a result they have about 23 = 8 times as much volume. Both animals must maintain a constant internal temperature throughout their bodily volume. All body heat that escapes must escape through the skin and polar bears do not have 8 times as much skin; they have only 4 times as much. This means that heat escapes from a polar bear's body only ½ as efficiently (4/8) as from a black bear's body. Polar bears have a clear advantage in their cold arctic habitat because of this reduced efficiency of heat loss. Of course, bears are not planets but this analogy may help in understanding the relevance of scaling effects on the rate of heat loss.

In this activity students will be given an opportunity to heat an equal mass of either small glass spheres or larger spheres (the number of spheres will not be the same: clearly there will be a larger number of small spheres than large ones). Each group may choose the size it will test. Since the mass is equal, the amount of heat absorbed by either type of sphere will be the same, as long as they are made of the same material. These heated spheres will be placed in a calorimeter containing a set amount of water. The amount of temperature gain by the water will be the same for both the small spheres and the large spheres but the larger spheres will require a longer time to heat the water fully. Different groups will each have only one size of sphere at first. After they have collected several trials' data a class discussion will be held in which the data are plotted together from different groups. Ideas about total change in temperature, rate of heat loss, and size will be discussed. Heat loss rate can be calculated at change in temperature over change in time:

∆T/∆t 1.

If one or more groups have very different plots then the issue should undergo troubleshooting. It is a great opportunity to talk about how important it is to run down sources of error. At times they reflect a blunder of one kind or another but at other times an outlier value may hint at some as-yet unexplored phenomenon.

Note for advanced users: First have students measure the heat capacity of the glass spheres and ensure that it is the same for the small and large spheres. Then have them control their experiments based on the amount of heat, not the mass of glass. Heat can be calculated using the following equation:

Q = mC∆T 2.

Where Q is the heat in joules (J), m is the mass in grams, C is the heat capacity in J/g-ºC, and ∆T is the change in temperature measured in ºC. Since the heat capacity of water is 4.18 J/g-ºC the amount of heat transferred to water can be calculated using this equation. Then, with the known value of heat, mass of glass, and ∆T the heat capacity of the glass can be calculated. This allows students to calculate heat loss rate as ∆Q/∆t rather than as ∆T/∆t (where the small t represents time in seconds). Ultimately, this may lead to more understandable results. than as ∆T/∆t (where the small t represents time in seconds). Ultimately, this may lead to more understandable results.

## Description and Teaching Materials

### In-class Activities

The teacher directs the class in the following steps with or without a written procedure, as per the preference of the instructor.

1. Students prepare their lab notebooks to collect temperature data as a function of time. Time intervals should be no longer than 20 sec and no shorter than 5 sec.
2. Students are given a choice of pre-measured samples of large or small spheres.
3. Students make a 100ºC hot water bath by boiling water over a burner or electric heater. They add the spheres to this bath carefully, to avoid splashing.
4. They allow the spheres to come up to the temperature of the water bath.
5. While they wait they prepare a styrofoam cup (which can be nested for further thermal isolation) with 50 mL of water and a thermometer. The thermometer can be clamped to a stand for convenience.
Once sufficient time has passed to ensure that the spheres have been heated through they are transferred to the styrofoam cup (this may require some creativity to enable this transfer with a minimum of time out of the water―perhaps one at a time with tongs or tweezers).
6. As soon as the glass is in the styrofoam cup students record the temperature as a function of time at intervals between 5 sec and 20 sec.
7. They plot this data as a graph in their lab notebooks with time on the x-axis and temperature on the y-axis.
8. Students should repeat this procedure a few times in each group to see whether they can get relatively consistent results. They should be encouraged to troubleshoot any trials which have big deviations.
9. Finally, they should calculate the change in temperature over the change in time (or change in heat, if the class is using that option).
10. The teacher can call the students together for a group discussion once all groups have collected a few trials' of data. First, a class-wide plot of data for each size of sphere should be made. Then the teacher can ask questions such as the following:
• Are there any problems with these data? What are they and how can we explain them or fix them?
• What is happening? (Heat is transferred from glass to water at different rates).
• What stays the same for both the small and large spheres?
• What is different?
• Let's try to explain both of these.
• Are there any experiments we can try to test our explanations? Let's go try them.
11. Go back to the lab and try the new experiments suggested by the data and discussion.
12. Conduct further discussion as necessary to work toward the conclusion that the rate of heat loss by the larger sphere is lower by a factor of 1/k where k is the number of times larger the radius of the larger sphere is than the radius of the smaller sphere.In-Class Activities

### At Home Assignments

Students will write a formal lab write-up as described at http://kaffee.50webs.com/Science/labs/Lab-Report.Writing.html

The report will describe the experiments done within the group and within the class. It will display the shared data. Each student must write their own interpretation of the data and discuss its relevance to the formation of planets and to comparing planets of different sizes.

### Materials

250-mL beakers
burners or heaters
large glass spheres (sufficient for several groups)
small glass spheres (sufficient for several groups)
styrofoam cups
thermometers
lab notebooks
safety goggles
tongs and/or forceps (small tweezers)

## Standards

This is not particularly relevant in my state.

## Teaching Notes and Tips

I calculated that 25 g of glass with a heat capacity of 0.75 J/g-ºC at 100ºC would be able to raise the temperature of 25 mL of water about 13.5º. If 50 mL of water is used then the ∆T would be about 7º. This seems do-able. Also, if the density of the glass is near 2.2 g/mL then it would have a volume of 11.4 mL and should be able to be submerged in 25 – 50 mL of water.

I have not yet determined whether I can get enough marbles (glass spheres) of sizes that would be useful in this activity. Ideally, the larger spheres should have a radius at least 3 times the radius of the smaller spheres.

It may also be problematic to arrange samples of large and small marbles that have the same mass. In that case it may be necessary to measure the heat capacity of the glass and use the heat content to calculate ∆Q/∆t rather than ∆T/∆t.

## Assessment

Assessment will be based on the formal lab write-up produced by individual students.

## Why is the Earth Still Hot Inside? --Discussion

G'Day Aaron:

Some random thoughts from a mathematician as I read through things:

1. SUMMARY: Is it worth making the sentence "... glass spheres of a different size but the same total mass" a little clearer? That is, you are willing to put many small spheres into the same cup to match the mass of the one big sphere in the other cup. (I assume this is what you mean? Or do you mean you are going to have single small spheres, one in a cup, but made of a denser material so that the mass of the small sphere is the same as the mass of a large sphere?

2. BACKGROUND: Writing math in an e-mail is indeed a pain. People write: k^2 and k^3 for k squared and k cubed.

I completely ignored the issue of how much food a polar bear needs to eat in order to maintain its muscle mass, c.f. a black bear. If you get into this dicussion with kids, they will bring this up. So being big in a cold climate means you get to "keep" more of the energy you produce, but you have to find a lot more to eat to be that efficient. It's a trade-off that you will need to be prepared to think about with kids.

3. I think all looks good - in this writing stage of things. Of course, I am interested to see if reality matches the ideal here. Let me know how it all turns out when you test-run all this for yourself.

Fun stuff!

- J

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Aaron -

Really neat idea, cleverly designed. I'll be very interested to see if the students understand the planetary importance; could be a real winner!

A few content notes: Most petrologists would put 1200C at the low end of upper mantle temperatures. A good range is 1200 - 1400C, with an accepted characteristic temperature of 1300 or 1350C.

"The date of the formation of the Earth is difficult to define but it is now generally agreed that the Earth was nearly fully formed by 4.567 billion years ago." This is actually not right - that age, 4.567 Ga, is the age of the first tiny spherical solids, the CAIs. The planetesimals that later assembled to form the Earth were just beginning to grow at that time. The protoEarth itself, the body that received the Moon-forming impact, was not assembled until at least 30 million or perhaps 60 - 90 million years later (4.53 to 4.47 Ga).

There's new evidence that Mercury's core may still be partly molten...not much more I can say about that. Also, Europa is thought to have a subsurface ocean, and something is melting the interior of Enceladus. Might be important to say something at the end about other influences?

I can now articulate more about your statement "...not all of the residual heat energy can be accounted for in this way. Instead, the Earth is still so hot inside because its surface is too small, relative to its volume, for this heat to escape." The surface-to-volume argument applies to all sources of heat energy, whether they are "fossil" accretionary heat or ongoing radiogenic heat. Your statement makes me, at least, think that radiogenic heat is a separate problem from surface heat loss. It may be better to say that there are two major sources of heat in planetary interior: accretionary heat not yet lost to space, and radiogenic heat, also dwindling with time. So why is the interior of the Earth still hot where Mars and the Moon are colder?

Little clean-up note: "As a result they have 23 = 8 times as much volume..." OK, I know you know 23 does not equal 8 ...can you write this out more?

How do you know polar bears produce eight times as much heat in their bodies? Not necessarily true. And do the two kinds of bears have the same normal body core temperature? I'd take care not to push this analogy too far. Also depends upon food sources...

Can't wait to hear how this turns out! I'd love to see the data.

all the best -

Lindy

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Jim and Lindy,
Thanks very much for your valuable comments. I was particularly glad to get the scientific details correct. I hope I have changed things with regard to those details so that they are more accurate. I hesitated to put in too much about the outer planets' moons because I thought it might muddy the waters. On the other hand, now that I've written it in it may be that it could lead to fruitful discussion.

You're right about the polar bear analogy, both of you. I added a sentence about it being 'only an analogy' so that readers will, I hope, not take it too far. I am making a mental note for questions about food: certainly the heat-efficiency is paid for with a food inefficiency.

I've fixed all the mathematical typos: I thought they were all set and didn't check them when I pasted the work into the site from the word processor. I had typed all the exponents as superscripts and foolishly assumed they had pasted that way. Now I believe they are formatted correctly so that I won't have to use the '^' key.

As for whether reality will match the ideal...a lot depends on what materials I can get to use for the glass spheres. If I can get two different sizes of pyrex glass spheres then I expect I can get decent data. If not, well, we'll see. As you may recall, I hesitated to even write this activity because I was so unsure about whether it would be practical in the lab. I'll certainly be testing things out myself before I set students loose on it. I am interested to see how it turns out, though, and I really expect that it will spark some interesting discussion in class.

Thanks again for your thoughts. This was a very stimulating class and my thoughts are still whirling as a result. If only I could teach an entire course about planetary science! I almost think we covered enough this week to fill it.

I'll be in touch once I've had a chance to run some tests and put it past my class(es). Nothing like this is ever really 'live' until you try it. Even so, I'm content for this activity to made live on the SERC site. Maybe someone else will get to it before I do and try it out!

Take care,

Aaron

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Hello Aaron,

I, too, find this an interesting activity and want to know what happens when you actually try it out. I’ll be interested to hear how the kids connect it to planet cooling, and also wonder what connections kids will make to related daily-life situations. (Perhaps they have been surprised by the amount of time it took for a pot heated on the stove to cool, or something like that.)

The level of detail you’ve provided throughout this write-up will be helpful to other teachers who consider it. It has also enabled you to continue developing your ideas about the science involved as others react to specifics.

About the bears: If you present this a little differently, you can avoid the problematic aspects of comparing different species, or bears with planets. You could just note that some scientists have observed that within a species, individuals in colder parts of the range tend to be larger. Others have noted that high surface area/volume parts like ears tend to be smaller in cold-dwelling critters (again, within a species) – larger in warmer parts of the range. I’ll bet there are exceptions to the rule, and it would be interesting for kids to think about.

Regarding the Standards: Eventually, you may opt to link your activity to the National Science Education Standards as that could be useful to other teachers who are considering it. Obviously not a high priority. I'm glad you have a good degree of freedom in designing your program.

Ellen

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