How Big is Big and How Small is Small

by Smith, Timothy Paul

  From the Sun’s perspective, the Earth occupies only a very tiny spot in the sky, so the Earth only intercepts a very tiny fraction of that outflow of energy, about 1.76 × 1017 J/s. By the time the light has come through the atmosphere about 30% has already been reflected back into space, in particular, most of the ultraviolet. Still, about 1000 J/s of energy strike each square meter of the Earth, if it is a cloudless day and the Sun is directly overhead. Modern commercial solar cells are about 15% efficient or perhaps a bit more, with laboratory cells measuring nearer to 40%. So if our square meter was covered with an average panel of solar cells we could expect about 150 J of energy every second, or 150 W of power. With that we could charge up batteries for our lights at night or drive a solar race car. So we have gone from the potential energy locked in the fuel of our star to the chemical potential in a battery. But that same sunlight can do a lot of other things as well.

  Most of the Sun’s light ends up heating the Earth. Because the Earth has a steady temperature (climate change is slow compared to the amount of energy that heats the Earth daily), all this energy eventually is radiated away. But for a while it dwells on Earth, drives the winds and stirs the oceans by heating different parts of our planet differently. Obviously the equatorial regions, with the Sun poised directly overhead, catch the most energy and are well heated, whereas the polar regions, with slanting sunlight, are only slightly warmed. In fact about 1% of the energy from the Sun ends up driving the wind and the currents on Earth.

  The sunlight can also smile upon plants. A little less than 0.08% of the energy in sunlight is absorbed by plants via the process of photosynthesis, transforming that energy into chemical energy embedded in the plant itself. We think of biofuels as the energy in the oils or starches of a plant, but the whole plant—the stalks, the roots and the leaves—is potential energy. The Earth produces about 150 billion tons of biomass each year. And all of that is potential energy.

  Over 20% of the Sun’s heat also drives the water cycle; evaporating water mainly from the oceans and then raining everywhere. If the rain falls on highlands we can collect it and run it through turbines that spin generators and produce electricity. If we are not tapping that energy, nature itself is using the rivers to sculpt mountains and valleys and to build up flood plains and deltas.

  At first it appears as if all of our energy comes from the Sun: heat, biomass, wind and hydro. Even the energy in fossil fuels, which is no more than ancient biomass, comes from the Sun. However there are a few non-solar energies in play on Earth.

  Nuclear power stations may be the most obvious non-solar energy form on Earth. The uranium that fuels these plants predates the Sun. The rarer elements resulted from a supernova, the death convulsions of an ancient star before our solar system even formed. But human-built power plants are not the only way to tap into this power source. The fact that the center of the Earth is at about 5400°C is in part due to radioactive material deep within our planet. It is not that there is a reactor with neutron-induced fission going on but rather a few atoms, sparsely distributed, are undergoing spontaneous radioactive decay and in that process giving up heat. It is not a lot of heat per decay. In fact there is no more heat in a kilogram of rock deep in the Earth than one on the surface. But there is a huge amount of material underfoot, and all that heat eventually must come up through the surface. It is like the city described in Chapter 2. There the volume of the city grew faster than its surface area, and so the traffic on the arterial roads became more intense. We see that energy traffic, that outflow, as volcanos, geysers and hot springs.

  One final non-solar energy source here on Earth is tidal power. It has only been tapped by humans in a very few unique locations, such as in the Bay of Fundy in Canada and Rance River, near St. Malo in France. Yet if you have ever watched the ocean rise and fall and contemplated the trillions of tons of sea water that is being shifted, it is clear that there is a lot of energy here.

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  Figure 7.1 Energy densities for chemical and nuclear fuels. (Top) Energy densities. Chemistry and nuclear fuels are well separated. Gravity does not really fit and so here it is offset. (Bottom) Detail of chemical fuel densities.

  Flow was an interesting way of looking at energy, and in the process we touched on a number of examples, but we still do not have that organizing principle. However, I think I now know what I do not want to plot. I know that a piece of wood contains a million joules of energy whereas a cord of firewood contains a billion joules. I do not want that to appear twice on our plot. This is like in Chapter 2, where I did not plot all elephants or whales, but rather only the largest. I would like to plot the energy for wood and the energy for uranium and the energy for a comet, and look for a single number that characterizes all pieces of wood or uranium or celestial ice balls, independent of their varied sizes. To do this I will concentrate on energy density (see Figure 7.1). How much energy does a gram of fuel or a flying soccer ball contain.

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  According to the nutritional panel on the side of my oatmeal box, one serving starts with 40 g (half a cup) of dry oatmeal and contains 150 Calories. These are food calories, and we will now return to the maze of energy units. One food calorie is one thousand heat calories, or one kilocalorie; the upper case “C” is just shorthand for this. So my 150 breakfast Calories is 150,000 heat calories or about 630,000 J, something we know because of James Joule’s experiments. So we find that oatmeal has 16,000 J/g.

  Pure carbohydrates and proteins are a bit higher that raw oatmeal, with 17,000 J/g, but fats and oils pack about twice the energy density, with about 37,000 J/g. A chocolate chip cookie, which is a combination of carbohydrates, proteins, fats and oils is someplace in the middle, at about 20,000 J/g, depending upon the particular recipe. Curiously enough, TNT only has about 3,000 J/g. It is hard to believe that a cookie actually packs seven times the energy of TNT, yet we do not use cookies to blast rocks, because of how fast they burn.

  Non-food fuels run across a wide range. But here again we have to pause. If you look up the energy in wood you might see it reported as “24 million BTUs per cord” for red oak. A cord is really a volume, defined as a stack of firewood 4 feet wide, 8 feet long and 4 feet high. The cord is a unit that makes sense to people hauling wagonloads of firewood. It can be converted to pounds and kilograms. But what about BTUs?

  A BTU is a British thermal unit, and is defined as the amount of heat that will raise 1 pound of water by 1 degree Fahrenheit. Its definition is very similar to a calorie, which is defined as the amount of heat that will raise 1 g of water by 1°C. Since there are 1000 g in a kilogram this also implies that 1 Calorie is the amount of heat to raise 1 kg by 1°C.

  Finally we can convert that “24 million BTUs per cord” for red oak into an energy density and find a figure of just under 15,000 J/g, very close to the value for oatmeal. People who burn wood tend to prefer oak, maple and ash over pine and hemlock because there are more BTUs per cord (and also because pine can be sooty). But a cord of oak actually weighs more than a cord of pine, and the energy density across all types of wood is nearly constant at 15,000 J/g.

  The energy density of coal is a bit higher than wood, at 24,000 J/g. Petroleum-based fuels all weigh in at about three times that density, with propane at 46,000 J/g and gasoline-petrol at 47,000 J/g, which is to be compared to the density of food fats and oils. Still, all of these fuels vary by only a factor of three.

  Nuclear is on a scale of its own. Uranium-235 packs 79,500,000,000 J/g (7.95 × 1010 J/g), an energy density about 2 million times greater than gasoline. That is also very close to the density of energy in hydrogen if it is undergoing fusion, as in the Sun. An ideal fuel would be deuterium and tritium, with an energy density of 3 × 1011 J/g.

  What about the weights on a cuckoo clock? At the beginning of the week, when the weights are raised, these may have an energy of 20 J/kg or 0.02 J/g. That might be enough energy to spin the hands of the clock, drive the bird out to recite its cuckoos and chime the hours, bu
t it is not a lot of energy.

  Another way in which gravity is used to store energy is with pump storage facilities. In these places water is pumped uphill to a high-elevation reservoir to store energy. When the energy is needed the water is released and it runs downhill, through a hydroelectric turbine that converts that gravity potential into electrical energy. Some of the largest of these are Bath County facility in Virginia and the Kannagawa project in Japan. Both of these can produce about 3 million watts of power, for about ten hours. Ten hours is a useful amount; it means you can store excess energy at night, when other power plants still produce it but there are few customers to consume it, and then use it in the day when the demand is high. As far as energy density is concerned, the Kannagawa project is the more impressive. It uses less water, but pumps it higher and achieves an energy density of 6.4 J/g.

  When we plot different systems or objects or fuels as a function of their energy density we find that on the far left are things driven by gravity, in the middle are things with energy stored in chemical bonds, and on the right are nuclear fuels. Each group is separated by a factor of nearly a million. It seems like a trend that will help us organize energy and also fits with our instinctive feeling that gravity is weaker than chemical bonds and that nuclear forces are stronger than everything else. But how do we reconcile this with the fact that a falling meteor contains up to 2.5 million J/g?

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  The meteor that struck the Earth about sixty-five million years ago had about 4 × 1023 J of energy. This was the meteor that formed the 180-km-wide Chicxulub crater, on the Yucatan peninsula in Mexico. Its impact and the resulting dust are credited with causing the demise of the dinosaurs. However, it is unfair to say that it was a typical meteor; it was a chunk of rock about ten kilometers on a side. Still, there are some things common to all meteors.

  If I were to drop a ball from the edge of the universe, by the time it hit the Earth it would be traveling at about 11 km/s. It does not matter the size of the object I drop, much like the two iron balls Galileo dropped from the top of a tower. If I drop a golf ball: 11 km/sec. If I drop a bulldozer: the same. But that is not up to the speed of meteors.

  If I dropped these same objects on the Sun they would be traveling at 62 km/s when they hit the Sun’s surface. This is because the Sun is much more massive than the Earth and exerts a much greater gravitational force. This is why meteors that hit the Earth average about 42 km/s, which seems very different to the 11 km/s cited in the last paragraph. Actually, meteors that hit the Earth are generally being attracted towards the Sun, but the Earth gets in their way. Their velocity can also be modified by the fact that the Earth is in motion around the Sun at about 29 km/s, and part of that velocity could add to or subtract from the 42 km/s of the meteor depending upon the relative directions of their motions.

  So the energy related to gravity depends upon the masses of the attractor and the attractee. The binding of the Moon or a satellite to Earth does not really characterize gravity in a general way; rather, it characterizes the particular nature of gravity on Earth.

  It is true that a rock falling to Earth releases potential energy just as the burning of wood or oil does, but it is not really a fuel. A fuel is a material, a substance, that contains energy intrinsically in its structure. Fuels also are stable. They just sit there until they are somehow ignited.

  When we burn a piece of wood or a teaspoon of gasoline, or when we metabolize a bit of bread, we are releasing energy by breaking chemical bonds. As an example we will look at the burning of hydrogen and oxygen;

  We start out with three molecules: two molecules of hydrogen (H2), each with two atoms (remember Dalton and Avagadro?) and a oxygen molecule (O2), as shown in Figure 7.2. These molecules are stable and will just sit there. I have to apply some energy to break them up. It takes 4.5 eV of energy (yes a new unit; just think energy on the atomic scale) to break up H2 → H + H and 5.1 eV for O2 → O + O, or 14.1 eV to break up all three molecules. Once they are broken up they will likely recombine into 2H2O and release 20 eV in the process. So there is a net release of about 6 eV. I went through this example to show why fuels do not burn spontaneously. They need a spark to get the process going.

  Figure 7.2 The energy that is released when hydrogen burns, at the molecular level. (A) Hydrogen and oxygen are broken by external energy. (B) Atoms regroup. (C) When the atoms recombine into H2O molecules, extra energy is released.

  If we turn to more complex fuels, like wood or oil, the molecules involved are different, but the principle is the same. Most foods and fuels are hydrocarbons, meaning that the molecules involved are primarily made of hydrogen and carbon. Some fuels, such as glucose (C6H12O6), also have some oxygen. One of the most important and well-known molecules in petroleum is octane. Octane chemically is C8H18, often written as CH3(CH2)6CH3, which is a chain of carbon atoms, with hydrogen filling any unfilled carbon bonds (see Figure 7.3). When octane burns it ideally takes two octane molecules and twenty-five oxygen molecules for the full reaction.

  Figure 7.3 The octane molecule. Octane is a hydrocarbon chain made up of eight (oct) carbon atoms and eighteen hydrogen atoms, which means that there are a lot of bonds with energy stored in them.

  Figure 7.4 Energy from nuclear fission. (A) A neutron hits235U. (B) The uranium is excited into236U, which is unstable. (C)236U decays into 92Kr + 141Ba + 3n + Energy.

  What makes octane such an attractive fuel is that it has a lot of chemical bonds and energy in a compact form.

  Nuclear fuel is similar. For example, the most common fuel in a nuclear power plant is uranium-235 (235U). It, like octane, will just sit there for a long time. However, in a reactor it is subjected to a barrage of neutrons. When the fuel 235U absorbs a neutron it becomes a new isotope called 236U, which is not stable. It will break up and decay. In fact there are many ways for it to decay, but a common one, illustrated in Figure 7.4, is;

  These reason energy is released is that the daughter particles, 92Kr and 141Ba, require fewer nuclear bonds to hold them together than 235U. Again, a fuel is characterized by the fact that is starts out stable, needs a “spark” to ignite it, but then is reconfigured into something with weaker or fewer bonds, releasing the excess energy. This excess must be greater than the “spark,” or in this case the number of neutrons, to sustain the burning.

  What we are seeing is that potential energy that is stored in a material, a fuel, is closely related to the forces that bind that material (see Table 7.1). The amount of energy in nuclear fuel is related to the binding of neutrons and protons by the nuclear force. The amount of energy in other fuels, such as oil or food, is related to the binding of atoms by chemical forces or bonds. Now try to imagine an analogous system for gravity. Maybe planets with satellites as our molecule. If two planet-plus-satellite systems were to collide could they rearrange themselves and give off energy? Yes they could, but the amount of energy would depend on the particular planetary system and the “burning” of this type of gravity-fuel would be uneven and unsatisfactory even to a cosmic titan. There really is not a good analogous gravitational fuel.

  Table 7.1 Comparison of nuclear and chemical forces and bonds.

  The fact that nuclear fuels have about a million times the energy density of other fuels is because the forces are about a million times stronger. But this still is not a comparison at the most fundamental level. Chemical and nuclear forces are not basic and fundamental forces in nature. They arise as a local expression of deeper forces.

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  Table 7.2 Comparison of the four fundamental forces in nature.

  Force

  Relative strength

  Range

  Gravity

  1

  ∞

  Weak

  1029−1035

  ~ 10−15 m

  Electromagnetic

  1036

  ∞

  Strong

  1038

  ~ 10−15 m

  The basic four forces of nature are gravit
y, and the weak, electromagnetic and strong forces (see Table 7.2). In this list the nuclear force and chemical force do not even appear. That is because nuclear and chemical forces are only residuals. For example, a neutral atom is made up of electrons and a nucleus. These are held together by the electromagnetic force and the atom is neutral. So one would not expect an atom to be electrically attracted to another neutral atom, but they are. Atoms are attracted to each other because the charges inside of them are not evenly distributed. So one atom’s positive region may be attracted to a second atom’s negative region and that attraction gives rise to chemical bonding. In a similar way the nuclear force between protons and neutrons is the leakage of the more basic strong force between quarks. We will describe this leakage in more detail in Chapter 10 for atoms and chemistry, and in Chapter 11 for the nucleus.

  Right now, however, we will try to compare these fundamental forces to see if we can make some sense of the scales of energy. Is the strong/nuclear force intrinsically stronger than everything else? Is it really a million times stronger than the electromagnetic force? How weak is gravity?

  The most common way of comparing forces is to find a place in nature where the two forces are at work and then compare their magnitudes in that situation. So to compare the strong and electromagnetic forces we can look at two quarks inside of a proton or neutron. If we do that we find that the strong force is about 50–100 times stronger than the electromagnetic force. There are two reasons that I can only give a range for comparing their strengths. Firstly, the ratio depends upon which quarks I pick; different quarks have different electric charges but the same strong binding. Secondly it depends upon the distance separating the quarks, and these forces act differently as distances grow. The electromagnetic force tails off over long distances, whereas the strong force grows as the quarks are pulled farther apart. In that respect the strong force is like a spring, harder to stretch the farther you pull it apart.