Understanding E = mc2
Ed. note: A few weeks ago, I had the pleasure of hearing William Tucker speak at a conference in Washington, DC. His explanation of E = mc2 was the best I had ever heard. Even better, Tucker explained how Einstein”s equation applied to renewable energy sources like wind, solar, and hydro. His lecture was a revelation. It showed that the limits of renewable energy have nothing to do with politics or research dollars, but rather with simple mathematics. During a later exchange of emails with Tucker, I praised his lecture and suggested he write an article that explained E = mc2 and its corollary, E = mv2.
To my delight, he informed me that he”d already written such an essay and he agreed that we could publish it in Energy Tribune.
I love this essay. And I”m proud that Tucker has allowed us to run it.
Prof. Albert Einstein delivers the 11th Josiah Willard Gibbs lecture at the meeting of the American Association for the Advancement of Science in the auditorium of the Carnegie Institue of Technology Little Theater at Pittsburgh, Pa., on Dec. 28, 1934. Photo by AP
When I was in college, I took a course in the great political philosophers. We studied them in order – Hobbes, Locke, Rousseau, Kant, John Stuart Mill and Karl Marx.
In my mind, I had placed them with the historical eras they had influenced – Hobbes and the 18th century monarchs, Locke and the American Revolution, Rousseau and 19th century Romanticism, Kant and the 19th century nation-states, Marx and 20th century Communism.
Then one day I saw a time-line illustrating when they had all lived and died. To my astonishment, each had lived a hundred years before I had placed them in history. The implicated seemed clear. “It takes about a hundred years for a new idea to enter history.”
Almost exactly 100 years ago, Albert Einstein posited the equation E = mc2 in his “Special Theory of Relativity.” The equation suggested a new way of describing the origins of chemical energy and suggested another source of energy that at that point was unknown in history – nuclear energy. Nuclear power made its unfortunate debut in history 40 years later in the form of an atomic bomb. But 100 years later, Americans have not quite yet absorbed the larger implications of Einstein’s equation – a new form of energy that can provide almost unlimited amounts of power with a vanishingly small impact on the environment.
E = mc2. Who has not heard of it? Even Mariah Carey named her last album after it. “E” stands for energy, “m” for mass, and “c” is the speed of light – that’s easy enough. But what does it really mean? (The answer is not “relativity.”)
What E = mc2 says is that matter and energy are interchangeable. There is a continuum between the two. Energy can transform into matter and matter can transform into energy. They are different aspects of the same thing.
This principal of the equivalence of energy and matter was a completely unexpected departure from anything that had gone before. In the 18th century, Antoine Lavoisier, the great French chemist, established the Conservation of Matter. Performing very careful experiments, such as burning a piece of wood, he found that the weight of the resulting gases and ashes were always exactly equal to the weight of the original material. Matter is never created nor destroyed, it only changes form.
Then in the 19th century a series of brilliant scientists – Count Rumford, Sadi Carnot, Rudolf Clausius, Ludwig Boltzman – established the same principal for energy. Energy can take many forms – heat, light, motion, potential energy – but the quantity always remains the same. Energy is never created nor destroyed either.
Now at the dawn of the 20th century, Albert Einstein posited a third principal that united the other two in a totally unexpected way. Einstein stated a Law of Conservation between matter and energy. Nothing like this had ever been imagined before. Yet the important thing is that co-efficient – the speed of light squared. That is a very, very large number, on the order of one quadrillion.
We really don’t have a reference point for a factor of one quadrillion. We know what a trillion is – that’s the federal budget deficit. But a quadrillion is still a bit beyond our ken. What it means, though, is that a very, very large amount of energy transforms into a very, very small amount of matter and a very, very small amount of matter can transform into a very, very large amount of energy.
Perhaps the way to understand the significance of Einstein’s equation is to compare it to another equation, the formula for kinetic energy:
Kinetic energy is the energy of moving objects, “E” once again standing for energy, “m” indicating mass and “v” representing the velocity of the moving object. If you throw a baseball across a room, for example, its energy is calculated by multiplying the mass of the ball times the square of its velocity – perhaps 50 miles per hour.
The two formulas are essentially identical. When brought into juxtaposition, two things emerge:
- For any given amount of energy, mass and velocity are inverselyrelated. For an identical amount of energy, the higher velocity goes, the less mass is required and vice versa.
- When compared to the velocities of moving objects in nature – wind and water, for instance – the co-efficient in Einstein’s equation is fifteen orders of magnitude larger – the same factor of one quadrillion.
How is this manifested in everyday life? Most of what we are calling “renewable energy” is actually the kinetic flows of matter in nature. Wind and water are matter in motion that we harness to produce energy. Therefore they are measured by the formula for kinetic energy.
Let’s start with hydroelectricity. Water falling off a high dam reaches a speed of about 60 miles per hour or 80 feet per second. Raising the height of the dam by 80 or more feet cannot increase the velocity by more than 20 miles per hour. The only way to increase the energy output is to increase the mass, meaning we must use more water.
The largest dams – Hoover and Glen Canyon on the Colorado River -stand 800 feet tall and back up a reservoir of 250 square miles. This produces 1000 megawatts, the standard candle for an electrical generating station. (Lake Powell, behind Glen Canyon, has silted up somewhat and now produces only 800 MW.)
Environmentalists began objecting to hydroelectric dams in the 1960s precisely because they occupied such vast amounts of land, drowning whole scenic valleys and historic canyons. They have not stopped objecting. The Sierra Club, which opposed construction of the Hetch-Hetchy Dam in Yosemite in 1921, is still trying to tear it down, even though it provides drinking water and 400 megawatts of electricity to San Francisco. Each year more dams are now torn down than are constructed as a result of this campaign.
Wind is less dense than water so the land requirements are even greater. Contemporary 50-story windmills generate 1- 1/2 MW apiece, so it takes 660 windmills to get 1000 MW. They must be spaced about half a mile apart so a 1000-MW wind farm occupies 125 square miles. Unfortunately the best windmills generate electricity only 30 percent of the time, so 1000 MW really means covering 375 square miles at widely dispersed locations.
Tidal power, often suggested as another renewable resource, suffers the same problems. Water is denser than wind but the tides only move at about 5 mph. At the best locations in the world you would need 20 miles of coastline to generate 1000 MW.
What about solar energy? Solar radiation is the result of an E = mc2 transformation as the sun transforms hydrogen to helium. Unfortunately, the reaction takes place 90 million miles away. Radiation dissipates with the square of the distance, so by the time solar energy reaches the earth it is diluted by almost the same factor, 10-15. Thus, the amount of solar radiation falling on a one square meter is 400 watts, enough to power four 100-watt light bulbs. “Thermal solar” – large arrays of mirrors heating a fluid – can convert 30 percent of this to electricity. Photovoltaic cells are slightly less efficient, converting only about 25 percent. As a result, the amount of electricity we can draw from the sun is enough to power one 100-watt light bulb per card table.
This is not an insignificant amount of electricity. If we covered every rooftop in the county with solar collectors, we could probably power our indoor lighting plus some basic household appliances – during the daytime. Solar’s great advantage is that it peaks exactly when it is needed, during hot summer afternoons when air conditioning pushes electrical consumption to its annual peaks. Meeting these peaks is a perennial problem for utilities and solar electricity can play a significant role in meeting the demand. The problem arises when solar enthusiasts try to claim solar power can provide base load power for an industrial society. There is no technology for storing commercial quantities of electricity. Until something is developed – which seems unlikely – wind and solar can serve only as intermittent, unpredictable resources.
There is only so much energy we can draw from renewable sources. They are limited, either by the velocities attained, or by the distance that solar energy must travel to reach the earth. So is there anyplace in nature where we can take advantage of that “c2” co-efficient and tap transformations of matter into energy? There is one that we have used through history. It is called “chemistry.”
Chemical energy is commonly described in terms of “valences.” A sodium atom has a valence of +1, meaning it is missing an electron in its outer shell. Meanwhile, a chlorine atom has a valence of -1, meaning it has an extra electron. Together they “mate” to form sodium chloride (table salt). All chemical reactions are either “endothermic” or “exothermic,” meaning energy is either absorbed or released in the process. The Bunsen burner in chemistry class is a way of adding energy to a reaction. The other thing that can happen occasionally in chemistry lab is a sudden release of energy called an “explosion.”
The great achievement of 20th century quantum physics has been to describe chemical reactions in terms of E = mc2.
When we burn a gallon of gasoline, one-billionth of the mass of the gasoline is completely transformed into energy. This transformation occurs in the electron shells. The amount is so small that nobody has ever been able to measure it. Yet the energy release is large enough to propel a 2000-pound automobile for 30 miles – a remarkable feat when you think of it.
Still, electrons make up only 0.01 percent of the mass of an atom. The other 99.99 percent is in the nucleus of the atom. And so the question arose, would it be possible to tap the much greater amount of energy stored in the nucleus the way we tap the energy in the electrons through chemistry?
For a long time many scientists doubted it could be done. Einstein himself was skeptical, saying that splitting an atom would be like “trying to hunt birds at night in a country where there aren’t many birds.” But other pioneering scientists – Enrico Fermi, George Gamov, Lise Meitner and Leo Szilard – discovered it could be done. By the late 1930s it had become clear that energy in unprecedented quantity could be obtained by splitting the unstable uranium atom.
Unfortunately, World War II pre-empted the introduction of nuclear power. This is a historical tragedy. The atom bomb stands in the same relation to nuclear energy as gunpowder stands to fire. While gunpowder has played an important role in history, fire’s role has been far more essential. Would we want to give up fire just because it led to guns? Yet the atom bomb continues to cast a shadow over the equally important discovery of nuclear energy.
The release of energy from splitting a uranium atom turns out to be 2 million times greater than breaking the carbon-hydrogen bond in coal, oil or wood. Compared to all the forms of energy ever employed by humanity, nuclear power is off the scale. Wind has less than 1/10th the energy density of wood, wood half the density of coal and coal half the density of octane. Altogether they differ by a factor of about 50. Nuclear has 2 million times the energy density of gasoline. It is hard to fathom this in light of our previous experience. Yet our energy future largely depends on grasping the significance of this differential.
One elementary source of comparison is to consider what it takes to refuel a coal plant as opposed to a nuclear reactor. A 1000-MW coal plant – our standard candle – is fed by a 110-car “unit train” arriving at the plant every 30 hours – 300 times a year. Each individual coal car weighs 100 tons and produces 20 minutes of electricity. We are currently straining the capacity of the railroad system moving all this coal around the country. (In China, it has completely broken down.)
A nuclear reactor, on the other hand, refuels when a fleet of six tractor-trailers arrives at the plant with a load of fuel rods once every eighteen months. The fuel rods are only mildly radioactive and can be handled with gloves. They will sit in the reactor for five years. After those five years, about six ounces of matter will be completely transformed into energy. Yet because of the power of E = mc2, the metamorphosis of six ounces of matter will be enough to power the city of San Francisco for five years.
This is what people finds hard to grasp. It is almost beyond our comprehension. How can we run an entire city for five years on six ounces of matter with almost no environmental impact? It all seems so incomprehensible that we make up problems in order to make things seem normal again. A reactor is a bomb waiting to go off. The waste lasts forever, what will we ever do with it? There is something sinister about drawing power from the nucleus of the atom. The technology is beyond human capabilities.
But the technology is not beyond human capabilities. Nor is there anything sinister about nuclear power. It is just beyond anything we ever imagined before the beginning of the 20th century. In the opening years of the 21st century, it is time to start imagining it.
William Tucker is the author, most recently, of Terrestrial Energy: How Nuclear Power Will Lead the Green Revolution and End America’s Energy Odyssey.