Understanding E = mc2

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.
By William Tucker
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 betweenmatter 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:
1.
For any given
amount of energy, mass and velocity are inversely related. For
an identical amount of energy, the higher velocity goes, the less mass is
required and vice versa.
2.
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 1-½ 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
any place 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.