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U235: A Lesson in Plausibility & Proportion

He had made an estimate, and George Uhlenbeck, who shared an office with him in Pupin Hall, was there one day to overhear him. Fermi was standing at his panoramic office window high in the physics tower looking down the gray winter length of Manhattan Island, its streets alive as always with vendors and taxis and crowds. He cupped his hands as if he were holding a ball. “A little bomb like that,” he said simply, for once not lightly mocking, “and it would all disappear.”

Richard Rhodes (from The Making of the Atomic Bomb, 1986)

No one who was unfamiliar with experimental developments in nuclear physics in late 1939, and even some physicists who were, would have said that Enrico Fermi’s offhand remark, “A little bomb like that (cupping his hands) and it would all disappear” was plausible. How could something you could hold in your hands contain enough energy to destroy a city! It was just too remote from human experience, too wildly out of proportion. But physicists had known for some time that a single atom of a certain rare variety (or isotope) of uranium, known as U235, would release 200 million electron volts (eV) of energy when it fissioned (split) into two smaller atoms. That’s a lot of energy for something as small as a uranium atom. And the conviction was growing on some physicists that it might actually be possible to cause a very large number of U235 atoms to fission in a few millionths of a second, a process that was known as a fast neutron chain reaction. In which case Fermi’s estimate, if not precise, was definitely in the ballpark. Here are the data and simple calculations—two numbers are either multiplied or divided—that make the implausible, plausible.

There are 6.24 billion billion eV of energy in a joule. “But how much is a joule?” you ask. Lift a three pound weight three inches—that took roughly a joule of energy (or work). Drop the weight from three inches onto a book to visualize how much energy a joule is. It is not a negligible amount—imagine the weight falling on your head instead of a book. A single U235 atom releases 200 million eV on fissioning. That is 0.000000000032 joules of energy. It would require 31.2 billion U235 atoms to fission to release a single joule of energy. If you placed each of those atoms in the centre of a little box one square millimetre in area ( ¤ ), you would need a square that measured just over 176 metres on the side to accommodate all the little boxes, each with its U235 atom. Looked at in that way it doesn’t seem that fissioning U235 atoms are anything to worry about. A few more simple calculations will show that there is plenty to worry about! Even an atom of uranium, the heaviest naturally occurring element with 92 protons and 140 to 146 neutrons in its nucleus—there are six isotopes of uranium, U232, U233, U234, U235, U236 and U238—is an incredibly small object. How small? Well, a U235 atom weighs 235.0439242 amu (atomic mass units). One amu weighs:

0.00000000000000000000000000166053873 kg (kilograms)

For comparison, a proton weighs slightly more than an amu, about 1.0073 times more, while a neutron weighs about 1.0087 times more.

Thus, a U235 atom weighs:

0.000000000000000000000000390299539 kg

Not very much you say. But that’s the problem! It is such a tiny quantity of matter that you can hold an absurdly large number of U235 atoms in your ‘cupped hands.’ “So what!” you reply. Well, here’s the problem: the energy released by a fissioning U235 atom is out of all proportion to its mass on the high side! Consider this: the 0.000000000032 joules of energy released by one fissioning U235 atom may seem insignificant, but suppose it was sufficient to cause something that was visible to the naked eye to move. A simple calculation will settle the matter. The circular area of this sheet of paper covered by a period, ( . ), is 0.34 mm in diameter. That means it has an area of 0.0908 square millimetres, or 0.0000000908 square metres. But this kind of paper (24 lb cover stock) weighs 90 grams or 0.090 kg per square metre. Therefore the amount of paper covered by a period weighs 0.000000008172 kg. Now that period-sized paper disk is a colossally massive object compared with a U235 atom. Nevertheless, the energy of a single fissioning U235 atom can make that behemoth visibly move. Here’s the calculation. Work (or energy) equals force times distance: W = Fd. Use F = mg (a special instance of Newton’s second law, F = ma, force equals mass times acceleration) to convert the mass of the paper disk into weight (i.e. force) by multiplying its mass by g (g = 9.8067m/s2, the acceleration due to the earth’s gravity). Multiplying 0.000000008172 kg by g gives 0.00000008014 N (Newtons), which is how much the paper disk weighs for scientific purposes, even though we use units of mass (kilograms) in everyday life. Turning again to W = Fd, we divide the force we just calculated into W, the work or energy that the U235 atom releases on fissioning (see above). The resulting d, or distance, is 0.00039 metres or 0.39 millimetres. That much movement could easily be seen with the naked eye, assuming the U235 atom “exploded” efficiently under the paper disk, lifting it or perhaps shoving it to one side.

Still not impressed? Ever heard the saying “There’s strength in numbers”? There is also energy in numbers, large numbers of U235 atoms in this case. Remember that each fissioning U235 atom is a miniature atom bomb. Detonate enough of those tiny atom all at once and you’ve got a bomb that is anything but tiny. How many atoms of U235 are we talking about here? Well, the bomb that destroyed fifty thousand buildings in Hiroshima involved the fissioning of 1.78 trillion trillion U235 atoms. To get a feel for that number calculate the surface area of planet earth in square millimetres. You should get 510 billion billion. Then divide that number into 1.78 trillion trillion atoms. The result is 3490, the number of earths-sized planets required to accommodate that many U235 atoms if each atom (or miniature atom bomb) was allotted one square millimetre of surface area. One such tiny bomb is of no consequence, but that many tiny bombs going off all at once changes the nature of war and of international affairs. And it’s all because each U235 atom is so tiny compared with familiar objects—as we’ve already seen, it only weighs:

0.000000000000000000000000390299539 kg

So when we multiply the above number by our 1.78 trillion trillion U235 atoms, we only come up with 0.695 kg, or a bit more than one and a half pounds. And such a small mass is what makes the atomic bomb a practical weapon. It is portable enough to be carried by rocket, airplane, artillery shell or even a soldier or two. And the amount of U235 that has to be separated from naturally occurring uranium, despite the enormous cost and the tremendous technical ingenuity required, is quite manageable.

So, how many joules of energy do we get from our 0.695 kg of fissioned U235? The calculation is simple. You recall that one U235 atom yielded 200 million eV or 0.000000000032 joules of energy. Therefore we multiply that number by our 1.78 trillion trillion atoms to get 57 trillion joules. But a number like that is conceptually awkward. Nuclear physicists prefer to work in units of kt (1000 tonnes) of the high explosive TNT. And a kt of TNT yields 3.8 trillion joules of energy. If we divide that number into 57 trillion joules we get 15 kilotons, which is to say that the bomb that destroyed Hiroshima was equivalent to 15,000 tonnes of TNT.

Calculations based on experimental data, such as those above, convinced Fermi and other physicists that what at first had seemed wildly implausible was, in fact, quite plausible: an object you could hold in your cupped hands could, under the right conditions, release enough energy to obliterate Manhattan. Indeed, the uranium core of the bomb that destroyed Hiroshima weighed 64 kilograms, making it only about 1 percent efficient. But the 21 kt bomb that was dropped on Nagasaki had a plutonium core which fissioned about 20 percent of its Pu239 isotopes—each Pu239 atom releases 207 million eV on fissioning—and that 3.6 inch diameter spherical core only weighed 6.2 kilograms, making it slightly smaller than a softball.

A Bold Analogy?

It may seem like a bit of a stretch, but I submit that miracles—defined as an interference with Nature by supernatural power—are similar to atomic bombs in three respects. They seem implausible because they fly in the face of ordinary experience. Believing that they exist depends (for most of us) not on experience but on human testimony, and on evidence, such as photographs, films, x-rays, doctor’s records, etc., that entails trusting those who present the evidence. If the possibility or impossibility of miracles could be established beyond doubt, then the result would be the utter destruction of one of the cases for the two most common—but opposite—world views.

If miracles ever occur, then naturalism is obviously false. Thus, a proof of miracles would constitute a crushing blow to naturalism—though some naturalists might accommodate it by allowing a preternatural realm of nature, not to be identified with traditional notions of the supernatural. If miracles never occur, then revealed religion, such as Christianity, is obviously false. (A supernatural realm might still exist, but there would be no intelligent reason for anyone to believe in it.) Thus, a disproof of miracles would constitute a crushing blow to religion—though religion would probably live on in a very reduced position of power and influence.

However, demonstrating a universal negative—in this case, that miracles never happen—is usually conceded to be impossible. Demonstrating theoretically that they can happen—the way it was demonstrated theoretically that a hand held object could contain enough energy to destroy a city before receiving empirical confirmation—is equally impossible. In fact, the thinking required to address the plausibility of miracles with total intellectual integrity turns out to be much more subtle and demanding than for almost any scientific question—provided you don’t cross over the controversial line that divides science from metaphysics. Perhaps that is why the question of miracles is neglected or shied away from by naturalists and supernaturalists alike. And when it is addressed it usually receives very superficial treatment. Potentially, however, it remains a question of explosive intellectual force.

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