If I hammered and flattened a penny enough, could I cover the entire earth with it?
Category: Physics Published: June 15, 2015
No. If you spread out the atoms from a single penny over the entire surface of the earth, you would no longer have a single piece of solid material since the atoms would be too far apart to bond to each other. Let's do some careful calculations to show this result. A modern United States penny has a mass of 2.500 grams according to the US Mint. Since a penny is composed of 97.50% zinc and 2.50% copper, it therefore contains 2.4375 grams of zinc and 0.0625 grams of copper. At a molar mass of 65.380 grams per mole for zinc and 63.546 grams per mole for copper, a penny therefore contains 0.037282 moles of zinc and 0.00098354 moles of copper. Since a mole of atoms contains 6.0221 × 1023 atoms, there are 2.2452 × 1022 zinc atoms and 5.9230 × 1020 copper atoms in a penny, for a total of 2.3044 × 1022 atoms in a penny.
The earth has a surface area of 510,072,000 square kilometers, or 5.10072 × 1032 square nanometers. The surface area of the earth really depends on what you include in your definition of surface. For instance, if we wish to cover the area of every leaf on every tree and shrub with atoms from the penny, then this will change our answer. Surprisingly, it will not change our answer very much. Most of the earth is covered in relativity flat oceans, sandy deserts, snow fields, barren rocks, and meadows. Trees, shrubs, buildings, and other irregularly-shaped objects only cover a very small percentage of the earth (trees and buildings seem common to most of us humans because most of us live near crowded concentrations of trees and/or buildings). At any rate, we must pick some definition of earth's surface area to make any calculations. The number cited above does not include the surface area of tree leaves and other small irregularities. In the context of trying to cover the earth with a flattened penny, you can think of this definition of surface area as us lowering a sheet of zinc so that it drapes along the tops of the trees, but does not wrap around any of the leaves or branches of the trees. The thinnest we could ever hammer a sheet of material is one atom thick. We therefore assume that we are creating a one-atom-thick planar sheet of material. Using the above value for earth's surface area, we divide it by the number of atoms in a penny to find how much area each atom will occupy when the atoms are spread evenly across earth's surface. We get the value of 2.21347 × 1010 square nanometers per atom, or 0.0000343 square inches per atom. This may seem like a small area, but it is huge compared to the types of areas spanned by simple molecules.
From here on out we will assume that all of the atoms in the penny are zinc atoms. This is a good assumption because almost all of the atoms in the penny are zinc atoms (97.5%). Also, in terms of atomic size and bonding distance, zinc and copper are nearly identical. When allowed to bond into a solid piece of material, zinc atoms arrange themselves into stacks of planar hexagonal grids. Therefore, in creating our one-atom thick sheet of zinc, we will arrange our zinc atoms along a planar hexagonal grid. If the penny's atoms are spread out uniformly on a hexagonal grid covering earth's surface, then each atom will have to be 159,870 nanometers away from its next nearest atoms in order to cover the entire earth (for a hexagonal grid of objects, the distance between nearest-neighbor objects is 1.0746 times the square root of the area that each object has to itself). In other words, taking the atoms of a single penny and spreading them out over the entire earth in a hexagonal arrangement will cause each atom to be about 0.16 millimeters away from its neighboring atoms. In order to form a solid chunk of material, atoms have to be close enough to form stable bonds. In regular pieces of zinc metal, stable chemical bonds are formed when each zinc atom is a distance of 0.26649 nanometers away from its next nearest zinc atoms. Therefore, the atoms of our smashed penny will be almost exactly 600,000 times too far apart to maintain stable bonds and constitute a solid piece of metal. With this kind of separation, we don't have a solid penny at all. We have a very dilute zinc gas spread over the earth. These widely separated atoms would blow around, dissolve into the ocean, mix with the clouds, and react with other atoms, so that we no longer have any type of distinct object that we would say is covering the earth. For this reason, you cannot hammer and flatten a penny until it covers the entire. The earth is simply too big and a penny has simply too few atoms to accomplish this task. Even if we break each zinc atom into 30 hydrogen atoms (ignoring all the messy details of the nuclear reactions involved), the atoms are still about a hundred thousand times too far apart to form stable chemical bonds. Besides, hydrogen atoms don't bond to form a solid chunk of material under normal conditions.
These thoughts lead to another question: How much area can a smashed penny cover and still remain a solid chunk of material? To calculate this, we again realize that the thinnest a material can get is one atom thick. Also, the distance that zinc atoms need to be from each other and still stay bonded as a solid is 0.26649 nanometers, as already mentioned. This means that as part of a hexagonal planar arrangement of atoms, each zinc atom needs to occupy 0.0615 square nanometers of area. Multiply this number by the 2.3044 × 1022 atoms in a penny and we get a total area of 1417 square meters, which is about a quarter of the size of an American football field. In other words, if you had a special machine that carefully hammered a penny until it was everywhere just an atom thick, it would only cover a quarter of a football field. Keep in mind that at this thickness, you would be hard pressed to even see the penny and you would rip the penny when walking on it without feeling any resistance (think of walking on aluminum foil, but much, much thinner). To cover the entire earth's surface with one-atom-thick smashed pennies, you would need at least 360 billion pennies.