Tuesday, December 08, 2015

Ask a Physicist: Gild the Moon

Dr. Madhav Pakare from Mumbai wants to know:

How much gold would it take to cover the moon in a layer one atom thick?

Gold has been renowned for its malleability since ancient times and, while we've yet to create graphene-like structures that are a single atom thick, you can come close with tools no more complex than a hammer and some parchment; it's possible to expand a gold nugget to 20,000 times its initial surface area, creating sheets less than a thousandth of a millimeter in width. Right now, scientists working on the construction of the James Webb Space Telescope are using innovative manufacturing techniques to produce ultra-thin gold coatings for the telescope's mirrors, a process that you can learn more about in NASA's "Behind the Webb" series.

So let's start with the surface area of the moon. If we treat it as a rough sphere, with an average radius of 1737.5 kilometers, the equation:
yields a surface area of ~3.8 x 10^7 km^2, or close to 38 million square kilometers. Obviously this is only a rough estimate; an incarnation of the coastline paradox is going to bedevil you if you try this in real life, but it's a good enough start.

Now, we've got to determine how much a monoatomic layer of gold would weigh, in terms of area. Rather than trying to figure out the bond length between two gold atoms and multiplying from there, it might be easiest to find out how much gold leaf weighs per area, and divide that by its thickness in number of atoms.

Some chemistry calculations from the good people in the forums at Cooking the Books reveal that a 19 gram piece of gold will hammer out into a thousand sheets of gold leaf, each 3.375" square and 0.134 microns thick. Further dimensional analysis reveals that 0.134 microns of gold is equivalent to 521 atoms (an astoundingly small number) for the thickness of the sheets. 

3.375" is 85.725 millimeters, which means each of the thousand sheets of gold leaf that comes from our 19 grams has a surface area of about 7348.775 square millimeters. To multiply that figure by a thousand, we can just move the decimal place over three spaces, yielding a total of 7,348,775 square millimeters, about 7.35 square meters. 

But since we're shooting for a layer only one atom thick, we can multiply the 7.35 square meters by a factor of 521 to determine how much square area 19 grams of gold would cover if it were stretched literally as thin as possible, finding that figure to be 3829.35 square meters.

We're now just two steps away from reaching our solution! Dividing the surface area of the moon we found earlier by the surface area covered by a monoatomic layer of gold weighing 19 grams, we find we'd need ~9.92 billion of these sheets to cover the moon. Since each one weighs 19 grams, we multiply those quantities to arrive at our final answer of:
or 189 million kilograms. While this sounds like an awful lot, it'd only be enough to fill about four olympic-sized swimming pools. Condensed into a single solid cube, it'd only be 70 feet to an edge! According to Wolfram Alpha, this is also roughly 1.3 times the amount of gold ever mined.

...someone keep an eye on Dr. Pakare. He might be up to something.

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