The arithmetic is one of the oldest and most reliable demonstrations in physics, and the conclusion is one of the harder facts about the size of atoms to actually believe. A standard 250-millilitre glass of water contains roughly 25 trillion trillion atoms, written as 2.5 × 10²⁵, accounting for the three atoms in each water molecule. The world’s oceans, according to NOAA’s National Ocean Service, contain approximately 1.335 billion cubic kilometres of water. Convert that into 250-millilitre glasses and the answer comes out to about 5.3 × 10²¹ glasses of water in all the oceans on Earth combined. The number of atoms in your single glass exceeds the number of glasses in all the oceans by a factor of roughly 4,700.
The implication is the second half of the demonstration. Pour the glass into the sea, wait for the oceans to mix completely, and scoop another glass anywhere on Earth: that glass will contain several thousand of the original atoms. Even after a single glass has been dispersed throughout every ocean basin on the planet, the atoms involved are so numerous that they cannot be diluted out of detection. Some fraction of the water you drink today is, mathematically, water that has passed through every ocean on Earth at least once.
The brainteaser is older than people think
The thought experiment is generally attributed to William Thomson, the first Baron Kelvin, the nineteenth-century British physicist whose name attached to the absolute temperature scale and whose work on the transatlantic telegraph cable made him a household figure in late-Victorian science. Erwin Schrödinger, in his 1944 book What is Life?, attributed the demonstration directly to Kelvin and reproduced its essential form: “Suppose that you could mark the molecules in a glass of water; then pour the contents of the glass into the ocean and stir the latter thoroughly so as to distribute the marked molecules uniformly throughout the seven seas; if then you took a glass of water anywhere out of the ocean, you would find in it about a hundred of your marked molecules.” The geophysicist Matt Hall, writing for Agile Scientific in 2011, has traced the modern statement of the problem back through Schrödinger’s quotation and noted that Kelvin’s “about a hundred” is approximately correct for the glass-and-ocean dimensions Kelvin would have had in mind.
Kelvin’s “about a hundred” was based on a smaller glass and less accurate ocean-volume estimates than the modern figures. The current arithmetic, using a 250-millilitre glass and modern ocean-volume measurements, gives a figure of about 1,500 marked molecules per glass after complete mixing, which corresponds to several thousand marked atoms. The order of magnitude has not changed dramatically in the century and a half since Kelvin first proposed the problem. What has changed is the precision available for the answer.
Why this works
The reason the demonstration works is the staggering disparity between the size of an atom and the size of any object made of atoms. A water molecule is approximately 0.3 nanometres across, or about three ten-billionths of a metre. In a single 250-millilitre glass of water, the number of molecules is on the order of 10²⁵, which is roughly forty times the estimated number of stars in the entire observable universe. Each of those molecules is doing the same kind of physics as every other water molecule on the planet. Each one is a candidate to be marked, dispersed, and counted again.
The world’s oceans, by contrast, contain only a finite number of any particular unit of volume. The most recent satellite-based measurement of total ocean volume, made by Matthew Charette of Woods Hole Oceanographic Institution and Walter Smith of NOAA in 2010, gave a value of 1.332 billion cubic kilometres, close to but slightly under the long-standing NOAA figure of 1.335 billion. Either way, the oceans hold a vast volume of water and a small number of glasses’ worth, compared with the count of atoms in any reasonably sized container of liquid. The ratio between the two figures is what produces the surprising result. Atoms are not just smaller than everyday objects. They are smaller in a way that even physicists find hard to keep intuitive.
What Schrödinger was using it to argue
Schrödinger’s use of the Kelvin brainteaser in What is Life? was not just a curiosity. He was building toward a serious biological argument: that the orderly behaviour of living organisms requires biological molecules to be made of enormous numbers of atoms. If life were built out of just a few atoms per molecule, the statistical fluctuations from random thermal motion would overwhelm any orderly behaviour. The fact that biological molecules are, in practice, made of millions or billions of atoms each is what allows them to behave reliably on average. Schrödinger’s point was that atoms have to be small relative to organisms, otherwise organisms could not work.
The marked-glass demonstration is the same principle reversed. A glass of water can be marked at the atomic level and still be detectable after dilution into the largest body of liquid on Earth, because the number of atoms in the glass is so much larger than the number of glass-sized volumes available to receive the dilution. Both ends of the comparison are doing the same kind of work. The atomic scale is unintuitive in both directions.
What you actually drink
One of the more entertaining applications of the demonstration is to extend it through time. The water on Earth is roughly 4.5 billion years old. The total amount of it has been approximately constant for most of that history, with small additions from cometary impacts and small losses from atmospheric escape balancing out over geological timescales. Water that fell as rain on a Mesozoic forest is still in circulation. Water that was drunk by ancient humans is still in circulation. The atoms in any glass have been through the oceans, through the atmosphere, through clouds, glaciers, rivers, and through the bodies of an enormous number of living things, on the way to the glass.
The marked-glass thought experiment is, in a small way, an inversion of this. The atoms that pass through your body today are statistically connected to the atoms that have passed through every other body and every other ocean. The reason the math works is the same in either direction. There are far more atoms in any container of water than there are containers of water available in the world to dilute them into. Even on a planetary scale, atoms are the larger number.
