A pint of ocean. Let’s say you poured a pint of water into the ocean, then stirred the ocean thoroughly, or waited long enough for the ocean currents to mix the molecules of water of your pint with those of the ocean. If you travelled to the opposite side of the Earth and scooped up another pint from the ocean, how many molecules from the original glass of water would you now have? None? One? Well, the answer is actually closer to eight thousand—provided that the world’s oceans were thoroughly mixed, which doesn’t sound very physically realistic. What many may found baffling, though, is that there exist far more molecules of water in one pint glass than pints of (salty) water in the world’s oceans. For those of you not put off by a few lines of maths, the relevant back-of-the-envelope estimate follows.
The mass in grams of one mole of water, that is, the volume of water containing 6.022×1023 molecules (the so-called Avogadro constant), is: 2 × atomic weight of hydrogen + atomic weight of oxygen, since each molecule contains two hydrogen atoms (H) for each oxygen (O).
An imperial pint contains 1000 grams/1.760 = 568 grams of water. Therefore, the number of moles of water in a pint is: mass of a pint of water/mass of one mole of water = 31.543 moles, from which we derive the number of water molecules in a pint glass: moles in a pint × Avogadro constant = 1.90×1025.
The mass of water contained in the oceans is approximately 1.3×1024 grams, corresponding to a number of molecules of water equal to: mass of the oceans/mass of one mole of water × Avogadro constant.
Let us assume that the dissolved salt is sodium chloride (NaCl), our regular table salt, and present in ionized form (the salt doesn’t significantly change the final result). One mole of NaCl weighs: (atomic weight of sodium + atomic weight of chlorine)/2. The ocean has 3.5% average salinity, from which, by remembering that 1 cm3 contains 1 g of water:
number of moles of salt per cm3 = salinity/mass of one mole of NaCl,
number of moles of water per cm3 = 1/mass of one mole of water.
The number of molecules in the oceans is then given by:
molecules of water (1 + moles of salt per cm3/moles of water per cm3) = 4.44×1046. (Had we neglected the salt, the result would have been 4.35×1046—the difference between the two numbers is neglectable.)
The number of pints of water in the ocean is given by: mass of the ocean/mass of one pint of water = 2.29×1021, a number much smaller than that of water molecules that fill a pint glass.
For each molecule of the original pint, there are 2.34×1021 ocean molecules, from which:
molecules of the original pint, for each pint in the ocean = 8100.
