In general, if you were to try make something hard in a liquid medium (especially water) it is quite difficult to make the material solid.

Things such as mollusks though, have no problem generating hard shells in a wet medium.

What exactly are the physics behind this formation?

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    $\begingroup$ You know cement hardens under water. And your body is a pretty wet place but your bones are still solid. So molluscs aren't particularly unique. $\endgroup$ – Chris White Dec 7 '14 at 3:07
  • $\begingroup$ Should the question be rephrased to what is the physics behind making hard substances under water? $\endgroup$ – gansub Dec 7 '14 at 3:13
  • $\begingroup$ I think this would be a relevant video from Minute Earth. $\endgroup$ – fibonatic Dec 7 '14 at 3:32
  • $\begingroup$ Have you noticed lime deposits? hard stuff sticking to the bath , the kettle etc? They come out of the water. It is called precipitation of dissolved elements, and they form a lattice given a seed crystal and the proper temperatur to do so. This is a chemistry question really. Cells in sea organisms utilize the chemical possibilities of building lattices out of available solvents or food elements. $\endgroup$ – anna v Dec 7 '14 at 7:56

Carbon dioxide from the atmosphere (or from decaying matter in the ocean) reacts with water to form carbonic acid:

$$ \ce{ CO2 + H2O -> H_2CO_3 }$$

and this reacts with calcium ions to form calcium carbonate:

$$ \ce{H2CO3 + Ca^{2+} -> CaCO3 + 2H+ }$$

The solubility of calcium carbonate is about $13~\mathrm{mg \over L}$, so if the concentration of calcium carbonate is greater than this the excess will precipitate out as solid calcium carbonate.

Shell-forming organisms actively absorb calcium from the water around them, so they are able to increase the concentration of calcium carbonate to above $13~\mathrm{mg\over L}$ and precipitate the excess to form their shells. They can get the carbon dioxide from the water around them or from their own metabolism. The actual details of shell formation is far more complex than this as it's controlled by processes with the cells of the organism rather than just being uncontrolled precipitation. I'm not sure how well the details are understood even today.

Incidentally, this is why shelly fauna are not keen on the acidification of the oceans that results from increased atmospheric carbon dioxide. The solubility of calcium carbonate is strongly $\mathrm{pH}$-dependent and rises sharply as the water gets more acid. Given that the shell is usually intended to stop other animals eating you, having your shell dissolve is generally not good for life expectancy.

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