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I got in my car this morning and the temperature read 13 F (that's around -10.5 C.) While driving down the road, I noticed a bottle of water in my cup holder was frozen solid as expected.

Another bottle of water (cheap, thin-walled, plastic disposable water bottle) contained liquid water! I shook the bottle, not a single crystal of ice!

Shortly thereafter, I noticed the water became slushy, so it had begun freezing quite quickly!

The interior of the bottle was hardly perfectly smooth given the quality of the bottle and the water was half drunk so it was probably not super pure.

What caused this? Was this supercooling?

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You may have just experienced supercooling.

Ordinary crystallization upon reaching the freezing point of a liquid requires nucleation sites - places for crystals to form. In absence of nucleation sites, water can be cooled down to lower temperatures until crystal homogeneous nucleation occurs. In water, this can occur as low as $224.8\ \mathrm{K\ (-48.3\ ^\circ C,\ -55\ ^\circ F)}$.

Nucleation can also be induced by disrupting the system, which happened when you shook the bottle. Even so, the fact that it only became slush may attest to the presence of impurities. A $\mathrm{19\ ^\circ F,\ 10\ ^\circ C}$ freezing point depression is not unreasonable depending on the amount of saliva that might have ended up back in the bottle.

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