The high water content helps a lot. Water has a pretty high “specific heat capacity” which means it has to lose a lot of heat energy to cool down. Its “big oval lump” shape also gives it a very small surface area relative to its large volume.
But also, potato is a *not that great* conductor of heat. Meaning that even if the outside of the potato is cooled by the air around it (or the hand holding it), it could still be very hot just a cm or two below the surface; it takes a long time for heat to travel outward from the inside, to replace the heat of the cooled outer layer.
Most important of all, a whole potato does not provide a lot of opportunity for the hottest watery parts in the middle, to contact any air. This means there isn’t as much chance for evaporative cooling to happen. Mashed potato lets off a lot more steam, and cools down faster, than whole baked potato.
French fry has a lot of surface area relative to its volume, so it loses heat *very* fast.
It’s a mix of high density, high water content, and high starch content. Starch in particular influences it more than other veggies; cellulose breaks down more easily than starch (which is why other water dense veggies like carrots or beets get mushy as they cook), which in turn allows water/steam to escepe the plant more easily.
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