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If I have 2 pots of water of the identical size and identical volume of water contained within. If I boil both pots of water to the same temp. For example 100 degrees C. If I boil one pot using a lower flame and the other pot using a higher flame. Then I turn off the flames. Will the hot water contained in both pots cool down at the same rate or will one retain its heat longer than the other?

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Will the hot water contained in both pots cool down at the same rate

Yes

or will one retain its heat longer than the other?

No

The cool down is really just a function of how much thermal energy is in water/pots and what the thermal resistance to the ambient temperatures. As long as that's the same for both pots they will cool down at the same rate.

The pot with the lower flame will take more time to get to boiling, but once both pots are boiling, they are at the same temperature and that's all that matters.

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    $\begingroup$ Correct, and here is an added thought that might be worth mentioning: if the two pots are the only things around, then there is nothing more to say. But cooling is also dependent on the temperature of the surroundings. If in the two cases the surroundings were at different temperatures then the cooling times would be different. For example, heating the pot more slowly would be liable to make the surroundings a bit warmer too. (But it may be that the question was not intended to bring in complications like that). $\endgroup$ Oct 12, 2021 at 13:07

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