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From cold, sure. But he's talking about an urn that started full of hot coffee, and then saying energy input required to maintain that temperature will decrease as the coffee volume decreases. What's the mechanism for that supposed to be?

The more I think about it, the more I think he's just wrong about the coffee thing. Contents of the urn have no effect on energy required to maintain its temperature.

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I think OP is correct. Think about the extremes of the urn, full vs 1mL in it. It will require much less energy to heat the 1mL state than the full state.

Regarding the contents, the thermal conductivity of liquid (Water mostly for coffee) is higher than that of air, so it transfers heat out of the urn faster than an urn full of air. It also has higher heat capacity which requires more heat energy per degree change of temperature.

Heat capacity is given by (C) = heat absorbed (Q) / temperature change (ΔT); heat capacity increases with volume

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