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I have a cup and I can only pour hot water inside, I wanna know whether the heat will dissipate more quickly with more water or less water? How about the occasion when my cup is well covered?

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The amount of thermal energy in the water is proportional to the volume of water (which is proportional to its size cubed: $r^3$). The rate at which heat can be dissipated (the rate it cools) is proportional to the surface area of the water (which is proportional to its size squared: $r^2$). Thus, the more water you have, the lower the ratio of surface-area to volume ($A / V \propto r^2 / r^3 = 1/r )$, and the longer it will take to cool.

Aside: this is why larger mammals have more trouble cooling themselves than smaller animals. This is why elephants have giant ears: they drastically increase the surface area1 to increase the rate of heat dissipation.

If you cover the cup, it increases its insulation --- which is its resistance to exchanging heat with the environment. More insulation means slower heat transfer, and in this case, slower cooling.


1: The ears also have excellent blood-flow, which keeps transferring heat from the body to the ears---just like the radiator in a car.

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This is very helpful!! Thanks!! –  Junyan Mar 14 '13 at 11:26
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You could also put black food-coloring in the hot water to make it radiate faster, because of course black bodies radiate faster. It's the same reason black coffee cools down faster than coffee which has added milk.

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That's not how it works –  zhermes Mar 14 '13 at 4:28
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Uh, yes it is... –  elfmotat Mar 14 '13 at 6:00
    
A 'black' body, does not refer to its color per se, but its spectrum. For example, the sun (clearly not black) is very close to a black-body. See, e.g., this link –  zhermes Mar 14 '13 at 6:14
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@zhermes or to put it another way, you're right in that the "black" in "black body" usually refers mostly to the spectral properties in the infra-red range - but most pigments that are black in the visual range are also black in the IR, so in practice painting things black does increase their rate of thermal radiation emission or absorption. –  Nathaniel Mar 14 '13 at 8:07
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I am sorry, but I think black color absorb the heat, so my understanding in the occasion of cup and water, if the cup is black, then the liquid tends to cool down faster, while if the liquid is black(black coffee for instance), it takes more time to cool down than with milk added. p.S.. I am really really an idiot in physics, and thank you guys for all the eyes-opening information! –  Junyan Mar 14 '13 at 11:12
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The question could perhaps benefit from some clarification. The more water that is in the cup, the larger the surface that will be transferring heat from the water to the cup and to the surroundings, and more heat will be dissipated to the surroundings.

Covering the cup will do two things: 1) the most important is that it will reduce the heat loss due to evaporation, as the steam will condense inside the cup keeping its heat there rather than letting escape, 2) reduce heat loss due to radiation and convection by insulation.

A more interesting question might be whether the temperature of the water drops more quickly in the larger or smaller volume. This question is answered above in the comparison of surface area and volume.

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