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An object painted black will heat-up faster than the same object painted white. Is this solely due to the darker paint absorbing more wavelengths of light and thereby converting more light to heat in the darker object, or does painting a solid actually alter its heat capacity (even if by a tiny amount)?

What if instead of paint on its surface the entire object was black versus white (e.g., candles)? If lattice vibrations largely determine heat capacity of solids, does this mean mixing in "black paint" throughout the entire solid alters the physical lattice structure of the atoms in a black candle compared to a white candle? Or does an object's color have nothing to do with its heat capacity?

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The heat capacity depends on the material not the color or the source of the heat. Adding a significant amount of a black material could change the heat capacity but due to the change in the chemistry of the material not because it is black

Both the bulk material (eg water has a high heat capacity so adding water to almost anything is likely to increase the heat capacity) or the nature of the chemical bonds if the addition makes a new compound

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