Need to start with how candles work: The wax is the fuel that burns. The heat from the flame melts some wax, the liquid wax gets sucked up the wick string like how water sucks up into a paper towel, and then up there right beside the flame, the liquid wax can get hot enough that it evaporates. *It’s this vaporized/evaporated wax rising up off the wick that’s burning to make the candle flame.* That flame melts more wax, it gets sucked up and evaporated, and the process continues.
Now for your question: The “smoke” from a recently blown out candle isn’t just smoke, it’s unburnt wax vapor. That’s why it looks different from the smoke that comes off the same candle when its lit and burning away. When it’s lit there shouldn’t be much smoke at all, and the wax vapor is what’s burning. When you blow out the candle, for a minute or so the wick is still hot enough to be evaporating the wax that gets sucked up into it. But there’s no flame, so there’s just a trail of flammable wax vapor rising up into the air. You can light that and the flame follows the fuel path back to the wick and starts the normal self-sustaining operation from there.
When something burns, particles from the original substance react with the air when heated to produce more heat, light, and also smoke as biproducts. When you blow out a candle, that reaction is no longer taking place. Residual smoke rises from the wick because the wick is still losing particles as well as wax, but they have not reacted with oxygen to produce light. You can heat those up with a lighter, and the reaction continues like a chain down to the original wick
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