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I know that complete combustion is a process where a fuel reacts with $\ce{O2}$ to become $\ce{CO2}$ and $\ce{H2O}$. But what is incomplete combustion?

According to the Internet, $\ce{CH4 + O2}$ and $\ce{C6H12O6 + O2}$ are examples of incomplete combustion. What makes them so?

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Complete combustion is, as the name suggests, complete. That on its own doesn’t mean much, but I can tell you that the completeness implies all atoms that were put into combustion leave it fully oxidised (or fully reduced in the case of oxygen, fluorine, chlorine; or unchanged in the case of nitrogen).

An incomplete combustion is one where molecules remain that may still be oxidised, i.e. can react with further oxygen. Examples would be carbon monoxide or elemental carbon (colloquially ‘soot’). Incomplete combustion often results in a yellow flame stemming from carbon particles, while complete combustion typically yields hotter, colourless flames.

In your average Bunsen burner, you can choose between complete and incomplete combustion by regulating the amount of air (meaning: oxygen) that reaches the flame. Complete combustion would follow the equation $\ce{CH4 + 2 O2 -> CO2 + 2 H2O}$. There is no unique equation for incomplete combustion, but one possibility would be $\ce{CH4 + O2 -> C + 2 H2O}$.

By the way, the way living beings burn carbohydrates in cellular respiration is, by definition, a complete combustion since every carbon atom ends up in $\ce{CO2}$ and a set of two hydrogen atoms always ends up as $\ce{H2O}$. The overall equation is $\ce{C6H12O2 + 6O2 -> 6CO2 + 6H2O}$. If glucose is combusted with less than 6 equivalents of $\ce{O2}$, then incomplete combustion results.

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