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I cannot find a definitive equation that describes the incomplete combustion of a hydrocarbon. Equations $1$, $2$ and $3$ are some of the variations I have found.

Is it,

$(1)$ $\ce{C_xH_y + O_2 \rightarrow CO + H_2O}$,

$(2)$ $\ce{C_xH_y + O_2 \rightarrow CO2 + CO + H_2O}$ or,

$(2)$ $\ce{C_xH_y + O_2 \rightarrow CO2 + CO + C + H_2O}$?

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There is no definitive equation for an incomplete combustion. Sometimes incomplete combustion of hydrocarbons may even form species such as polyaromatic compounds and other hydrocarbons. The result may differ by variation of factors such as temperature, oxygen concentration, and the hydrocarbon undergoing combustion. Basically what you're asking is like asking "is there a definitive number between 0 and 1?" You may already know that numbers between 0 and 1 is an infinite set. Incomplete combustion may not be an infinite set, because nobody proved that it is, but it's still long ways off from a definitive one equation solve it all thing.

There is only a definitive equation for complete combustion; like there is only two numbers within a set of {0, 1}.

It's easier to recognize that any combustion that is not a complete combustion as a incomplete combustion.

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