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If the colour of flame produced by the atom of an element is the property of that element then why do unsaturated and saturated carbon compounds give different colours of flame even though both contain the element carbon and hydrogen?

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The flames that you see when you burn carbon-containing compounds have colour for a completely different reason that if you add ionic compounds (e.g. sodium chloride) to a flame. Ionic compounds won’t combust, their bonds will merely be broken and the metallic atom then has an electron that can be excited and that can re-relax under photon emission.

For carbon, we are actually talking about proper combustion, i.e. a reaction of the carbon compound with oxygen to give carbon dioxide and water. This comes in two flavours: complete and incomplete.

  • In complete combustion, all carbon atoms are fully oxidised to carbon dioxide due to sufficient oxygen amounts. We can write a nice equation e.g. for methane:

    $$\ce{CH4 + 2 O2 -> CO2 + 2 H2O}\tag{1}$$

    Complete combustion is often shown with methane as it often tends to combust completely.

  • Incomplete combustion, as the name says, is incomplete, i.e. some carbon atoms won’t make it all the way to carbon dioxide. It is impossible to give an exact equation for incomplete combustion, but one possible one could be the following:

    $$\ce{2 CH4 + 3 O2 -> CO2 + C + 4 H2O}\tag{2}$$

    Incomplete combustion is commonly shown with unsaturated compounds because they have a greater tendency to display it.

The key factor in incomplete combustion is the fact that elemental carbon (soot) is generated; it is that soot whose electronic transitions cause the yellow colour of a flame. This is not a feature of unsaturated versus saturated hydrocarbons: as the equation above shows, methane can burn both in a complete and in an incomplete manner depending on how much oxygen is added.

Likewise, the most common example for incomplete combustion, ethyne, can be combusted completely if only enough oxygen is added, see this question.

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  • $\begingroup$ Thanks Jan for catching my mistake. I completely coasted and focused on the "different colours" wording in the question and my mind went to ionic compounds. $\endgroup$ – J. Ari Mar 1 '17 at 14:14

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