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Aromatic aldehydes do not give Fehling's test, which makes me believe that for the same reason acrolein should not. It has a double bond conjugated with the carbon-oxygen double bond as in benzaldehyde. Is my assumption correct? Or is extended conjugation a factor?
Glyoxal(CHO-CHO) and Glyoxalic acid(CHO-COOH) also don't give Fehling's test. Any reasons for that?
It has nothing to do with aromaticity. It is related to this answer.
Why does benzaldehyde not respond to Fehling's test?
Any aldehyde without an alpha hydrogen will not give Fehling's test. Acrolein also doesn't have an alpha hydrogen, so it will not give Fehling's test.
Please refer, Article by Ralph daniels (JCE, Vol.37, No.4, 1960 link: https://pubs.acs.org/doi/abs/10.1021/ed037p205). According to this paper, even acetaldehyde is not oxidised by Fehlings solution. They claim these are errors in the text books are carried over for many years.
In this journal, the authors observations are: Category-1 :Positive fehling test: Glucose and other reducing sugars,glyoxal & pyruvic aldehyde (both of them does not have alpha H).
Category-II: Acetaldehyde, crotnaldehyde, cinnamaldehyde, these aldehydes show a colour change, but they have not observed the presence of Cu+1 in the reaction, (the aldehyde group is not oxidised,hence it is false positive test.)
Negative fehling test: Benzaldehyde other aromatic aldehydes, isobutraldehyde.
My best guess would be that the electron withdrawing groups attached to the aldehydes increase the reduction potential of the half reaction (make it harder to oxidize) since the electrons are more delocalized in the molecule. In acrolein, the inductive effect of the vinyl group isn't too big, plus there is an electron donating effect through resonance, so I'd expect it to be reactive to Fehling's reagent.
