Acetone contains a carbonyl group -- a carbon with a double bond connecting it to an oxygen atom -- and two methyl groups connected to the carbonyl carbon. Methyl carbon is a term that refers to the carbons in these methyl groups. Alpha-halogenation refers to adding a halogen like chlorine to the first, or alpha, carbon. Generally, you need an acid or base to make this reaction take place at an appreciable rate.
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In the acid-catalysed process, the acid donates a hydrogen ion to carbonyl oxygen, making the two methyl groups more acidic. One of the methyl groups loses a hydrogen ion to a base like a water molecule, creating an intermediate called an enol. The deprotonated alpha-carbon in the enol now attacks one of the chlorine molecules, forming a bond with it and creating a chloride ion. Finally, the carbonyl oxygen donates the hydrogen connected to it back to a base in the solution, regenerating the catalyst.
In the base-promoted mechanism, the base removes a hydrogen ion from one of the two methyl groups. This is only possible because the alpha-carbons in a ketone are unusually acidic. The loss of the hydrogen ion creates an intermediate called an enolate ion. The deprotonated carbon in the enolate ion now attacks the chlorine molecule, forming a bond with one of the two chlorine atoms and creating a chloride ion in the process.
The acid-catalysed and base-promoted mechanisms proceed by different mechanisms, although they initially appear to have the same result -- a chlorine atom substituting for one of the hydrogen atoms on either of the two methyl groups. However, there is an important difference. The chlorine atom is electron-withdrawing, so it makes the carbonyl oxygen less electron-rich and hence less basic, while it also makes the alpha-carbon less electron-rich and hence more acidic.
In the acid-catalysed mechanism, once either of the two methyls has been halogenated, the carbonyl oxygen is a poorer base and thus less likely to accept a hydrogen ion again, so the acetone will typically only be halogenated once in an acidic solution. In the base-mediated mechanism, by contrast, the more halogens that are attached to them, the more acidic the alpha-carbons become, so the acetone will typically be chlorinated multiple times. With acids, you will generally have only one chlorine-for-hydrogen substitution. With bases, multiple hydrogen ions on the alpha carbon will be exchanged for chlorines, leaving behind starting material that has not reacted.
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