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I saw an ad on tv last night for Mitchum deodorant that showed big text on the screen that said "OXYGEN eliminates all odors!" or something similar. Their website indeed claims "Our exclusive formula releases pure oxygen, a powerful odor fighter...".

How does oxygen fight odors, and how can this aspect of the deodorant be more effective than the abundance of pure O$_2$ in the air?

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In my estimation, this is basically marketing nonsense. If you examine the list of ingredients, the primary active agent is invariably some aluminum or aluminum-zirconium salt, which act as anti-perspirants primarily by the mechanical actions of shrinking pores (due to their astringency) and clogging ducts in the apocrine glands.

Listed among the inactive ingredients is hydrogen peroxide, $\ce{H2O2}$. At sufficient concentrations, $\ce{H2O2}$ can be a potent oxidizing agent, and probably the source of the oxygen the ad refers to. It might indeed be able to neutralize some odors by oxidizing the molecules which cause them, thereby transforming those molecules into odorless ones. However, as it's listed among the inactive ingredients, it's almost certainly present in very low concentration. This is to be expected, since prolonged skin contact will invariably cause chemical burns unless the mixture is very dilute. Furthermore, much of the odor of sweat is the result of metabolic byproducts from skin-dwelling bacteria, chief among them certain short-chain carboxylic acids that are already maximally oxidized. The only malodorous substances in sweat that can be neutralized by oxidation, as far as I can tell, are certain amines and thiols. That said, I doubt the concentration of $\ce{H2O2}$ is sufficient to be of any use here. $\ce{H2O2}$ does also have anti-septic properties, so perhaps its purpose is bacteriostatic, but there I'm just speculating. Ultimately, I'm uncertain of its purpose in the formulation, but I'm extremely dubious that it helps substantially in eliminating odor.

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