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In my understanding, by bonding to the free radical species, say by one of the double bonds, a new free radical on the $\beta$-carotene develops.

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$\beta$-Carotene is a highly delocalised $\pi$ system. According to George Britton there are numerous mechanisms to achieve this.

  • Oxidation
    Oxidising radicals can remove an electron from the carotene molecule. $$\ce{CAR + .R+ -> .CAR+ + :\!R}$$

  • Reduction
    Analogous it can accept an additional electron. $$\ce{CAR + .R- -> .CAR- + R}$$

  • Hydrogen Abstraction
    It is possible to abstract a hydrogen atom from a saturated carbon in allylic position. $$\ce{H-CAR + .R -> .CAR + HR}$$

  • Addition
    You are correct in assuming, that a free radical may form a bond with that system. This is most common for hydroxyl radicals. $$\ce{CAR + .OH -> .CAR-OH}$$

You are also correct in assuming, that another radical will always be created. However, the single electron will be very delocalised (resonance stabilised) throughout the whole system and not nearly as reactive as the free radical before.

The remaining molecule can then of course attract more free radicals to form other singlet species, which the body can digest or excrete.

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    $\begingroup$ Good point, there most likely are more than just one radical species. $\endgroup$ – TMOTTM Jun 24 '14 at 10:03
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The extended, conjugated series of alternating double-bonds in $\beta$-carotene comprises a system of highly delocalized $\pi$ electrons. If a radical forms by hydrogen abstraction at one of the two terminal allylic positions, that lone electron is delocalized over the entire $\pi$ electron system, and thereby stabilized. Similarly, radical addition across any of the double bonds will result in a highly stabilized radical (though addition to one of the internal double bonds will actually disrupt the conjugation, but is nevertheless quite possibile for reasons of sterics and the greater lability of $\pi$-bonds generally). The stability of the resulting radical drastically lowers the energy of activation for the reaction, hence making $\beta$-carotene so liable to react. In essence, its antioxidant activity is (at least in large measure) a direct consequence of the ease with which it is itself sacrificially oxidized. The same factors are responsible for, e.g., the propensity of unsaturated fats to spoil more readily than saturated ones.

That said, as is typical of radical reactions, the resulting $\beta$-carotene radical, while reasonably stable, can still go on to propagate the radical chain reaction. There are various other antioxidants which form highly stable end products after initial radical formation, and thereby terminate the chain reaction.

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