Introduction:
In order to determine the relative stability of allyl, benzyl, and tert-butyl free radicals, we need to consider their structural characteristics and the factors that influence radical stability. Stability is primarily governed by the extent of delocalization of the unpaired electron in the radical, which can occur through resonance or hyperconjugation.

Allyl Free Radical:
The allyl free radical is formed by the removal of a hydrogen atom from one of the carbon atoms in the allyl group (-CH2-CH=CH2). It possesses a delocalized π-electron system due to resonance. The unpaired electron is delocalized over all three carbon atoms, resulting in stabilization. This resonance stabilization contributes significantly to the stability of the allyl free radical.

Benzyl Free Radical:
The benzyl free radical is formed by the removal of a hydrogen atom from the benzyl group (-CH2-C6H5). It also possesses a delocalized π-electron system due to resonance. The unpaired electron is delocalized over the benzene ring, providing stabilization. However, the extent of resonance in the benzyl radical is slightly less compared to the allyl radical due to the presence of an additional carbon atom, which reduces the contribution of resonance to stability.

Tert-Butyl Free Radical:
The tert-butyl free radical is formed by the removal of a hydrogen atom from one of the methyl groups in tert-butyl (CH3)3C-). Unlike the allyl and benzyl radicals, the tert-butyl radical does not possess a delocalized π-electron system. However, it exhibits hyperconjugation, which involves the overlap of a filled σ-orbital with an adjacent empty p-orbital or π-orbital. The unpaired electron in the tert-butyl radical can be stabilized by hyperconjugation with the three adjacent methyl groups, leading to increased stability.

Relative Stability:
Based on the above information, we can rank the radicals in terms of stability as follows:

1. Tert-butyl free radical (most stable): The tert-butyl radical exhibits hyperconjugation, which provides significant stabilization.
2. Allyl free radical: Resonance delocalization of the unpaired electron contributes to its stability.
3. Benzyl free radical (least stable): Although it also exhibits resonance, the additional carbon reduces the extent of resonance and hence, the stability.

Conclusion:
The tert-butyl free radical is the most stable among the allyl, benzyl, and tert-butyl free radicals due to the presence of hyperconjugation. The allyl radical is relatively stable due to resonance, while the benzyl radical is the least stable due to reduced resonance caused by an extra carbon atom.

Order of stability of radicals :Benzyl radical>Allyl radical> tertiary radical > secondary radical>primary radical >methyl radical>vinyl radical.