Question

how can hyperconjucation effect and inductive effect explain the stability of primary , secondary and tertiary free radicals.

Answer 1

Hyperconjugation is the interaction of the electrons in a sigma bond of an alpha C-H bond with an adjacent empty (or partially filled) non-bonding p-orbital, antibonding π orbital, or filled π orbital, to increases the stability of the system.
Stability of carbon radical:
The stability of the carbon radical increases with the increase in th number of alkyl groups attached to the carbon carrying the odd electron by hyperconjugation.

Inductive effect is polarisation of sigma bond due to electron withdrawing or electron donating effect of adjacent groups or atoms.
The stability of radicals increases with increase in number of alkyl groups due to their +I effect. The alkyl groups releases electrons to the carbon carrying odd electron and thus stabilise the free radical.

Thus, the increase in stability of free radical is given as:
primary < secondary < tertiary