The reactivity of primary, secondary, and tertiary hydrogen atoms towards bromination depends on the stability of the free radical intermediate formed during the reaction.

Understanding Radical Bromination

Bromination of alkanes is typically a free radical substitution reaction. This reaction involves the abstraction of a hydrogen atom from the alkane by a bromine radical, forming an alkyl radical and HBr. The stability of the resulting alkyl radical significantly affects the activation energy of this step and thus the rate of reaction.

• A primary hydrogen is bonded to a carbon atom that is bonded to only one other carbon atom (e.g., the hydrogen in RCH$_3$). Abstraction forms a primary radical (RCH$_2\bullet$).
• A secondary hydrogen is bonded to a carbon atom that is bonded to two other carbon atoms (e.g., the hydrogens in R$_2$CH$_2$). Abstraction forms a secondary radical (R$_2$CH$\bullet$).
• A tertiary hydrogen is bonded to a carbon atom that is bonded to three other carbon atoms (e.g., the hydrogen in R$_3$CH). Abstraction forms a tertiary radical (R$_3$C$\bullet$).

Reactivity and Radical Stability

The stability of alkyl free radicals increases in the order: primary < secondary < tertiary.

The order of stability is:

Tertiary Radical (>CH$_3$)$_3$C$\bullet$ > Secondary Radical (>CH$_3$)$_2$CH$\bullet$ > Primary Radical >CH$_3$CH$_2\bullet$

According to the Hammond postulate, the transition state for the hydrogen abstraction step resembles the product (the alkyl radical). A more stable radical corresponds to a lower energy transition state, which means a lower activation energy and a faster reaction rate.

Therefore, the ease of abstracting a hydrogen atom is greatest for tertiary hydrogens, followed by secondary, and then primary hydrogens.

This leads to the reactivity order:

Tertiary Hydrogen > Secondary Hydrogen > Primary Hydrogen

In terms of the molecules provided in the options:

• RCH$_3$ has primary hydrogens.
• R$_2$CH$_2$ has secondary hydrogens.
• R$_3$CH has a tertiary hydrogen.

Thus, the reactivity order for bromination is R$_3$CH > R$_2$CH$_2$ > RCH$_3$.