Animated Solution for Chemistry - Organic Chemistry: The increasing order of stability of the following free radicals is

Resonance vs. Hyperconjugation in Free Radicals

Imagine a free radical as a highly energetic, slightly desperate molecule. It has an unpaired electron, meaning its octet is incomplete. Because it is electron-deficient, it constantly seeks ways to stabilize itself by pulling in electron density from its surroundings. In organic chemistry, the two primary superheroes that come to the rescue of free radicals are Resonance and Hyperconjugation.

In this problem, we are tasked with arranging four different free radicals in increasing order of their stability. To do this, we must evaluate the structural features of each radical and determine which stabilizing forces are at play.

The Power of Hyperconjugation

Let's first examine the two alkyl radicals: the isopropyl radical, $({\text{CH}}_3{)}_2\dot{\text{C}}\text{H}$, and the tert-butyl radical, $({\text{CH}}_3{)}_3\dot{\text{C}}$.

Notice that neither of these molecules contains any $\pi$ bonds or aromatic rings. Therefore, resonance is completely absent. Instead, these radicals rely entirely on hyperconjugation—the delocalization of electrons from adjacent $\text{C}-\text{H}$ $\sigma$ bonds into the half-empty $p$-orbital of the radical carbon.

The rule for hyperconjugation is simple: the more $\alpha$-hydrogens (hydrogens attached to the carbon directly adjacent to the radical center), the more hyperconjugative structures can be drawn, and the greater the stability.

- The isopropyl radical has two methyl groups attached to the radical center, giving it $3+3=6$ $\alpha$-hydrogens. - The tert-butyl radical has three methyl groups attached, giving it $3+3+3=9$ $\alpha$-hydrogens.

Because $9>6$, the tert-butyl radical enjoys significantly more hyperconjugative stabilization than the isopropyl radical. Thus, we can establish our first inequality:

The Dominance of Resonance

Now, let's turn our attention to the heavyweights: the diphenylmethyl radical, $({\text{C}}_6{\text{H}}_5{)}_2\dot{\text{C}}\text{H}$, and the triphenylmethyl radical, $({\text{C}}_6{\text{H}}_5{)}_3\dot{\text{C}}$.

In these molecules, the carbon bearing the unpaired electron is directly attached to benzene rings. This structural setup allows the odd electron to delocalize extensively into the extended $\pi$ system of the aromatic rings. This phenomenon is known as resonance.

It is a fundamental principle in organic chemistry that resonance stabilization is vastly superior to hyperconjugation. The ability to spread an electron over a large, conjugated $\pi$ network lowers the energy of the system much more effectively than the weak overlap of $\sigma$ bonds. Therefore, both of these phenyl-substituted radicals will be more stable than our alkyl radicals.

But how do they compare to each other?

- The diphenylmethyl radical can delocalize its unpaired electron across two phenyl rings. - The triphenylmethyl radical can delocalize its unpaired electron across three entire phenyl rings!

More rings mean a greater number of resonance structures, which translates to a more dispersed electron cloud and maximum stability. Consequently, the triphenylmethyl radical is the most stable species in our lineup.

Combining the Orders

By synthesizing our logical deductions, we can construct the final, comprehensive order of stability. We know that the resonance-stabilized radicals outrank the hyperconjugation-stabilized radicals. Within their respective categories, we have also determined the internal hierarchy.

Putting it all together, the increasing order of stability is:

This elegant sequence perfectly demonstrates how molecular structure dictates chemical stability, a cornerstone concept in organic chemistry.