Introduction to Hyperconjugation

Hyperconjugation is a stabilizing interaction that involves the delocalization of $\sigma$ electrons of a C-H or C-C bond into an adjacent empty or partially filled p-orbital or a $\pi$-orbital. It is also known as "no-bond resonance". The stability it provides is related to the number of contributing hyperconjugative structures, which in turn depends on the number of $\alpha$-hydrogens.

Step-by-Step Solution

1. Stability of tert-butyl cation $(CH_3)_3C^+$

• Structure: The tert-butyl cation consists of a central carbon atom with a positive charge, which is $sp^2$ hybridized. This $sp^2$ hybridized carbon atom has an empty p-orbital perpendicular to the plane of the three C-C bonds.
• Condition for Hyperconjugation: This empty p-orbital is adjacent to three methyl groups ($-CH_3$). The C-H bonds in these methyl groups are $\sigma$ bonds. The carbons of the methyl groups are called $\alpha$-carbons.
• Mechanism: The electrons from the C-H $\sigma$ bonds of the methyl groups can delocalize into the adjacent empty p-orbital of the positively charged carbon atom. This delocalization stabilizes the carbocation.
• Orbital Interaction: The interaction involves the overlap of the filled $\sigma$ orbital of the C-H bond with the empty p orbital of the carbocation.
• Notation: This type of electron delocalization is represented as $$\sigma \to p$(empty)$.

$\text{H}-\text{C}_\alpha \text{H}_2 - \text{C}^+ \longleftrightarrow \text{H}^+ \text{H}_2\text{C}_\alpha = \text{C}$

For the tert-butyl cation, there are nine such $\alpha$-hydrogens, leading to nine hyperconjugative structures and significant stability.

2. Stability of 2-butene $CH_3-CH=CH-CH_3$

• Structure: 2-butene is an alkene with a carbon-carbon double bond (C=C). The carbons of the double bond are $sp^2$ hybridized. The C=C bond consists of a $\sigma$ bond and a $\pi$ bond. The formation of the $\pi$ bond also results in a corresponding empty antibonding $\pi^*$ molecular orbital.
• Condition for Hyperconjugation: The C=C double bond is flanked by two methyl groups ($-CH_3$). The C-H bonds in these methyl groups are $\sigma$ bonds, and they are adjacent to the $\pi$ system.
• Mechanism: The electrons from the C-H $\sigma$ bonds of the methyl groups can delocalize into the adjacent empty $\pi^*$ antibonding orbital of the double bond. This delocalization strengthens the C-C single bond and slightly weakens the C=C double bond, leading to overall stabilization of the molecule.
• Orbital Interaction: The interaction involves the overlap of the filled $\sigma$ orbital of the C-H bond with the empty $\pi^*$ orbital of the C=C bond.
• Notation: This type of electron delocalization is represented as $\sigma \to \pi^*$.

$\text{H}-\text{C}_\alpha \text{H}_2 - \text{CH}=\text{CH}- \longleftrightarrow \text{H}^+ \text{H}_2\text{C}_\alpha = \text{CH}-\overline{\text{C}}\text{H}-$

For 2-butene, there are six $\alpha$-hydrogens (three on each methyl group), leading to six hyperconjugative structures.

3. Evaluation of Options

• For tert-butyl cation: The stabilization is due to $$\sigma \to p$(empty)$ delocalization.
• For 2-butene: The stabilization is due to $\sigma \to \pi^*$ delocalization.

Let's analyze the given options based on our findings:

• A: $$\sigma \to p$(empty)$ and $\sigma \to \pi^*$ electron delocalisations: This matches our analysis for both tert-butyl cation and 2-butene, respectively. This is the correct option.
• B: $\sigma \to \sigma^*$ and $\sigma \to \pi$ electron delocalisations: Incorrect. $\sigma \to \pi$ delocalization is not possible because the bonding $\pi$ orbital is already filled.
• C: $$\sigma \to p$(filled)$ and $\sigma \to \pi$ electron delocalisations: Incorrect. The carbocation has an empty p-orbital, not a filled one. Delocalization into a filled orbital would be destabilizing.
• D: p (filled) \to \sigma^*$$ and \sigma \to \pi^*$ electron delocalisations: Incorrect. The tert-butyl cation does not have a filled p-orbital to donate electrons from. This describes negative hyperconjugation, which is not relevant here.

Conclusion

The correct description for the hyperconjugative stabilities of tert-butyl cation and 2-butene is $$\sigma \to p$(empty)$ and $\sigma \to \pi^*$ electron delocalisations, respectively.