To understand why a tertiary butyl carbocation is more stable than a secondary butyl carbocation, we need to consider the factors that stabilize carbocations.$\\$A carbocation is a positively charged carbon atom with three bonds and an empty p orbital. The stability of a carbocation is influenced by:$\\$• Hyperconjugation$\\$• Inductive effect$\\$• Resonance (if applicable)$\\$In this case, the most relevant factor is hyperconjugation.$\\$Hyperconjugation is the delocalization of electrons from adjacent C-H bonds (sigma bonds) into the empty p orbital of the carbocation. This delocalization helps to stabilize the positive charge.$\\$Let's analyze the two types of carbocations:$\\$1. Tertiary butyl carbocation:$\\$• This carbocation has three methyl groups (-CH$_3$) attached to the positively charged carbon.$\\$• Each methyl group can participate in hyperconjugation, providing multiple C-H bonds that can stabilize the positive charge.$\\$• The more hyperconjugative structures available, the more stable the carbocation.$\\$2. Secondary butyl carbocation:$\\$• This carbocation has only two alkyl groups that can participate in hyperconjugation.$\\$• Fewer hyperconjugative structures are available compared to the tertiary carbocation, making it less stable.$\\$Therefore, the tertiary butyl carbocation is more stable due to the greater number of hyperconjugative structures available.$\\$The correct answer is Option 3: Hyperconjugation.