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Q. The hyperconjugative stabilities of tert-butyl cation and $2$-butene, respectively, are due to

JEE AdvancedJEE Advanced 2013Organic Chemistry – Some Basic Principles and Techniques

Solution:

Spreading out charge by the overlap of an empty $p$-orbital with an adjacent $\sigma$-bond is called hyperconjugation. This overlap (the hyperconjugation) delocalises the positive charge on the carbocation, spreading it over a larger volume, and this stabilises the carbocation.
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tertiary butyl carbocation has one vacant $p$-orbital, hence, it is stabilised by $\sigma-p$ (empty) hyperconjugation.
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In $2$-butene, stabilisation is due to hyperconjugation between $\sigma-\pi^{*}$ electron delocalisation.