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

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Solution:

Spreading out charge by the overlap of an empty -orbital with an adjacent -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 -orbital, hence, it is stabilised by (empty) hyperconjugation.
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In -butene, stabilisation is due to hyperconjugation between electron delocalisation.