Section 21 on AST

Have issued a section 21 notice to a tenant before 1st December 2022. The date that the tenant has to vacate by is 13th February 2023. 

He has yet to advise of the date he will vacate, and the subsequent monthly rent is due on 13th January. 

Please clarify, can he leave any time between the serving of the section 21 notice and 13th February? In doing so, to what extent is his liability to pay rent? Upon paying the rent on 13th January, is he entitled to a refund on the days he did not occupy, i.e. if he vacates between 13th January & 13th February?

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