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In the Bronsted definition, an acid is a proton donor while a base is a proton acceptor. In the Lewis definition instead, an acid is an electron acceptor while the base is the electron donor. While the definition may be different, both Lewis and Bronsted pretty much refer to the same thing.
The two are essentially the same but they relate the ions through different means. The Lewis definition relates two ions using the transfer of electrons. A Lewis acid is an electron acceptor while a Lewis base is the electron donor. The Bronsted definition relates the two through protons in which a Bronsted acid is a proton donor and a Bronsted base is a proton acceptor.
In addition to the different definitions, lewis classification is more general (all acids or bases can be classified as a lewis acid or base) while bronsted classification is more specified. Not all lewis acids/bases will also be bronsted acids/bases.
The simple definition for Lewis is Lewis acids are electron pair acceptors and Lewis bases are electron pair donors. For Bronsted acids, these are proton donors (H+ donor) and a Bronsted base is a proton acceptor (H+ acceptor). Lewis acids and bases are more general terms since every Bronsted acid/bases are Lewis acids/bases but not every Lewis acid/bases are bronstead acids/bases. Bronsted acids/bases can be thought of as a subcategory for Lewis acids and bases.
Lewis acids/bases are a more general concept than the Bronsted acids/bases. Bronsted acids are proton donors and bronsted bases are proton acceptors. Lewis acids are electron acceptors and Lewis bases are electron donors.
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