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To differentiate the three theories of acids and bases, it would be better if we can define them one by one.
Arrhenius theory: This theory says that acid is a hydrogen ion (`H^(+)` ) donor and base is hydroxide ( `OH^(-)` ) donor. The reaction of hydrogen and hydroxide ions produces water. One limitation of this theory is that not all acid-base reactions produces water and is neutral. There are reactions that are acidic and basic even after the reaction.
Bronsted-Lowry theory: An acid is any substance that produces proton (`H^(+) or H3O^(+)` ) and bases are anything that accepts the proton. This theory is limited only to protic solvents. Meaning, it cannot explain an acid-base phenomenon of reactions with no proton transfer. There are substances (AlCl3 and BCl3) that act as acid but cannot produce/donate protons.
Lewis acid theory: It says that an acid is an electron acceptor and base is the electron donor or giver. This explains why BCl3 can act as an acid. It is because BCl3 can accommodate a pair of electron in its shell.
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