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What is the Bronsted-Lowry acid and base versus Lewis?
Here's how I understand it. We will look at some straight definitions, and then we will put them into context right afterwards.
A Brønsted-Lowry acid is a proton donor, while a Lewis base is an electron donor.
When a proton (##"H"^(+)##) accepts electrons donated from a Lewis base (like ##"OH"^(-)##), it exceeds its valency and must break its bond with a given atom on a Brønsted-Lowry acid (like the nitrogen on ##"NH"_4^(+)##).
If a bond between a Brønsted-Lowry acid and a proton breaks, the Brønsted-Lowry acid donates the proton.
Thus, a Lewis base can donate electrons to acquire a proton, while the Brønsted-Lowry acid had donated that proton. They can work together!
From the other perspective, a Brønsted-Lowry base is a proton acceptor, while a Lewis acid is an electron acceptor.
If a Brønsted-Lowry acid (like ##"NH"_4^(+)##) accepts electrons from a Lewis base (like water in this case), then it is also a Lewis acid by definition.
Furthermore, if that proton donates itself onto the Lewis base (water in this case), that base is also a Brønsted-Lowry base (water in this case), because it accepted the proton.
Thus, a Lewis acid that possesses a proton can accept electrons to donate a proton, while the Brønsted-Lowry base accepts that proton. Again, they work together!