The Brønsted definition explains everything that involves a proton. The Lewis definition explains everything else — and in doing so it reveals that most of the reactions in this course, including ones that look nothing like acid–base chemistry, are acid–base reactions in disguise.
The Lewis definition
A Lewis acid is an electron-pair acceptor. A Lewis base is an electron-pair donor.
Notice that the proton has disappeared from the definition entirely. That is the point. Every Brønsted base is also a Lewis base — it donates its lone pair to a proton — but a Lewis acid needs no proton at all, only an empty or accessible orbital. BF₃, AlCl₃, FeBr₃, ZnCl₂, TiCl₄ and every carbocation qualify.
The relationship between the two definitions is one of nesting, not of competition. Lewis is the broader category; Brønsted is the special case where the electron pair is donated specifically to H⁺. Both are correct, and chemists use whichever is more convenient for the reaction at hand.
Boron in BF₃ has only 6 valence electrons — an incomplete octet, one of the exceptions from Module 1 — and therefore a genuinely empty p orbital perpendicular to its trigonal planar framework.
One arrow: tail on nitrogen's lone pair, head on boron. The lone pair becomes a new B–N bond, and no existing bond breaks.
Formal charges after: nitrogen has 4 bonds, so +1; boron has 4 bonds and 3 valence electrons, so 3 − 0 − 4 = −1. The product, F₃B–NH₃, is a neutral adduct with an internal charge separation. Boron has gone from trigonal planar to tetrahedral, and now has a full octet.
A bond formed from a pair supplied entirely by one atom is sometimes called a dative or coordinate bond. Once formed it is an ordinary covalent bond in every respect; the name records only its origin.
Why this matters for mechanisms
The Lewis framework earns its keep in three places you will meet repeatedly.
Carbocations (Module 6) are textbook Lewis acids: empty p orbital, full positive charge, ready to accept a pair from anything nucleophilic within reach. Describing SN1's second step as a Lewis acid–base reaction is not a stretch — it is the most accurate description available.
Lewis acid catalysts activate otherwise unreactive electrophiles. In Friedel–Crafts acylation (Module 13), AlCl₃ accepts a lone pair from the chlorine of an acyl chloride, which weakens the C–Cl bond enough to generate a genuinely free acylium ion — an electrophile powerful enough to attack benzene, which the acyl chloride alone is not. The catalyst did not supply the electrophile; it manufactured one.
Carbonyl activation without acid. Coordinating a Lewis acid such as ZnCl₂ or TiCl₄ to a carbonyl oxygen pulls electron density off the carbon, making it far more electrophilic — the same result protonation achieves, but under neutral conditions that an acid-sensitive substrate can survive.
Hard and soft, briefly
Lewis acids and bases are often sorted as hard (small, charge-dense, not very polarizable — H⁺, Li⁺, BF₃, F⁻, RO⁻) or soft (large, diffuse, polarizable — I⁻, RS⁻, Ag⁺, Cu⁺). The useful generalization is that hard prefers hard and soft prefers soft.
You do not need the full theory. Its immediate use is that it resolves something Module 2 left slightly unsatisfying: thiolate (RS⁻) is a better nucleophile than alkoxide (RO⁻) while being the much weaker base of the two. Size and polarizability were the explanation given there, and hard/soft is the same observation generalized — a large, diffuse, polarizable nucleophile pairs well with a large, diffuse electrophilic carbon, whatever their basicities happen to be.
What carries forward
Lewis acid catalysis appears in Friedel–Crafts chemistry (Module 13), in epoxide opening (Module 8), and throughout carbonyl chemistry wherever a substrate cannot tolerate strong protic acid. More broadly, recognizing that nucleophilic attack is Lewis acid–base chemistry means the whole course has fewer independent ideas in it than the list of named reactions suggests.