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Arg vs Lys

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There's a question on a NS exam asking why arginine is a stronger base than lysine. The answer is because the electron donating group (which i'm assuming is the double bond NH group) stabilizes the conjugate acid. While I see why this would stabilize the conjugate acid, wouldn't that mean it would be a weaker base? A more stable CA would mean you wouldn't want to protonate the NH2 group because this would lead to loss of hybridization right?

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