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I believe for something like this we would need the pKb value but we can also use a rule of thumb that most compounds with N are going to be a weak base and also due to the negative charge on hypobromite we can figure it will be a stronger acid as this allows it readily accept a proton.
We know that BrO- is stronger base since it has a negative charge. This means that when it accepts a proton H+, it will become neutral whereas for C17H19O3N, it will become positively charged. Neutral charge is more favorable.
BrO- is more likely to pick up a hydrogen because it is the conjugate base of a weak acid and equilibrium will be more likely to lie the left. This makes it a STRONGER base than C17H19O3N since it is considered a weak base and is not too likely (but still can) pick up a proton with the lone pair on nitrogen.
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