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In the hydrolysis of propanamide ($\ce{CH3CH2CONH2}$) how do I know which molecules react with the water?

$$ \ce{CH3CH2CONH2 + H2O -> ?} $$

I don't know how to split the molecule. In the videos I've watched (example) it seems as they just assume you know this.

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I think you mean which atoms react with the water, as clearly the molecule that reacts is propanamide (it's the only non-water molecule in your question).

The product of the reaction is propanoic acid and ammonia.

$$\ce{CH3CH2CONH2 + H2O -> CH3CH2COOH + NH3}$$

The carbonyl carbon in amides (and esters and acyl halides) has a partial positive charge because of the electron withdrawing effect of the carbonyl oxygen. (Oxygen is very electronegative.) This makes the carbon atom is activated toward nucleophilic attack.

This Wikipedia page has a good explanation and different mechanisms for acid vs. base conditions.

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