The unknown in panacetin is either acetanilide or phenacetin. Regardless of which it is either one is more soluble in dichloromethane than aqueous sodium bicarbonate. Why?
Panacetin is a pseudo drug which is a mixture of three compounds: acetylsalicylic acid (aka aspirin), acetanilide, and starch.
Panacetin or acetanilide does not react with aqueous sodium bicarbonate and therefore are not converted to salts. That is why low solubility in aqueous sodium bicarbonate.
Sodium bicarbonate is a base. The only way to dissolve any compound in aq.sodium bicarbonate is it should be acidic one. (As example salicylic acid.) Only acid-base reaction in case of sodium bicarbonate will help to dissolve solutes. Neither panacetin nor acetanilide are acidic. Therefore, they will not be much soluble in aqueous sodium bicarbonate.
On the other hand, polarity of dichloromethane is very low. Which helps dichloromethane to dissolve panacetin and/or acetanilide which are fairly nonpolar themselves.
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