According to the Argentina consulate your Argentina Reciprocity Fee is valid for 10 years, regardless of the validity of your passport.
You have to bring your old passport as proof of payment to Argentina so that immigrations officers can verify the payment. Take the original expired passport with you, a copy will not be accepted.
NOTE: Lately passengers have indicated they are getting turned away at the airport because their passports changed. This is probably due to uninformed immigration officials. For this reason, we strongly suggest passengers apply for a new Reciprocity Fee despite the Argentina consulate confirming the above rule.
For more info go here: https://www.visatoargentina.com