AppleID problem

Hi.

My mom bought a new iPhone a while ago and now she has an old one which I want start using. My question is, because her every app and AppleID is still in that phone, can I log out from the phones AppleID without logging out of every other device also? Basically, she's not home in a few days but she gave me an order to get her old phone in use. She has to sell my old phone next week. She doesn't really remember her AppleID password and also she needs her phone on the road so I don't want to cause trouble if her phone "stops" working. Anyone knows the answer?