Q: I have an iPhone 5 that constantly shuts off after 5-10 minutes..

You should get it checked by visiting an Authorized Apple Service Provider or Apple Store and see if the battery needs to be replaced.

Did you check the moisture indicators to see if the device did get damaged by liquid?

Liquid Contact Indicator location

Yes, depending on the age and use of the phone by the previous owner, this could be a battery issue.

Hi. You can check the complete charge cycles at Apple Store or Apple Authorized Service Provider, the standard as below:

iPhone Owners

Your battery is designed to retain up to 80% of its original capacity at 500 complete charge cycles. The one-year warranty includes service coverage for a defective battery. If it is out of warranty, Apple offers a battery service for $79, plus $6.95 shipping, subject to local tax.