Find and Replace across multiple worksheets?

I have a workbook with two sheets, "Patients" and "Prescriptions." Prescriptions is the primary worksheet that contains all the info I need.

due to a normalization issue, I created a new record in my primary sheet, and it created a duplicate record of each patient I selected. So I've got multiple records for the same patient in my "Patients" sheet.

I want to find replace everything in the "patient_id" column in "Prescriptions", to match up with the actual Patient name, not an ID number... rather than performing a find and replace one-by-one... Anybody know how to accomplish this?

I've attached a sample of my sheet so you know what I'm talking about. The actual sheet contains over 5000 entries so a way to automate this process as much as possible is ideal.