Final answer:
The essential piece of equipment for administering TPN solution via a central line is an Electronic infusion pump, as it ensures precise and continuous delivery of nutrients.
Step-by-step explanation:
When preparing to hang the first bag of TPN solution via a central line, the most essential piece of equipment a nurse should obtain is an Electronic infusion pump. TPN, or Total Parenteral Nutrition, is a method of feeding that bypasses the gastrointestinal tract. A high level of precision is required when administering TPN to ensure the correct rate and amount of nutrients are delivered. The Electronic infusion pump helps regulate the flow rate of TPN solution into the central line accurately and continuously, which is crucial for patient safety and the effectiveness of the nutrition therapy.