Imagine some helium in a cylinder with an initial volume of 1 litre and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 litres, in such a way that its pressure rises in direct proportion to its volume. (a) Sketch a graph of pressure vs volume for this process. (b) Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done. (c) Calculate the change in the helium’s energy content during this process. (d) Calculate the amount of heat added to or removed from the helium during this process. (e) Describe what you might do to cause the pressure to rise as the helium expands.
Answer
Helium Gas Expansion – Thermodynamics Explained
🔷 Given Data
- Initial Volume: V₁ = 1 liter
- Final Volume: V₂ = 3 liters
- Initial Pressure: P₁ = 1 atm
- Relation: Pressure is directly proportional to volume → P ∝ V
🔹 (a) Final Pressure
Since P ∝ V, we have:
🔹 (b) Work Done on the Gas
Given P = kV and P₁ = 1 atm at V₁ = 1 L, so k = 1.
W = -4 × 101.325 = -405.3 J
🔹 (c) Change in Internal Energy
Using the ideal gas energy relation:
ΔU = 12 × 101.325 = 1,215.9 J
🔹 (d) Heat Added
First Law of Thermodynamics:
🔹 (e) Cause of Pressure Rise
The pressure increases due to:
As volume expands, added heat increases molecular motion, raising pressure accordingly.
✅ Final Summary
- (a) Final pressure = 3 atm
- (b) Work done = –405.3 J
- (c) ΔU = 1,215.9 J
- (d) Heat added = 1,621.2 J
- (e) Cause = Controlled Heating