Q.61
Consider a piston-cylinder assembly shown in the figure below.
The walls of the cylinder are insulated.
The cylinder contains 1 mole of an ideal gas at 300 K and the piston is held initially
at the position z₁ using a stopper.
After the stopper is removed, the piston suddenly rises against atmospheric pressure
(1.013 × 105 Pa) to the new position z₂ where it is held by another stopper.
The heat capacity (CV) of the gas is
12.5 J mol−1 K−1.
The cross-sectional area of the cylinder is
10−3 m2.
Assume the piston is weightless and frictionless.
If z₂ − z₁ = 1 m, the final temperature of the gas (rounded off to nearest integer)
is ___________ K.
Introduction
Piston–cylinder thermodynamics questions frequently appear in GATE Mechanical, Chemical, and Biotechnology papers.
These problems test concepts like adiabatic work, ideal gas behavior, heat capacity, and sudden expansion.
In this solved example, the piston rises suddenly when released, causing the gas to expand against atmospheric pressure.
Because the walls are insulated, no heat exchange occurs — making this an adiabatic process.
Let’s compute the final temperature scientifically.
Problem Summary
- Gas = 1 mole ideal gas
- Initial temperature = 300 K
- Volume change = A × (z₂ − z₁)
- Area A = 10⁻³ m²
- Lift distance = 1 m
- ⇒ ΔV = 10⁻³ × 1 = 10⁻³ m³
- External pressure = 1.013 × 10⁵ Pa
- Cv = 12.5 J/mol·K
- Piston frictionless, insulated → adiabatic
Theory
Since no heat transfer occurs:
Q = 0
By the First Law of Thermodynamics:
ΔU = –W
Where:
- ΔU = nCv (T₂ – T₁)
- W = Pext (V₂ – V₁)
Step-by-Step Calculation
1️⃣ Work Done by the Gas
\( W = P_{\text{ext}} \Delta V = (1.013 \times 10^5) \times (1 \times 10^{-3}) = 101.3 \text{ J} \)
2️⃣ Internal Energy Change
ΔU = nCv(T₂ – T₁)
n = 1, Cv = 12.5 J/mol·K
3️⃣ Apply First Law
12.5(T₂ - 300) = -101.3
T₂ - 300 = -8.104
T₂ = 300 - 8.104 = 291.896 K
Final temperature ≈ 292 K
Correct Answer
The final temperature of the gas = 292 K
Option-by-Option Analysis (Likely MCQ Choices)
| Option (K) | Explanation |
|---|---|
| 300 | Incorrect — assumes no temperature change; ignores expansion work |
| 292 (Correct) | Correct — energy lost as work reduces temperature |
| 285 | Incorrect — miscalculated work or wrong Cv |
| 250 | Incorrect — would require free/vacuum expansion, not atmospheric |
Conclusion
The sudden lifting of the piston causes the gas to perform work, reducing its internal energy and temperature.
Since the cylinder is insulated, the process is adiabatic — no heat enters to compensate.
Thus, the final temperature of the gas is 292 K, consistent with first-law thermodynamics.
