GATE 2022 BT – Question 61
An insulated cylinder contains 1 mol ideal gas at 300 K. A weightless frictionless piston suddenly rises by 1 m against atmospheric pressure $1.013\times10^5$ Pa between two stoppers. Cylinder area is $10^{-3}\ \mathrm{m^2}$ and $C_v=12.5$ J/(mol K). Final temperature (nearest integer K) is ___________.
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Correct answer: 292
Explanation
The cylinder is insulated, so the process is **adiabatic** ($Q=0$). The gas pushes the piston up against the constant atmospheric pressure and does work on the surroundings. By the first law the internal energy falls by the work done.
**Step 1: work done by the gas.**
$$W=P_{atm}\,\Delta V=P_{atm}\,A\,\Delta z=1.013\times10^5\times10^{-3}\times1=101.3\text{ J}.$$
**Step 2: first law.** $\Delta U=-W$, and $\Delta U=n\,C_v\,\Delta T$:
$$1\times12.5\times(T_2-300)=-101.3\;\Rightarrow\;T_2-300=-8.10 .$$
$$T_2=300-8.10=291.9\approx\mathbf{292\ K}.$$