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JEE Advanced 2021 Paper 2 · Chemistry · Chemical Thermodynamics · hard

Two moles of a monoatomic ideal gas at 300 K occupy 10 L and are allowed to expand to 20 L by three different processes: (I) reversible isothermal…

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Two moles of a monoatomic ideal gas at 300 K300\ \mathrm{K} occupy 10 L10\ \mathrm{L} and are allowed to expand to 20 L20\ \mathrm{L} by three different processes: (I) reversible isothermal expansion, (II) free expansion into an evacuated chamber inside a thermally insulated container, (III) reversible adiabatic expansion. Take R=8.314 J mol−1K−1R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}, ln⁡2=0.693\ln 2=0.693, 22/3=1.5872^{2/3}=1.587. The correct statement(s) is/are
  1. A. ΔSsystem\Delta S_{\text{system}} is the same, about 11.5 J K−111.5\ \mathrm{J\,K^{-1}}, for processes I and II
  2. B. ΔSuniverse\Delta S_{\text{universe}} is zero for process I and about +11.5 J K−1+11.5\ \mathrm{J\,K^{-1}} for process II
  3. C. In process III the final temperature is about 189 K189\ \mathrm{K} and the work done by the gas is about 2.77 kJ2.77\ \mathrm{kJ}
  4. D. The work done by the gas in process I is smaller than that in process III
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Answer: A, B, C

Entropy is a state function, and processes I and II connect the same initial and final states of the gas (same TT and volume), so ΔSsys=nRln⁡(V2/V1)=2(8.314)(0.693)=11.5 J K−1\Delta S_{sys}=nR\ln(V_2/V_1)=2(8.314)(0.693)=11.5\ \mathrm{J\,K^{-1}} for both: A is correct. In I the surroundings give heat reversibly, ΔSsurr=−11.5\Delta S_{surr}=-11.5, so ΔSuniv=0\Delta S_{univ}=0; in II no heat flows, ΔSsurr=0\Delta S_{surr}=0, so ΔSuniv=+11.5\Delta S_{univ}=+11.5: B is correct. In III, T2=300(1/2)γ−1=300(1/2)2/3=300/1.587=189 KT_2=300(1/2)^{\gamma-1}=300(1/2)^{2/3}=300/1.587=189\ \mathrm{K} with γ=5/3\gamma=5/3, and wby=nCV(T1−T2)=2(1.5)(8.314)(111)=2.77w_{by}=nC_V(T_1-T_2)=2(1.5)(8.314)(111)=2.77 kJ: C is correct. D claims the work in I is smaller than in III, but wI=nRTln⁡2=3.46w_I=nRT\ln2=3.46 kJ exceeds 2.772.77 kJ, since the adiabatic gas cools and does less work. D is wrong. Answer: A, B, C.
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