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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2020 · Paper I

Q2(a). Heat absorbed in an isothermal expansion against constant pressure

  • 10 marks
  • Answered
Question

1.0 mol of a monoatomic ideal gas expands to double of its initial volume against a constant pressure under isothermal condition of 300 K. Calculate the heat involved.

Answer in brief

Δ U=0, so q=-w=pₑₓₜ(V₂-V₁). With pₑₓₜ equal to the final pressure, q=12nRT=+1247 J (heat absorbed).

Key points

  • U of an ideal gas depends on T only; at constant T, Δ U=0, so q=-w.
  • Boyle's law at 300 K, V₂=2V₁: p₂=(p₁)/(2)=(nRT)/(2V₁)
  • w=-pₑₓₜ(V₂-V₁)=-(nRT)/(2V₁) (2V₁-V₁)=-(nRT)/(2)
  • w=-((1.0)(8.314)(300))/(2)=-1247 J; q=+1247 J (absorbed by the gas)

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

Reading: the paper does not give the value of the constant pressure. A gas expands against a constant external pressure until its own pressure falls to that value, so pₑₓₜ is taken as the final pressure p₂ (one-step expansion).

  1. First law: U of an ideal gas depends on T only; at constant T, Δ U=0, so q=-w.
  2. Final pressure: Boyle's law at 300 K, V₂=2V₁: p₂=(p₁)/(2)=(nRT)/(2V₁)
  3. Work against constant pressure: w=-pₑₓₜ(V₂-V₁)=-(nRT)/(2V₁) (2V₁-V₁)=-(nRT)/(2)
  4. Substitution: w=-((1.0)(8.314)(300))/(2)=-1247 J; q=+1247 J (absorbed by the gas)
  5. The initial volume cancels: the result holds for any starting pressure (for example V₁=24.6 L at 1 atm, pₑₓₜ=0.5 atm).
  6. Comparison with the reversible path: qᵣₑᵥ=nRTln(V₂)/(V₁)=(8.314)(300)2=+1729 J. The one-step expansion absorbs less heat and does less work (hatched area) than the reversible one (whole area under the isotherm).

Monatomic: the word fixes C_V=32R, but it does not enter: no temperature change means no C_VΔ T term.

Application. the difference 1729-1247=482 J is the work lost in the irreversible expansion; multi-stage expansion against falling pressures approaches the reversible limit, the principle of staged expansion in compressors and expansion turbines.

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Model answer to Q2(a), 2020 Paper I: solution page 7
Solution page 7 of 45
Read the whole 2020 solution2020 question paper

All parts of Question 2