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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

Q1(c). C_V in terms of U; (∂ C_V/∂ V)_T=0 for an ideal gas

  • 10 marks
  • Answered
Question

Express heat capacity at constant volume (C_V) of a substance in terms of its internal energy (U) and show that ((∂ C_V)/(∂ V))_(T,n)=0; for ideal gas.

Answer in brief

C_V=(∂ U/∂ T)_V. Then (∂ C_V/∂ V)_T=∂/∂ T (∂ U/∂ V)_T, and for an ideal gas (∂ U/∂ V)_T=0, so (∂ C_V/∂ V)_T=0.

Key points

  • at constant volume no expansion work is done, so dU=dq_V and C_V=((dq)/(dT))_V=((∂ U)/(∂ T))_(V,n)
  • ((∂ C_V)/(∂ V))_T=(∂)/(∂ V)((∂ U)/(∂ T))_V. U is a state function, so dU is exact and the order of differentiation can be changed: ((∂ C_V)/(∂ V))_T=(∂)/(∂ T)((∂ U)/(∂ V))_T
  • from dU=T dS-p dV and the Maxwell relation (∂ S/∂ V)_T=(∂ p/∂ T)_V: π_T=((∂ U)/(∂ V))_T=T((∂ p)/(∂ T))_V-p
  • ((∂ p)/(∂ T))_V=nR/V, so π_T=(nRT)/(V)-p=0. There are no intermolecular forces, so moving the molecules apart costs no energy (Joule's experiment).

The full model answer

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

  1. Heat capacity at constant volume: at constant volume no expansion work is done, so dU=dq_V and C_V=((dq)/(dT))_V=((∂ U)/(∂ T))_(V,n)
  2. Differentiate with respect to V: ((∂ C_V)/(∂ V))_T=(∂)/(∂ V)((∂ U)/(∂ T))_V. U is a state function, so dU is exact and the order of differentiation can be changed: ((∂ C_V)/(∂ V))_T=(∂)/(∂ T)((∂ U)/(∂ V))_T
  3. Internal pressure: from dU=T dS-p dV and the Maxwell relation (∂ S/∂ V)_T=(∂ p/∂ T)_V: π_T=((∂ U)/(∂ V))_T=T((∂ p)/(∂ T))_V-p
  4. Ideal gas, p=nRT/V: ((∂ p)/(∂ T))_V=nR/V, so π_T=(nRT)/(V)-p=0. There are no intermolecular forces, so moving the molecules apart costs no energy (Joule's experiment).
  5. Result: (∂ U/∂ V)_T=0 at every temperature, so its derivative with respect to T is also zero: ((∂ C_V)/(∂ V))_(T,n)=0

Meaning: U of an ideal gas depends on T only (for a monatomic gas U=32nRT, C_V=32nR at every volume). In the figure the two isotherms are horizontal, and raising T by dT costs the same C_V dT at every volume.

Application. for a van der Waals gas π_T=an²/V², which does not depend on T, so (∂ C_V/∂ V)_T=0 there too: attraction changes U with volume but not C_V.

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Model answer to Q1(c), 2020 Paper I: solution page 4
Solution page 4 of 45
Read the whole 2020 solution2020 question paper

All parts of Question 1