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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 · 2018 · Paper I

Q5(a). Half-life of the neutralisation H⁺ + OH⁻

  • 10 marks
  • Answered
Question

The rate constant k₁ for the reaction described by H+(aq) + OH-(aq) ⇌H₂O(l) is 1.4×10¹¹ dm³.mol⁻¹.s⁻¹.
Calculate the half life of the above reaction if the initial conditions are
(i) [H⁺]₀=[OH⁻]₀=0.10 mol.dm⁻³ (ii) [H⁺]₀=[OH⁻]₀=1.0×10⁻⁷ mol.dm⁻³

Answer in brief

Second order with equal concentrations, t_(1/2)=1/k₁[A]₀: (i) 7.1×10⁻¹¹ s; (ii) 7.1×10⁻⁵ s (a formal value: these are the equilibrium concentrations of pure water, so no net change is seen).

Key points

  • Rate =k₁[H⁺][OH⁻]. With [H⁺]₀=[OH⁻]₀=[A]₀, the two stay equal, so the rate is k₁[A]² and the integrated law is (1)/([A])-(1)/([A]₀)=k₁t.
  • (2)/([A]₀)-(1)/([A]₀)=k₁t_(1/2), t_(1/2)=(1)/(k₁[A]₀)
  • (i) t_(1/2)=1(1.4×10¹¹ dm³ mol⁻¹ s⁻¹)(0.10 mol dm⁻³) t_(1/2)=7.1×10⁻¹¹ s
  • (ii) t_(1/2)=1(1.4×10¹¹)(1.0×10⁻⁷) t_(1/2)=7.1×10⁻⁵ s

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. Rate =k₁[H⁺][OH⁻]. With [H⁺]₀=[OH⁻]₀=[A]₀, the two stay equal, so the rate is k₁[A]² and the integrated law is (1)/([A])-(1)/([A]₀)=k₁t.
  2. At t_(1/2), [A]=[A]₀/2: (2)/([A]₀)-(1)/([A]₀)=k₁t_(1/2), t_(1/2)=(1)/(k₁[A]₀)
  3. (i) t_(1/2)=1(1.4×10¹¹ dm³ mol⁻¹ s⁻¹)(0.10 mol dm⁻³) t_(1/2)=7.1×10⁻¹¹ s
  4. (ii) t_(1/2)=1(1.4×10¹¹)(1.0×10⁻⁷) t_(1/2)=7.1×10⁻⁵ s
  5. The half-life of a second-order reaction is inversely proportional to the initial concentration: diluting 10⁶ times lengthens it 10⁶ times. A first-order half-life would not change.
  6. 1.0×10⁻⁷ mol dm⁻³ is the equilibrium concentration in pure water at 25°C, so in case (ii) there is no net neutralisation: forward and back reactions balance. The value 7×10⁻⁵ s is the time scale on which the ions pair up and are replaced; the relaxation time τ=1/k₁([H⁺]+[OH⁻])+k₋₁ is 3.6×10⁻⁵ s, of the same order.
  7. k₁=1.4×10¹¹ is one of the largest second-order rate constants known: every encounter of H⁺ with OH⁻ reacts, so the rate is limited by how fast the ions diffuse together (diffusion control), helped by proton hopping along hydrogen-bonded water.

Application. reactions this fast are finished before any mixing method can follow them; Eigen measured k₁ by the temperature-jump relaxation method, in which a sudden heating shifts the equilibrium of water and the return is followed by conductivity.

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Model answer to Q5(a), 2018 Paper I: solution page 30
Solution page 30 of 55
Read the whole 2018 solution2018 question paper

All parts of Question 5