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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

Q5(c). Half-life of two parallel first-order reactions

  • 10 marks
  • Answered
Question

Show that half-life period (t_(1/2)) of the following parallel reaction is (ln(2))/(k₁+k₂).
A k₁ B, A k₂ C (Both reactions are elementary).

Answer in brief

A disappears by both paths at once: -d[A]/dt=(k₁+k₂)[A], a first-order decay with the rate constant k₁+k₂; so [A]=[A]₀e^(-(k₁+k₂)t) and t_(1/2)=2/(k₁+k₂).

Key points

  • (d[B])/(dt)=k₁[A], (d[C])/(dt)=k₂[A]
  • Loss of A by both paths: -(d[A])/(dt)=k₁[A]+k₂[A]=(k₁+k₂)[A]
  • ∫_([A]₀)^([A])(d[A])/([A])=-(k₁+k₂)∫₀^tdt, ln([A]₀)/([A])=(k₁+k₂) t
  • at t=t_(1/2), [A]=[A]₀/2: 2=(k₁+k₂) t_(1/2), t_(1/2)=(2)/(k₁+k₂)=(0.693)/(k₁+k₂)

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 laws (unimolecular elementary steps): (d[B])/(dt)=k₁[A], (d[C])/(dt)=k₂[A]
  2. Loss of A by both paths: -(d[A])/(dt)=k₁[A]+k₂[A]=(k₁+k₂)[A]
  3. Integrate from [A]₀ at t=0: ∫_([A]₀)^([A])(d[A])/([A])=-(k₁+k₂)∫₀^tdt, ln([A]₀)/([A])=(k₁+k₂) t
  4. Half-life: at t=t_(1/2), [A]=[A]₀/2: 2=(k₁+k₂) t_(1/2), t_(1/2)=(2)/(k₁+k₂)=(0.693)/(k₁+k₂)
  5. Independent of [A]₀, as for any first-order reaction; it is shorter than the half-life of either path alone, because A is consumed by both.
  6. Products: dividing the two rate laws, d[B]/d[C]=k₁/k₂, so [B]/[C]=k₁/k₂ at every instant: [B]=(k₁)/(k₁+k₂)[A]₀(1-e^(-(k₁+k₂)t)). The measured k₁+k₂ and the product ratio give k₁ and k₂ separately.

Application. the nitration of toluene to o-, m- and p-nitrotoluene, and the decay of a nucleus by two branches (for example ⁴⁰K to ⁴⁰Ca and ⁴⁰Ar), follow this scheme: one overall half-life, with the branching ratio fixed by k₁/k₂.

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Model answer to Q5(c), 2020 Paper I: solution page 25
Solution page 25 of 45
Read the whole 2020 solution2020 question paper

All parts of Question 5