UPSC Combined Geo-Scientist, Mains · 2026 · Mains Paper II
Q1(g). Half-life of a nuclide with two parallel first-order decays
Question
A radioactive nuclide X has two decay modes (both first order) X→ B (k₁) and X→ C (k₂); k₁ and k₂ are decay constants. Express t_(1/2) of X in terms of k₁ and k₂.
Answer in brief
t_(1/2)=(2)/(k₁+k₂).
The full model answer
Worked out step by step, as it should go in the answer booklet. The solution sheet follows it.
-(d[X])/(dt)=k₁[X]+k₂[X]=(k₁+k₂)[X] ⇒ [X]=[X]₀e^(-(k₁+k₂)t)
At t=t_(1/2), [X]=12[X]₀: (k₁+k₂)t_(1/2)=2 ⇒ t_(1/2)=(0.693)/(k₁+k₂). The branching ratio is [B]/[C]=k₁/k₂; the fraction going to B is k₁/(k₁+k₂).
Coversparallel reactionsHalf-life of a nuclide with two parallel first-order decays
All parts of Question 1
- 1(a) Compressibility factor and its significance5 marks
- 1(b) Andrew's isotherms and the critical-point conditions5 marks
- 1(c) First-order Bragg angle for the (110) planes5 marks
- 1(d) Intrinsic semiconductor and its conversion to extrinsic5 marks
- 1(e) Fugacity against pressure of a non-ideal gas5 marks
- 1(f) Kₚ and K_c for A+B⇌ C5 marks
- 1(g) Half-life of a nuclide with two parallel first-order decays5 marks
- 1(h) Soap and detergent5 marks
- 1(i) Galvanic cell for 2 Cu⁺leftharpoonsCu²⁺+Cu5 marks
- 1(j) pH when a weak base is 65% neutralised5 marks
- 1(k) Uncertainty in an energy eigenstate5 marks
- 1(l) Expectation value and eigenvalue5 marks
- 1(m) Normal modes of vibration5 marks
- 1(n) Raman activity and polarisability ellipsoids of CO₂5 marks
- 1(o) The incorrect statement on electronic spectra5 marks
- 1(p) Non-radiative decay competing with fluorescence5 marks