Nuclei: Speed-Solving Techniques (3)

hard • 2 min read
Tags Nuclei

Question

A radioactive sample has a half-life of T1/2=20T_{1/2} = 20 days. After 60 days, what fraction of the original sample remains? Also find the activity ratio between t=0t = 0 and t=60t = 60 days. Solve in under 30 seconds.

Solution — Step by Step

Time elapsed = 60 days, half-life = 20 days. So number of half-lives passed = 60/20=360/20 = 3.

After nn half-lives, fraction remaining = (12)n\left(\dfrac{1}{2}\right)^n. With n=3n = 3:

N(t)N0=(12)3=18\frac{N(t)}{N_0} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}

So 1/8 of the original sample remains.

Activity A=λNA = \lambda N, and λ\lambda is constant. So A(t)/A(0)=N(t)/N0=1/8A(t)/A(0) = N(t)/N_0 = 1/8. Activity at 60 days is 1/8 of the initial activity.

Why This Works

The half-life formula N(t)=N0(1/2)t/T1/2N(t) = N_0 (1/2)^{t/T_{1/2}} is much faster than the exponential form N=N0e−λtN = N_0 e^{-\lambda t} when tt is a clean multiple of T1/2T_{1/2}. JEE and NEET examiners almost always pick numbers where this shortcut works.

Activity is proportional to the number of nuclei present, since each unstable nucleus has a fixed probability of decay per unit time. So activity decays at the same rate as the number of nuclei.

Three speed-solving moves for nuclei:

  1. Count half-lives. If tt is a multiple of T1/2T_{1/2}, just halve repeatedly.
  2. Mean life τ=T1/2/ln⁡2≈1.44 T1/2\tau = T_{1/2}/\ln 2 \approx 1.44 \, T_{1/2}. This is the average lifetime of a single nucleus.
  3. Decay constant λ=ln⁡2/T1/2≈0.693/T1/2\lambda = \ln 2 / T_{1/2} \approx 0.693/T_{1/2}. Use when activity is given.

Alternative Method

Use the exponential form: N=N0e−λtN = N_0 e^{-\lambda t} with λ=ln⁡2/T1/2=0.0347\lambda = \ln 2 / T_{1/2} = 0.0347 per day. Then N/N0=e−0.0347×60=e−2.08≈0.125=1/8N/N_0 = e^{-0.0347 \times 60} = e^{-2.08} \approx 0.125 = 1/8. Same answer, but takes 3x longer with a calculator.

When tt is not a clean multiple of T1/2T_{1/2}, students still try the half-life shortcut and get fractional powers of 1/21/2 wrong. Use the exponential form when t/T1/2t/T_{1/2} is not an integer.

Final answer: 1/8 of sample remains; activity ratio = 1/8.

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