Physics quiz question: activity decrease of bismuth-209

Physics 205B Quiz 7, spring semester 2017
Cuesta College, San Luis Obispo, CA

Bi(209,83) has the longest known half-life of any isotope that undergoes alpha decay, with a half-life of 1.9×10¹⁹ years[*].

It will take __________ for a sample of Bi(209,83) to decrease its activity by 1.0%.
(A) 1.9×10¹⁷ years.
(B) 2.8×10¹⁷ years.
(C) 2.7×10¹⁹ years.
(D) 3.8×10¹⁹ years.

[*] Pierre de Marcillac, Noël Coron, Gérard Dambier, Jacques Leblanc and Jean-Pierre Moalic, "Experimental Detection of α-Particles from the Radioactive Decay of Natural Bismuth," Nature, vol. 422 (April 24, 2003), 
pp. 876-878 nature.com/nature/journal/v422/n6934/full/nature01541.html.

Correct answer (highlight to unhide): (B)

The activity of a sample is given by:

R = R₀·(1/2)^(t/T_1/2),

where T_1/2 is the half-life. Solving for the time t it would take for the activity to drop down to R = 0.990 (assuming R₀ started at exactly 1, as it "decreases its activity by 1.0%" down to 99.0%), then:

(R/R₀) = (1/2)^(t/T_1/2);

ln(R/R₀) = (t/T_1/2)·ln(1/2);

t = T_1/2·ln(R/R₀)/ln(1/2);

t = (1.9×10¹⁹ years)·(ln(0.990/1)/ln(1/2) = 2.754918242×10¹⁷ years,

or to two significant figures, it will take this sample 2.8×10¹⁷ years to decrease its activity by 1.0%.

(Response (A) is 1% of T_1/2; response (C) is the time constant τ = 1/(T_1/2·ln(2)); response (D) is the time it would take for the sample to decrease its activity by 99% (or decrease its activity down to 1%.)

Sections 30882, 30883
Exam code: quiz07N4ci
(A) : 12 students
(B) : 7 students
(C) : 5 students
(D) : 2 students

Success level: 27%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.57