Gallium-65, a radioactive isotope of gallium decays by first order kinetics. The half-life of this isotope is 15.2 minutes. How long would it take for
7

8
of a sample of this isotope to decay (i.e., for
7

8
of the sample to react)?

Question

Gallium-65, a radioactive isotope of gallium decays by first order kinetics. The half-life of this isotope is 15.2 minutes. How long would it take for 
7

8
 of a sample of this isotope to decay (i.e., for 
7

8
 of the sample to react)?

BlackTom AI · Accepted Answer

To approach this, recognize that gallium-65 decays by first-order kinetics, so the fraction remaining after time t follows N/N0 = (1/2)^(t / t1/2), where t1/2 is 15.2 minutes.
Option by option reasoning:
- Option 1: 45.6 min. Here, compute the remaining fraction: N/N0 = (1/2)^(45.6 / 15.2) = (1/2)^3 = 1/8 = 0.125. That means 7/8 of the sample has decayed, which matches the problem statement. This is consistent and correct.
- Option 2: 15.2 min. At t = t1/2, the remaining fraction is N/N0 = (1/2)^(1) = 1/2, i.e., only 1/2 has decayed. This would correspond to 1/2 decayed, not 7/8, so this is incorrect.
- Option 3: 2.93 min. This is roughly t1/2 divided by 5.2, but more importantly the exponent t/15.2 would be about 0.193, giving N/N0 ≈ 0.87 remaining, meaning about 13% decayed, far from the required 87% decayed. Therefore incorrect.
- Option 4: 30.4 min. This gives N/N0 = (1/2)^(30.4 / 15.2) = (1/2)^2 = 1/4 remaining, i.e., 3/4 decayed. That’s 75% decayed, not the requested 87.5%. So this is not correct either.
In summary, only 45.6 min yields a remaining fraction of 1/8, corresponding to 7/8 decayed, aligning with the problem’s requirement.

Gallium-65, a radioactive isotope of gallium decays by first order kinetics. The half-life of this isotope is 15.2 minutes. How long would it take for 7 8 of a sample of this isotope to decay (i.e., for 7 8 of the sample to react)?单项选择题

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