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 of a sample of this isotope to decay (i.e., for 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  of a sample of this isotope to decay (i.e., for  of the sample to react)?

BlackTom AI · Accepted Answer

Question restatement: The problem states that Gallium-65 decays by first-order kinetics with a half-life of 15.2 minutes, and asks how long it would take for seven-eighths (7/8) of a sample to decay (i.e., for 7/8 of the sample to react).

Step-by-step analysis:
- For first-order decay, the amount remaining N at time t is given by N = N0 * (1/2)^(t / t1/2), where t1/2 is the half-life.
- If 7/8 of the sample decays, then the remaining fraction is 1/8. That is, N/N0 = 1/8.
- Substitute into the equation: 1/8 = (1/2)^(t / 15.2).
- Recognize that (1/2)^(t / 15.2) = 1/8 equals (1/2)^3, so the exponents must be equal: t / 15.2 = 3.
- Solve for t: t = 3 * 15.2 = 45.6 minutes.

Why this makes sense conceptually:
- Each half-life reduces the amount by a factor of 2. After 1 half-life, you have 1/2 remaining; after 2 half-lives, 1/4 remaining; after 3 half-lives, 1/8 remaining. Since we want 7/8 to decay (leaving 1/8), it takes three half-lives, i.e., 3 × 15.2 minutes = 45.6 minutes.

Notes on missing options:
- The prompt here does not provide alternative answer choices to evaluate against. In a standard multiple-choice setup, other distractors might correspond to decaying smaller or larger fractions (e.g., 15.2 min for half, 30.4 min for 3/4 decayed, or 2.92 min for 7/8 remaining), but those are not presented here. The calculation above shows why 45.6 minutes is the correct time for 7/8 decay given a 15.2-minute half-life.

Chem 1C General Chemis... Self Assessment Quiz 2

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 of a sample of this isotope to decay (i.e., for of the sample to react)?

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