**AP Chemistry Question 160: Answer and Explanation**

### Test Information

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**Question: 160**

**2.** Approximately how many half-lives will it take to reduce 10.0 kg of a radioactive substance to 1.0 microgram of that substance?

- A. 8
- B. 13
- C. 29
- D. 34

**Correct Answer:** D

**Explanation:**

(D) Mathematically, the fraction left is 1 microgram/10 kilograms, which is a ratio of (10^{-6})/(10 × 10^{3}) or 1/10^{-10}. For half-lives, the fraction remaining is (1/2)^{n}, where n is the number of half-lives. By equating (1/2)^{n} = (1/10^{-10}), we can take the logarithm of both sides of the equation to get n log (1/2) = -10. We can invert the fraction within the log term to get n log (2) = 10. (Note that this operation changes the sign.) Finally, n = 10/(log 2) = 33.2. This needs to be rounded to the next highest half-life, 34. This may be solved without a calculator by recalling that log (2) = 0.3.