### Half-Life Problems: Part One

Reminders:

(starting amount) x (1/2)number of half-lives = ending amount (sometimes remaining rather than ending is used)

(1/2)number of half-lives is the decimal fraction which remains (1.000 is the original starting amount, 0.500 at the end of one half-life, 0.250 at the end of two, 0.125 at the end of three, etc.)

number of half-lives that have occurred can be expressed as (total time elasped ÷ length of half-life)

1) The half-life of Zn-71 is 2.4 minutes. If one had 100.0 g at the beginning, how many grams would be left after 7.2 minutes has elapsed?

7.2 / 2.4 = 3 half-lives

(1/2)3 = 0.125 (the amount remaining after 3 half-lives)

100.0 g x 0.125 = 12.5 g remaining

2) Pd-100 has a half-life of 3.6 days. If one had 6.02 x 1023 atoms at the start, how many atoms would be present after 20.0 days?

3) Os-182 has a half-life of 21.5 hours. How many grams of a 10.0 gram sample would have decayed after exactly three half-lives?

(1/2)3 = 0.125 (the amount remaining after 3 half-lives)

10.0 g x 0.125 = 1.25 g remain

10.0 g - 1.25 g = 8.75 g have decayed

Note that the length of the half-life played no role in this calculation.

4) After 24.0 days, 2.00 milligrams of an original 128.0 milligram sample remain. What is the half-life of the sample?

5) U-238 has a half-life of 4.46 x 109 years. How much U-238 should be present in a sample 2.5 x 109 years old, if 2.00 grams was present initially?

(2.5 x 109) / (4.46 x 109) = 0.560 (the number of half-lves that have elapsed)

(1/2)0.560 = 0.678 (the decimal fraction of U-238 remaining)

2.00 g x 0.678 = 1.36 g remain

6) How long will it take for a 40.0 gram sample of I-131 (half-life = 8.040 days) to decay to 1/100 its original mass?

7) Fermium-253 has a half-life of 0.334 seconds. A radioactive sample is considered to be completely decayed after 10 half-lives. How much time will elapse for this sample to be considered gone?

8) At time zero, there are 10.0 grams of W-187. If the half-life is 23.9 hours, how much will be present at the end of one day? Two days? Seven days?

23.9 hours is 0.9958 of one day. We can make this problem easier by using one day for the half-life.

One day = one half-life; (1/2)1 = 0.50 remaining = 5.00 g

Two days = two half-lives; (1/2)2 = 0.25 remaining = 2.50 g

Seven days = 7 half-lives; (1/2)7 = 0.0078 remaining = 0.078 g

If you wish to use 23.9, then one day = 1.0042 half-lives (from 1/0.9958).

You may determine the values for 2 and 7 half-lives on your own.

9) 100.0 grams of an isotope with a half-life of 36.0 hours is present at time zero. How much time will have elapsed when 5.00 grams remains?

10) How much time will be required for a sample of H-3 to lose 75% of its radioactivity? The half-life of tritium is 12.26 years.

If you lose 75%, then 25% (0.25 as a decimal fraction) remains.

(1/2)n = 0.25

n = 2 (remember (1/2)2 = 1/4 and 1/4 = 0.25)

12.26 x 2 = 24.52 years

Comment: the more general explanation follows:

(1/2)n = 0.25

n log 0.5 = log 0.25

n = log 0.25 / log 0.5

n = 2

11) The half life of iodine-131 is 8.040 days. What percentage of an iodine-131 sample will remain after 40.2 days?

12) The half-life of thorium-227 is 18.72 days How many days are required for three-fourths of a given amount to decay?

13) If you start with 5.32 x 109 atoms of Cs-137, how much time will pass before the amount remaining is 5.20 x 106? The half-life of Cs-137 is 30.17 years.

14) The half-life of the radioactive isotope phosphorus-32 is 14.3 days. How long until a sample loses 99% of its radioactivity?

15) Find the number of disintegrations per minute emitted by 1.1 mol of 232-Th in 1 min. The half-life of 232-Th is 1.4 x 1010 year