Go to introductory half-life discussion

**Problem #11:** The half life of iodine-131 is 8.040 days. What percentage of an iodine-131 sample will remain after 40.20 days?

**Solution:**

40.20 d / 8.040 d = 5(1/2)

^{5}= 0.03125percent remaining = 3.125%

**Problem #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?

**Solution:**

3/4 = 0.75 <--- amount decayed1 - 0.75 = 0.25 <--- amount remaining

(1/2)

^{n}= 0.25n = 2

18.72 d times 2 = 37.44 d

**Problem #13:** If you start with 5.32 x 10^{9} atoms of Cs-137, how much time will pass before the amount remaining is 5.20 x 10^{6} atoms? The half-life of Cs-137 is 30.17 years.

**Solution:**

5.20 x 10^{6}/ 5.32 x 10^{9}= 0.0009774436 (the decimal amount remaining)(1/2)

^{n}= 0.0009774436n log 0.5 = log 0.0009774436

n = 9.99869892 half-lives

30.17 yr times 10 = 301.7 yr

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

**Solution:**

99% loss means 1% remaining1% = 0.01

(1/2)

^{n}= 0.01n log 0.5 = log 0.01

n = 6.643856

14.3 day times 6.643856 = 95.0 day

**Problem #15:** The half-life of Palladium-100 is 4 days. After 12 days a sample of Pd-100 has been reduced to a mass of 4.00 mg. (a) Determine the starting mass. (b) What is the mass 8 weeks after the start?

**Solution:**

12 day / 4 day = 3(1/2)

^{3}= 0.1254.00 mg / 0.125 = 32.0 mg

8 weeks = 56 days

56 d / 4 = 14 half-lives

(1/2)

^{14}= 0.00006103532.0 mg times 0.000061035 = 0.00195 mg (rounded to three figs)

**Problem #16:** Rn-222 has a half-life of 3.82 days. How long before only 1/16 of the original sample remains?

**Solution:**

recognize 1/16 as a fraction associated with 4 half-lives (from 1/2^{4}= 1/16)3.82 days x 4 = 15.3 days

**Problem #17:** U-238 has a half-life of 4.46 x 10^{9} years. Estimates of the age of the universe range from 9 x 10^{9} years to 23 x 10^{9} years (Cauldrons in the Cosmos: Nuclear Astrophysics, C.E. Rolfs and W.S. Rodney, Univ. of Chicago, 1988, p. 477). What fraction of this isotope present at the start of the universe remains today? Calulate for both minimum and maximum values, as well as a median value of 16 x 10^{9} years.

**Solution:**

1) Calculation for the median value:

(16 x 10^{9}) / (4.46 x 10^{9}) = 3.587 half-lives

2) What fraction remains?

(1/2)^{3.587}= 0.08328.32% remains

**Problem #18:** A sample of Se-83 registers 10^{12} disintegrations per second when first tested. What rate would you predict for this sample 3.5 hours later, if the half-life is 22.3 minutes?

**Solution:**

210 min / 22.3 min = 9.42 half-lives (210 min is 3.5 hours)(1/2)

^{9.42}= 0.00146 (the decimal fraction remaining)10

^{12}x 0.00146 = 1.46 x 10^{9}disintegrations per second remaining

**Problem #19:** Iodine-131 has a half-life of 8.040 days. If we start with a 40.0 gram sample, how much will remain after 24.0 days?

**Solution:**

24.0 days / 8.040 days = 2.985 half-lives(1/2)

^{2.985}= 0.1263 (the decimal fraction remaining)40.0 g x 0.1263 = 5.05 g

**Problem #20:** If you start with 2.97 x 10^{22} atoms of molybdenum-99 (half-life = 65.94 hours), how many atoms will remain after one week?

**Solution:**

one week = 168 hours168 / 65.94 = 2.548

(1/2)

^{2.548}= 0.171 (the decimal fraction remaining)(2.97 x 10

^{22}) x 0.171 = 5.08 x 10^{21}

**Problem #21:** The isotope H-3 has a half life of 12.26 years. Find the fraction remaining after 49 years.

**Solution:**

49 / 12.26 = 3.9967(1/2)

^{3.9967}= 0.0626

**Problem #22:** How long will it take for a 64.0 g sample of Rn-222 (half-life = 3.8235 days) to decay to 8.00 g?

**Solution:**

8.00 / 64.0 = 0.125 (the decimal fraction remaining)(1/2)

^{n}= 0.125by experience, n = 3 (remember that 0.125 is 1/8)

3.8235 x 3 = 11.4705 days

**Problem #23:** A scientist needs 10.0 micrograms of Ca-47 (half-life = 4.50 days) to do an experiment on an animal. If the delivery time is 50.0 hours, how many micrograms of ^{47}CaCO_{3} must the scientist order?

**Solution:**

4.50 days x 24 hrs/day = 108 hrs50/108 = 0.463 half-lives

(1/2)

^{0.463}= 0.725 (the decimal portion of Ca-47 remaining after 50 hrs)10.0 mg / 0.725 = 13.8 mg

**Problem #24:** What precentage of the parent isotope remains after 0.5 half lives have passed?

**Solution:**

(1/2)^{n}= decimal amount remainingwhere n = the number of half-lives

(1/2)

^{0.5}= 0.707The question asks for a percentage, so 70.7%

**Problem #25:** Manganese-56 has a half-life of 2.6 h. What is the mass of manganese-56 in a 1.0 g sample of the isotope at the end of 10.4 h?

**Solution:**

10.4 / 2.6 = 44 half-lives = 0.0625 remaining

0.0625 g