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Half-life Practice

#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?

#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?

#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?

#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?

#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.

11) Fluorine-21 has a half life of approximately 5 seconds. What fraction of the original nuclei would remain after 1 minute?

12) Iodine-131 has a half life of 8 days. What fraction of the original sample would remain at the end of 32 days?

13) The half-life of chromium-51 is 28 days. If the sample contained 510 grams, how much chromium would remain after 56 days? How much would remain after 1 year? How much was present 168 days ago?

14) If 20.0 g of a radioactive isotope are present at 1:00 PM and 5.0 g remain at 2:00 PM, what is the half life of the isotope?

15) The half life of Uranium-238 is 4.5 billion years and the age of earth is 4.5 X 109 years. What fraction of Uranium-238 that was present when Earth was formed still remains?

16) Chromium-48 decays. After 6 half-lives, what fraction of the original nuclei would remain?

17) The half life of iodine-125 is 60 days. What fraction of iodine-125 nuclides would be left after 360 days?

18) Titanium-51 decays with a half life of 6 minutes. What fraction of titanium would remain after one hour?

19) A medical institution requests 1 g of bismuth-214, which has a half life of 20 min. How many grams of bismuth-214 must be prepared if the shipping time is 2 h?

20) The half life of radium 226 is 1602 years. If you have 500 grams of radium today how many grams would have been present 9612 years ago?

Answers

#1

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

20.0 / 3.6 = 5.56 half-lives

(1/2)5.56 = 0.0213 (the decimal fraction remaining after 5.56 half-lives)

(6.02 x 1023) (0.0213) = 1.28 x 1022 atoms remain

#3

(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. In addition, note that the question asked for the amount that decayed, not the amount that remaining.

#4

2.00 mg / 128.0 mg = 0.015625

How many half-lives must have elapsed to get to 0.015625 remaining?

(1/2)n = 0.015625 

n log 0.5 = log 0.015625 

n = log 0.5 / log 0.015625 

n = 6

24 days / 6 half-lives = 4.00 days (the length of the half-life)

#5

(2.5 x 109) / (4.46 x 109) = 0.560 (the number of half-lives 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

(1/2)n = 0.01

n log 0.5 = log 0.01

n = 6.64

6.64 x 8.040 days = 53.4 days

#7

0.334 x 10 = 3.34 seconds

#8

24.0 hr / 23.9 hr/half-life = 1.0042 half-lives

One day = one half-life; (1/2)1.0042 = 0.4985465 remaining = 4.98 g

Two days = two half-lives; (1/2)2.0084 = 0.2485486 remaining = 2.48 g

Seven days = 7 half-lives; (1/2)7.0294 = 0.0076549 remaining = 0.0765 g

#9: 

5.00 / 100.0 = 0.05 (decimal fraction remaining)

(1/2)n = 0.05

n log 0.5 = log 0.05

n = 4.32 half-lives

36.0 hours x 4.32 = 155.6 hours

#10

If you lose 75%, then 25% remains. Use 0.25 rather than 25%.

(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) Fluorine-21 has a half life of approximately 5 seconds. What fraction of the original nuclei would remain after 1 minute?

a. The answer is solved by creating the fraction [pic]. Where n = the

b. number of half lives. If each half life is 5 seconds, then in one minute

c. (60 seconds) there are 12 half lives. Therefore the answer is:[pic]

12) Iodine-131 has a half life of 8 days. What fraction of the original sample would remain at the end of 32 days?

a. Using the same fraction, you must figure out n. If the half life is 8 days,

b. then in 32 days, there are 4 half lives. Therefore the answer is:[pic]

13) The half-life of chromium-51 is 28 days. If the sample contained 510 grams, how much chromium would remain after 56 days? How much would remain after 1 year? How much was present 168 days ago?

a. In this problem, the fraction will be multiplied by the initial amount.

b. In the first problem each half life is 28 days, therefore in 56 days two half lives occur. This means that n=2. The solution is as follows:

[pic]

c. The second is solved the same way except that there are 13 half lives

d. over one year. This means n=13. The solution is as follows:

[pic]

e. The third is solved by recognizing there must be more of the

f. sample 168 days ago then there is now. 168 days represents

g. 3 half lives so n=3. The solution is:

[pic]

14) If 20.0 g of a radioactive isotope are present at 1:00 PM and 5.0 g remain at 2:00 PM, what is the half life of the isotope?

a. In this problem, you must figure out how many half lives have occurred.

b. After one half life 20.0g becomes 10.0g. After a second half life, 10.0g becomes 5.0g. This means that during the question, two half lives have occurred. Since this happened over the course of 1 hour, then each half life must be equal to:

c. 30 minutes.

15) The half life of Uranium-238 is 4.5 billion years and the age of earth is 4.5 X 109 years. What fraction of Uranium-238 that was present when Earth was formed still remains?

a. 4.5 billion is exactly the same as 4.5 x 109. Therefore, the age of the

b. Earth is equal to one half life of Uranium. This means that n=1. The solution is a follows:[pic]

16) Chromium-48 decays. After 6 half-lives, what fraction of the original nuclei would remain?

a. The answer is solved by creating the fraction [pic]. Where n = the

b. number of half lives. If there are 6 half lives, then n=6.Therefore the

c. answer is:[pic]

17) The half life of iodine-125 is 60 days. What fraction of iodine-125 nuclides would be left after 360 days?

a. The answer is solved by creating the fraction [pic]. Where n = the

b. number of half lives. If each half life is 60 days, then in 360 days

c. there are 6 half lives. Therefore the answer is:[pic]

18) Titanium-51 decays with a half life of 6 minutes. What fraction of titanium would remain after one hour?

a. The answer is solved by creating the fraction [pic]. Where n = the

b. number of half lives. If each half life is 6 minutes, then in 1 hour (60

c. minutes) there are 10 half lives. Therefore the answer is:[pic]

19) A medical institution requests 1 g of bismuth-214, which has a half life of 20 min. How many grams of bismuth-214 must be prepared if the shipping time is 2 h?

a. In this problem you must figure out the initial amount. If you use the

b. same set up as question 3, then you can solve for the initial amount. You

c. just have to figure out n. If each half life is 20 minutes, and 2 hours (120

d. minutes) go by, then n=6. The set up is as follows:

[pic]

e. Solving for x, x = 64g.

20) The half life of radium 226 is 1602 years. What fraction of a sample radium-226 would remain after 9612 years?

a. If each half life is 1602 years, then in 9612 years

b. there are 6 half lives. Therefore the answer is:

c. [pic]

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