### Half-Life: Examples and Problems only

Example #1: How many years will it take for 88.0 grams of tritium to decay to an 11.0 gram sample? (The half-life of tritium is 12.3 years.)

Example #2: How many years will it take for 84.0 grams of tritium to decay to a 23.5 gram sample? (The half-life of tritium is 12.3 years.)

Example #3: A 208 g sample of sodium-24 decays to 13.0 g of sodium-24 in 60.0 hr. What is the half-life of this radioactive isotope?

Example #4: Calculate the half life of a sample of ChemTeamium which decays from 27.5 g to 0.598 g in a period of 574 years.

Example #5: The radioactive element carbon-14 has a half-life of 5730. years. The percentage of carbon-14 present in the remains of plants and animals can be used to determine age. How old is an object that has lost 60% of its carbon-14?

Example #6: An element has a half-life of 25 years. You currently have 20 grams. How many grams did you have 100 years ago?

Example #7: P-32 is radioactive isotope with a half-life of 14.3 days. If you currently have 73.3 g of P-32, how much P-32 was present 4 days ago?

Example #8: If a sample contains 144 atoms of Au-179 (half-life = 7.5 s), approximately how many such atoms were present 30 seconds earlier?

Example #9: Find the half-life of an element which decays at a rate of 3.402% per day.

Example #10: The half-life of In-111 is 0.007685 years; how long (in hours) would it take for the amount of In-11 to decrease to 43.24% of its initial amount?

Bonus Example: A scrap of paper taken from a Dead Sea scroll was found to have a C-14/C-12 ratio of 0.795 times than found in plants living today. Estimate the age of the scroll.

Problem #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?

Problem #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?

Problem #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?

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

Problem #5: A radioactive isotope decayed to 17/32 of its original mass after 60 minutes. Find the half-life of this radioisotope.

Problem #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?

Problem #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?

Problem #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?

Problem #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?

Problem #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.

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?

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?

Problem #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 atoms? The half-life of Cs-137 is 30.17 years.

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?

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?

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

Problem #17:

Problem #17: One-eighth of a radioactive sample remains 9 days after it was brought into the lab. What is the half-life?

Problem #18: A sample of Se-83 registers 1012 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?

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?

Problem #20: If you start with 2.97 x 1022 atoms of molybdenum-99 (half-life = 65.94 hours), how many atoms will remain after one week?

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

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?

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 47CaCO3 must the scientist order?

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

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?

Bonus Problem: A rock is found contains uranium-238 and also lead-206. Scientist analyze the rock for these two elements and find that the total mass of uranium in the rock is 2.40 g, while the amount of lead is 1.11 g. How old is this rock?

Problem #26: The half life in two different samples, A and B, of radio-active nuclei are related according to T(1/2,B) = T(1/2,A)/2. In a certain period the number of radio-active nuclei in sample A decreases to one-fourth the number present initially. In the same period the number of radio-active nuclei in sample B decreases to a fraction f of the number present initially. Find f.

Problem #27: You have 20.0 grams of P-32 that decays 5% daily. How long will it take for half the original to decay?

Problem #28: A sample of radioactive isotopes contains two different nuclides, labeled A and B. Initially, the sample composition is 1:1, i.e., the same number of nuclei A as nuclei B. The half-life of A is 3 hours and, that of B, 6 hours. What is the expected ratio A/B after 18 hours?

Problem #29: The ratio of tritium, H-3, to hydrogen, H-1, in a sample of water was 1:1x1019. If the half life of tritium is 12.25 years, calculate the actual number of tritium atoms remaining in 10.0 g water after a period of 49 years.

Problem #30: The isotope Ra-226 decays to Pb-206 in a number of stages which have a combined half-life of 1640 years. Chemical analysis of a certain chunk of concrete from an atomic-bombed city, preformed by an archaeologist in the year 6264 AD, indicated that it contained 2.50 g of Ra-226 and 6.80 g of Pb-206. What was the year of the nuclear war?

Problem #31: A radioactive sample contains 3.25 x 1018 atoms of a nuclide that decays at a rate of 3.4 x 1013 disintegrations per 26 min.

(a) What percentage of the nuclide will have decayed after 159 days?
(b) What is the half-life of the nuclide?

Problem #32: The radioisotope potassium-40 decays to argon-40 by positron emission with a half life of 1.27 x 109 yr. A sample of moon rock was found to contain 78 argon-40 atoms for every 22 potassium-40 atoms. What is the age of the rock?

Problem #33: What is the age of a rock in which the mass ratio of Ar-40 to K-40 is 3.8? K-40 decays to Ar-40 with a half-life of 1.27 x 109 yr.

Problem #34: A radioactive isotope has a half-life of 4.5 days. What fractions of the sample will exist after 9 and 18 days?

(a) 1/2 and 1/4 of the original amount
(b) 1/9 and 1/18 of the original amount
(c) 1/4 and 1/16 of the original amount
(d) 1/4 and 1/8 of the original amount

Problem #35: Selenium-75 has a half-life of 120 days and is used medically for pancreas scans. (a) Approximately how much selenium-75 would remain of a 0.050 g sample that has been stored for one year? (b) How long would it take for a sample of selenium-75 to lose 99% of its radioactivity?

Problem #36: Natural samarium (average atomic mass 150.36) contains 14.99% of the radioactive isotope Sm-147. A 1 g sample of natural Sm has an activity of 89 decays per second. Estimate the half life of Sm-147.

Problem #37: You measure the radioactivity of a substance, then when measuring it 120 days later, you find that it only has 54.821% of the radioactivity it had when you first measured it. What is the half life of that substance?

Problem #38: Arsenic-74 is a medical radioisotopes with a half-life of 18 days. If the initial amount of arsenic-74 injected is 2.30 mCi, how much arsenic-74 is left in the body after 54 days?

Problem #39: Assume that today there is 10 grams of substance, while 1000 years ago there was 100 grams of it. If there is 15grams of substance today, how much will there be 600 years from now?

Problem #40: A radionuclide, cobalt-60, has a half-life of 5.27 years. How many hours would it take for the activity to diminish to one-thirty-secondth (3.125%) of its original value?

Problem #41: A chemist determines that a sample of petrified wood has a carbon-14 decay rate of 6.00 counts per minute per gram. What is the age of the piece of wood in years? The decay rate of carbon-14 in fresh wood today is 13.6 counts per minute per gram, and the half life of carbon-14 is 5730 years.

Problem #42: The carbon-14 decay rate of a sample obtained from a young tree is 0.296 disintegration per second per gram of the sample. Another wood sample prepared from an object recovered at an archaeological excavation gives a decay rate of 0.109 disintegration per second per gram of the sample. What is the age of the object?

Problem #43: The C-14 content of an ancient piece of wood was found to have three tenths of that in living trees (indicating 70% of the C-14 had decayed). How old is that piece of wood?

Problem #44: Carbon-14 is used to determine the age of ancient objects. If a sample today contains 0.060 mg of carbon-14, how much carbon-14 must have been present in the sample 11,430 years ago?

Problem #45: Determine the age of a sample of charcoal which is giving off 25 counts per hour, if carbon-14 from a just made piece of charcoal gives off 85 counts per hour. The half life of carbon-14 is 5730 years.

Problem #46: A living plant contains approximately the same isotopic abundance of C-14 as does atmospheric carbon dioxide. The observed rate of decay of C-14 from a living plant is 15.3 disintegrations per minute per gram of carbon. How many disintegrations per minute per gram of carbon will be measured from a 12900-year-old sample? (The half-life of C-14 is 5730 years.)

Problem #47: All current plants have a C-14 count of 15.3 cpm. How old is a wooden artifact if it has a count of 9.58 cpm?

Problem #48: Using dendrochronology (using tree rings to determine age), tree materials dating back 10,000 years have been identified. Assuming you had a sample of such a tree in which the number of C-14 decay events was 15.3 decays per minute before decomposition, what would the decays per minute be in the present day?

Problem #49: You read that a fossil dinosaur skull has been found in Montana and that it has been carbon-14 dated to be 73 million years old. Provide two (2) scientifically-based reasons to explain why C-14 dating cannot do this.

Problem #50: A mammoth skeleton has a carbon-14 decay rate of 0.0077 disintegrations per second per gram of carbon. How long ago did the mammoth live? (Assume that living organisms have a carbon-14 decay rate of 0.255 s¯11 and that carbon-14 has a half-life of 5730 y.)

Problem #51: All living things have a steady state C-14 activity of 15.3 atoms min¯11. Suppose a sample of cloth from an archeological site shows an activity of 0.372 atoms min¯11. How old is the cloth?

Problem #52: The smallest C-14 activity that can be measured is about 0.20%. If C-14 is used to date an object, the object must have died within how many years?

Problem #53: How long will it take for 25% of the C-14 atoms in a sample of C-14 to decay?

Problem #54:

Problem #55:

Problem #56: U-238 has a half-life of 4.468 x 109 years. How much U-238 should be present in a sample 2.50 x 109 years old, if 2.00 grams was present initially? (I found the U-238 half-life value here.)

Problem #57: A sample of rock is known to contain the isotopes U-238 and Pb-206 in the mass ratio of 2:1. What is the age of the sample assuming all the Pb-206 has originated from the decay of U-238? The half-life of U-238 is 4.468 x 109 years.

Problem #58: A sample of rock contains 64.27 mg of U-238 (isotopic mass 238.0508 amu) and 22.66 mg of Pb-206 (isotopic mass 205.9745 amu). Assuming that radioactive decay of U-238 is the only source of Pb-206, estimate the rock's age. Half life of U-238 radioactive disintegration is = 4.468 x 109 years.

Problem #59: A rock is found contains uranium-238 and also lead-206. Scientist analyze the rock for these two elements and find that the total mass of uranium in the rock is 2.40 g, while the amount of lead is 1.11 g. How old is this rock?

Problem #60: If the half-life of 238-U is 4.468 x 109 y and the half-life of 235-U is 7.1 x 108 y and the age of the Earth is 4.5 x 109 y and if the percentage of 238-U in the Earth is 99.3% and 235-U is 0.7% then what were their percentages when the Earth was formed?

Problem #61: The half-life for the following process is 4.468 x 109 yr.

U-238 ---> Pb-206

A mineral sample contains 43.20 mg of U-238 and 14.50 mg of Pb-206. What is the age of the mineral?

Problem #62: A rock from Australia was found to contain 0.435 g of Pb-206 to every 1.00 g of U-238. Assuming that the rock did not contain any Pb-206 at the time of its formation, how old is the rock? The half life of U-238 is 4.468 x 109 years.

Problem #63: How much Pb-206 will be in a rock sample that is 1.3 x 108 years old and that contains 3.25 mg of U-238?

Problem #64: U-238 has a half-life of 4.468 x 109 years. Estimates of the age of the universe range from 9 x 109 years to 23 x 109 years. 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 109 years.

Problem #65: Estimate the age of rocks in which the mole ratio of U-238 to Pb-205 is 0.75. The half life of U-238 is 4.468 x 109 yr.