Half Life Practice Problems #3 1. The half life of Cs-137 is 30.2 years. If the initial mass of the sample is 1.00kg, how much will remain after 151 years? 2. Carbon-14 has a half life of 5730 years. Consider a sample of fossilized wood that when alive would have contained 24g of C-14. It now contains 1.5g. How old is the sample? 3. A 64g sample of Germanium-66 is left undisturbed for 12.5 hours. At the end of that period, only 2.0g remain. What is the half life of this material? 4. With a half life of 28.8 years, how long will it take 1g of strontium-90 to decay to 125mg? 5. Co-60 has a half life of 5.3 years. If a pellet that has been in storage for 26.5 years contains 14.5g of Co-60, how much of this radioisotope was present when the pellet was put in storage? 6. A 1.000kg block of phosphorus-32, which has a half life of 14.3 days, is stored for 100.1 days. At the end of this period, how much phosphorus-32 remains?
7. Radon has a half life of 3.8 days. After 7.6 days, 6.5g remain. What was the mass of the original sample? 8. A 0.5g sample of radioactive Iodine-131 has a half life of 8.0 days. After 40 days, how much is left? 9. The half life of sodium-25 is 1.0 minutes. Starting with 1 kg of this isotope, how much will remain after ½ hour. 10. What is the half life of Po-214 if after 820 seconds, a 1.0g sample decays to 0.03125g?
IV) Radioactive Decay Equations A radioactive decay equation should always have one reactant (the element decaying), a particle emitted as a product (based on the decay mode) and the new element created as a product. The new element is determined by making sure the masses (top number) on both sides of the equation are equal and the charges (bottom number) on both sides of the equation are equal. The bottom number is also the atomic number and ultimately determines the identity of the new element. Given a radioisotope, you can look on Table N to find out its decay mode. We will discuss half-life later in this unit. Rules for Equation Writing 1. You will be given a radioisotope (also known as a nuclide). Since it decays on its own, write it down as the only reactant. You may have to determine the atomic number off of the Periodic Table. Add an arrow. 2. Using Table N, determine the decay mode. 3. Using Table O, determine the full notation of the emitted decay particle from Table N. 4. Write this notation for the particle to the right of the arrow as a product. 5. The top numbers (mass) on each side of the equation must be equal on both sides. Determine the top number of the new element. 6. The bottom numbers (nuclear charge) must be equal on both sides. Determine the bottom number of the new element. 7. Using the lower number (atomic number) and your Periodic Table, write the symbol of the new element. Examples: Read through the examples step by step so that you fully understand how to write radioactive decay equations. Writing Decay Equations Worksheet 1. Compare the decay modes of K-37 and K-42, according to Table N. 2. Write the decay equation for K-37.
3. Write the decay equation for K-42. 4. Which of the following would be more dangerous, radon-222 or cobalt-60 in the same dosage? Explain. 5. Write the decay equation for radon-222. 6. Write the decay equation for cobalt-60. V) Half-life A) What is it? The amount of time it takes for half of a sample s nuclei to decay. The amount of time for half of the sample to decay, regardless of amount, remains CONSTANT. It is important that you realize that half of the nucleus doesn t decay, but half of the atoms in a given sample decay during after one half-life. B) Who cares? What s interesting about half-life is that no matter how many atoms you start with, half of them will decay after one half-life. After a second half-life, half of the atoms that remain after the first decay, decay themselves. After a third halflife half of the atoms that remain after the second decay, decay themselves and so on. Note: the time that passes after each half-life does NOT change! (see the graph below) Although from a mathematical standpoint you wouldn t expect to ever reach zero where all atoms are decayed, in reality, eventually all atoms will decay and you WILL reach zero.
C) How do I know what a radioisotopes half-life is? Use Table N to look up the half-life for each nuclide. Each radioisotope has its own distinct half-life that cannot be changed. Changes in temperature, pressure, concentration, surface area, and catalysts do NOT affect the decay rate and thus do NOT affect the half-life. D) What does the value for half-life tell me? Each radioisotope has its own distinct half-life value which tells you how long it takes for half of the sample to decay. Radioisotopes with short half-lives are less dangerous because they decay more quickly. Radioisotopes with long half-lives are more dangerous because they decay slowly and thus, stay radioactive longer. This is a particular problem with radioactive waste. Highly recommended: check out the following web link for a nice review with good visuals. Try out the practice quiz for extra practice. http://www.darvill.clara.net/nucrad/hlife.htm Half-life Concepts Worksheet 1. What is half-life? 2. After three half-lives, what fraction of the original sample would remain? Explain. 3. A 50 gram sample of a radioisotope undergoes 2 half-lives. How many grams would remain at the end? Explain. 4. Name the element on Table N that has the shortest half-life that undergoes alpha decay.
5. Would the nuclide from #3 be dangerous to living things based on its half-life? Why or why not? 6. Which radioisotope on Table N has the longest half-life? 7. Would the nuclide from #5 be dangerous to living things based on its half-life? Why or why not? 8. A radioactive sample of C-14 is heated. How is the half-life affected? VI) Half-life Calculations Check out the following example problems shown below. Use them as models for the worksheet that follows. Ex 1) How many grams remain of a 10.0 gram sample of cobalt-60 after 15.78 years? Step 1: Determine the half-life from Table N: 5.26 years Step 2: Calculate the number of half-lives that have passed: # of half-lives = total time/1 half-life = 15.78 y/5.26 y = 3 half-lives Step 3: Set up a chart that shows the amount decaying by ½ with the passing of each half-life. Hint: Make sure that the initial amount is always before the first half-life at 0. # of half-lives 0 1 2 3 Fraction remaining 1 1/2 1/4 1/8 Grams remaining 10.0 g 5.00 g 2.50 g 1.25 g Step 4: State the final answer: After 15.78 years (3 half-lives), 1.25 grams of cobalt-60 remain. Ex 2) What mass of iodine-131 remains 32 days after a 100. g sample of the isotope is obtained. Step 1: Determine the half-life from Table N: 8.07 days Step 2: Calculate the number of half-lives that have passed: # of half-lives = total time/1 half-life = 32 d/8.07 d = 4 half-lives Step 3: Set up a chart that shows the amount decaying by ½ with the passing of each half-life. Hint: Make sure that the initial amount is always before the first half-life at 0. # of half-lives 0 1 2 3 4 Fraction remaining 1 1/2 1/4 1/8 1/16 Grams remaining 100. g 50.0 g 25.0 g 12.5 g 6.25 g
Step 4: State the final answer: After 32 days (4 half-lives), 6.25 grams of iodine-131 remain. Ex 3) A sample of an isotope decays from 100. grams to 25.0 grams in 20 days. What is the halflife of the unknown radioisotope? Step 1: Determine the number of half-lives with a chart showing the decay of the original amount to the final amount: # of half-lives 0 1 2 Fraction remaining 1 1/2 1/4 Grams remaining 100. g 50.0 g 25.0 g Thus, 2 half-lives have occurred. Step 2: Divide the total time by the number of half-lives to obtain the half-life: half-life = total time/# of half-lives = 20 d/2 half-lives = 10 days Step 3: State the final answer: The half-life (time for half to decay) is 10 days. Ex 4) If a sample of Sr-90 has a mass of 100. grams, how long will it take to decay such that only 12.5 grams of Sr-90 remains? Step 1: Determine the half-life from Table N: 28.1 years Step 2: Determine the number of half-lives with a chart showing the decay of the original amount to the final amount: # of half-lives 0 1 2 3 Fraction remaining 1 1/2 1/4 1/8 Grams remaining 100. g 50.0 g 25.0 g 12.5 g Thus, 3 half-lives have occurred. Step 3: Determine the total time by multiplying the half-life by the number of halflives: 28.1 y x 3 = 84.3 y State 4: State the final answer: It takes 84.3 years for Sr-90 to decay from 100. grams to 12.5 grams.
Half-life Calculations 1. How much of a 100. gram sample of 198 Au is left after 8.10 days? 2. A 50.0 gram sample decays to 12.5 grams in 14.4 seconds. What is its half-life? 3. How much of a 750.00 gram sample of K-42 is left after 62.00 hours? 4. What is the half-life of an element if a 500. gram sample decays to 62.5 grams in 639,000 years? 5. If there are 25.0 grams of a sample of thorium-232 left after 2.80 x 10 10 years, how many grams were in the original sample? 6. How long will it take for 16 grams of Ra-226 to break down until only 1.0 grams remain?