Niobium-91 has a half-life of 680 years. After 2,040 years, how much niobium-91 will remain from a $300.0-g$ sample?
A. 3 g
B. 18.75 g
C. 37.5 g
D. 100.0 g
Answer (2)
Determine the number of half-lives that have passed: 680 2040 = 3 .
Apply the half-life formula: R e mainin g A m o u n t = I ni t ia l A m o u n t × ( 2 1 ) N u mb er O f H a l f L i v es .
Substitute the given values: R e mainin g A m o u n t = 300.0 × ( 2 1 ) 3 .
Calculate the remaining amount: R e mainin g A m o u n t = 37.5 g . The final answer is 37.5 g .
Explanation
Problem Setup We are given that Niobium-91 has a half-life of 680 years. We start with a 300.0-g sample and want to find out how much remains after 2,040 years.
Calculating Number of Half-Lives First, we need to determine how many half-lives occur in 2,040 years. We can calculate this by dividing the total time (2,040 years) by the half-life (680 years): 680 years/half-life 2040 years = 3 half-lives So, 3 half-lives have passed.
Applying the Half-Life Formula After each half-life, the amount of Niobium-91 is halved. So, after 1 half-life, 1/2 of the original amount remains. After 2 half-lives, 1/2 of that amount remains, and so on. We can calculate the remaining amount using the formula: R e mainin g A m o u n t = I ni t ia l A m o u n t × ( 2 1 ) N u mb er O f H a l f L i v es In this case, the initial amount is 300.0 g, and the number of half-lives is 3.
Calculating Remaining Amount Now, we plug in the values into the formula: R e mainin g A m o u n t = 300.0 g × ( 2 1 ) 3 R e mainin g A m o u n t = 300.0 g × 8 1 R e mainin g A m o u n t = 37.5 g Therefore, after 2,040 years, 37.5 g of Niobium-91 will remain.
Final Answer After 2,040 years, 37.5 g of Niobium-91 will remain from the initial 300.0-g sample.
Examples
Half-life calculations are crucial in various fields, such as medicine and archaeology. For instance, in medicine, radioactive isotopes with specific half-lives are used for diagnostic imaging and cancer treatment. Understanding half-lives helps doctors determine the appropriate dosage and timing of these treatments to minimize harm to the patient. In archaeology, carbon-14 dating, which relies on the half-life of carbon-14, is used to estimate the age of ancient artifacts and fossils, providing insights into past civilizations and ecosystems. The principles of radioactive decay and half-life are fundamental in these applications, enabling precise measurements and informed decision-making.
After 2040 years, 37.5 g of Niobium-91 will remain from the initial 300.0 g sample. This is determined by calculating the number of half-lives that have passed and applying the half-life formula. The correct answer is therefore C. 37.5 g.
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