Textbook Question
A sample of 37Ar undergoes 8540 disintegrations/min initially but undergoes 6990 disintegrations/min after 10.0 days. What is the half-life of 37Ar in days?
Ch.20 - Nuclear Chemistry
Chapter 20, Problem 62
The electronic systems on the New Horizons spacecraft, which launched on January 19, 2006, and reached its closest approach to Pluto on July 14, 2015, were powered by elec-tricity generated by heat. The heat came from the radioac-tive decay of 238Pu in the 11 kg of 238PuO2 fuel onboard. The generator provided 240 W when the spacecraft was launched. If the power output is directly proportional to the amount of 238Pu in the generator, what was the power output when the spacecraft reached Pluto? The half-life of 238Pu is 87.7 y.
Verified step by step guidance1
Calculate the time elapsed from the launch of the spacecraft to its closest approach to Pluto. Subtract the launch date from the date of closest approach.
Use the half-life formula to determine the remaining amount of 238Pu after the elapsed time. The formula to use is: \( N = N_0 \times (1/2)^{\frac{t}{T}} \), where \( N_0 \) is the initial amount of 238Pu, \( t \) is the time elapsed, and \( T \) is the half-life of 238Pu.
Since the power output is directly proportional to the amount of 238Pu, set up a proportion based on the initial power output and the initial amount of 238Pu to find the power output at the time of closest approach. Use the formula: \( P = P_0 \times \frac{N}{N_0} \), where \( P_0 \) is the initial power output.
Substitute the values from the previous steps into the proportion formula to find the new power output when the spacecraft reached Pluto.
Verify the units and make sure the final power output is in watts (W).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radioactive Decay
Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive substance to decay. In this case, the half-life of 238Pu is 87.7 years, meaning that after this period, half of the original amount of 238Pu will remain.
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Rate of Radioactive Decay
Half-Life
Half-life is a key concept in nuclear chemistry that quantifies the time it takes for half of a sample of a radioactive substance to decay. For 238Pu, with a half-life of 87.7 years, this means that after 87.7 years, only 50% of the original 238Pu will be left. This concept is crucial for calculating the remaining amount of 238Pu in the New Horizons spacecraft after its journey to Pluto.
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Power Output and Proportionality
Power output in this context refers to the amount of electrical power generated by the decay of 238Pu. The problem states that the power output is directly proportional to the amount of 238Pu present. This means that as the quantity of 238Pu decreases due to radioactive decay, the power output will also decrease in a predictable manner, allowing for calculations based on the remaining amount of 238Pu after a certain time.
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Related Practice
Textbook Question
Radioactive decay exhibits a first-order rate law, rate = -∆N/∆t = kN, where N denotes the number of radio-active nuclei present at time t. The half-life of strontium-90, a dangerous nuclear fission product, is 29 years.(a) What fraction of the strontium-90 remains after three half-lives?
Textbook Question
Potassium ion, K+, is present in most foods and is an essen-tial nutrient in the human body. Potassium-40, however, which has a natural abundance of 0.0117%, is radioactive with t1/2 = 1.25 x 10^9 years. What is the decay constant of 40K? How many 40K+ ions are present in 1.00 g of KCl? How many disintegration/s does 1.00 g of KCl undergo?
Textbook Question
Uranium-238 has a half-life of 4.47 * 109 years and decays through a series of events to yield lead-206. Estimate the age of a rock that contains 105 mmol of 238U and 33 mmol of 206Pb. Assume all the 206Pb is from the decay of 238U.