Textbook Question
Which nuclide in each pair would you expect to have the longer half-life? a. Cs-113 or Cs-125 b. Fe-62 or Fe-70
Ch.20 - Radioactivity and Nuclear Chemistry
Chapter 20, Problem 46
A patient is given 0.050 mg of technetium-99m, a radioactive isotope with a half-life of about 6.0 hours. How long does it take for the radioactive isotope to decay to 1.0⨉10-3 mg? (Assume no excretion of the nuclide from the body.)
Verified step by step guidance1
Identify the initial amount of technetium-99m, which is 0.050 mg, and the final amount, which is 1.0 \times 10^{-3} mg.
Use the formula for radioactive decay: N(t) = N_0 \times (1/2)^{t/t_{1/2}}, where N(t) is the final amount, N_0 is the initial amount, t is the time elapsed, and t_{1/2} is the half-life.
Substitute the known values into the equation: 1.0 \times 10^{-3} = 0.050 \times (1/2)^{t/6.0}.
Solve for t by taking the natural logarithm of both sides to isolate the exponent: \ln(1.0 \times 10^{-3}/0.050) = (t/6.0) \times \ln(1/2).
Rearrange the equation to solve for t: t = 6.0 \times \ln(1.0 \times 10^{-3}/0.050) / \ln(1/2).
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 an unstable atomic nucleus loses 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. Understanding this concept is crucial for calculating the remaining quantity of a radioactive isotope over time.
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03:00
Rate of Radioactive Decay
Half-Life
The half-life of a radioactive isotope is the time it takes for half of the initial amount of the substance to decay. For technetium-99m, the half-life is approximately 6.0 hours. This concept allows us to determine how many half-lives have passed to find the remaining quantity of the isotope after a certain period.
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02:17Zero-Order Half-life
Exponential Decay Formula
The exponential decay formula describes how the quantity of a radioactive substance decreases over time. It is expressed as N(t) = N0 * (1/2)^(t/T), where N(t) is the remaining quantity, N0 is the initial quantity, t is the elapsed time, and T is the half-life. This formula is essential for calculating the time required for the isotope to decay to a specific amount.
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Related Practice
Textbook Question
Which nuclide in each pair would you expect to have the longer half-life? a. Cs-149 or Cs-139 b. Fe-45 or Fe-52
Textbook Question
One of the nuclides in spent nuclear fuel is U-235, an alpha emitter with a half-life of 703 million years. How long will it take for the amount of U-235 to reach 10.0% of its initial amount?
Textbook Question
A radioactive sample contains 1.55 g of an isotope with a halflife of 3.8 days. What mass of the isotope remains after 5.5 days?
Textbook Question
A sample of F-18 has an initial decay rate of 1.5⨉105/s. How long will it take for the decay rate to fall to 2.5⨉103/s? (F-18 has a half-life of 1.83 hours.)