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Ch.21 - Radioactivity & Nuclear Chemistry
Tro - Chemistry: A Molecular Approach 6th Edition
Tro6th EditionChemistry: A Molecular ApproachISBN: 9780137832217Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTro 6th EditionCh.21 - Radioactivity & Nuclear ChemistryProblema 48
Capitolo 21, Problema 48

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.)

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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).

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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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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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Zero-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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Beta Decay
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