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^-3mg? (Assume no excretion of the nuclide from the body.)

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780137832217

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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).

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.

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.

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.

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.)

Key Concepts

Radioactive Decay

Half-Life

Exponential Decay Formula

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^-3mg? (Assume no excretion of the nuclide from the body.)