A sample 99mTc used for a whole body bone scan has an activity of 600 MBq. If the half-life is 6.01 hours, what mass of 99mTc was injected? (a) 3.1 ng (b) 8.4 microgram (c) 67 microgram (d) 2.7 mg

Name: Chemistry
Author: McMurry
ISBN: 9781292336145

검증된 단계별 안내

Step 1: Understand the relationship between activity, decay constant, and number of atoms. The activity (A) of a radioactive sample is related to the number of radioactive atoms (N) and the decay constant (λ) by the equation A = λN.

Step 2: Calculate the decay constant (λ) using the half-life (t_{1/2}). The decay constant is given by the formula λ = \(\frac{0.693}{t_{1/2}\)}, where t_{1/2} is the half-life of the isotope.

Step 3: Rearrange the activity equation to solve for the number of atoms (N). Use the equation N = \(\frac{A}{λ}\), where A is the activity of the sample.

Step 4: Convert the number of atoms (N) to moles. Use Avogadro's number (6.022 \(\times\) 10^{23} atoms/mol) to convert the number of atoms to moles.

Step 5: Calculate the mass of 99mTc. Use the molar mass of 99mTc (approximately 99 g/mol) to convert the moles of 99mTc to mass.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Radioactive Decay and Half-Life

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. The half-life is the time required for half of the radioactive atoms in a sample to decay. Understanding half-life is crucial for calculating the remaining activity of a radioactive substance over time, which is essential in determining the initial amount of a radioactive isotope based on its current activity.

Activity and Its Units

Activity refers to the rate at which a sample of radioactive material decays, measured in becquerels (Bq) or megabecquerels (MBq). One megabecquerel equals one million disintegrations per second. In this question, the activity of 600 MBq indicates how many disintegrations are occurring, which can be used alongside the half-life to calculate the initial mass of the radioactive isotope present.

Mass-Energy Relationship in Radioactive Isotopes

The mass of a radioactive isotope can be calculated using its activity and half-life, applying the relationship between the number of radioactive atoms, their decay rate, and the isotope's molar mass. By using the decay constant derived from the half-life, one can determine the number of atoms present and subsequently convert this to mass using the molar mass of the isotope, which is essential for solving the problem.