In this problem, you will verify the statement (in Section...

In this problem, you will verify the statement (in Section 29.4) that the ${ }^{14} \mathrm{C}$ activity in a living sample is $0.25 \mathrm{~Bq}$ per gram of carbon. (a) What is the decay constant $\lambda$ for ${ }^{\mathrm{i} 4} \mathrm{C} ?$ (b) How many ${ }^{14} \mathrm{C}$ atoms are in $1.00 \mathrm{~g}$ of carbon? One mole of carbon atoms has a mass of $12.011 \mathrm{~g}$, and the relative abundance of ${ }^{14} \mathrm{C}$ is $1.3 \times 10^{-12}$. (c) Using your results from parts (a) and (b), calculate the ${ }^{14} \mathrm{C}$ activity per gram of carbon in a living sample.

Key Concepts

Isotopic Abundance

This refers to the relative proportion of a specific isotope present in a sample compared to the total amount of that element. It is important for determining the number of radioactive atoms in a mixture, especially when the isotope of interest is a trace component.

Avogadro's Number

Avogadro's number is a constant (approximately 6.022 x 10^23) that represents the number of atoms or molecules in one mole of a substance. It is essential for converting between atomic-scale measurements and macroscopic amounts such as grams or moles.

Activity

Activity is the measure of the rate at which a sample of radioactive material decays, expressed in decays per second (Becquerels). It is calculated as the product of the decay constant and the number of radioactive atoms present in the sample.

Half-life

The half-life of a radioactive isotope is the period over which half of a given number of nuclei decay. It is an intrinsic property of each radioactive element and is used to relate to the decay constant, offering a time scale for the decay process.

Decay Constant

The decay constant (?) is a parameter that quantifies the probability per unit time that a nucleus will decay. It is directly related to the half-life of the radioactive isotope by the formula ? = ln(2)/half-life, and it serves as a key factor in calculating the remaining quantity of the radioactive substance over a given period.

Radioactive Decay

This concept refers to the process by which unstable atomic nuclei lose energy by emitting radiation. The decay follows a statistical law where the probability of any given atom decaying is constant over time, leading to an exponential decrease in the number of undecayed nuclei.

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