In naturally occurring potassium, 0.0117 % of the nuclei are...

In naturally occurring potassium, $0.0117 \%$ of the nuclei are radioactive ${ }^{40} \mathrm{~K}$. (a) What mass of ${ }^{40} \mathrm{~K}$ is found in a broccoli stalk containing $300 \mathrm{mg}$ of potassium? (b) What is the activity of this broccoli stalk due to ${ }^{40} \mathrm{~K}$ ?

In naturally occurring potassium, $0.0117 \%$ of the nuclei are radioactive ${ }^{40} \mathrm{~K}$. (a) What mass of ${ }^{40} \mathrm{~K}$ is found in a broccoli stalk containing $300 \mathrm{mg}$ of potassium?
(b) What is the activity of this broccoli stalk due to ${ }^{40} \mathrm{~K}$ ?

Key Concepts

Isotopic Abundance

This key concept refers to the relative fraction of a specific isotope present in a naturally occurring element. In problems like the one provided, a given percentage (by number of atoms) is used to determine how much of the element is the radioactive isotope, which is essential when only part of the material contributes to radioactivity.

Mass Fraction Calculation

When a percentage is given, it can be used as a fraction to compute the mass of a specific isotope from the total mass of the element. This involves understanding how to convert a percent composition into a fraction that multiplies the total mass, even though some problems specify the percentage in terms of nuclei (number fractions) rather than mass, often assuming similar masses for isotopes.

Mole Concept and Avogadro's Number

These concepts are crucial when converting between mass and number of atoms. The mole relates the mass of a substance to the number of atoms it contains, using Avogadro's number as the bridge between atomic-scale counts and macroscopic mass quantities. This conversion is typically necessary when determining the number of radioactive atoms.

Radioactive Decay Constant

The decay constant (?) is a parameter that quantifies the probability per unit time that a single radioactive nucleus will decay. It is directly related to the half-life of the isotope by the formula ? = ln(2)/T?/?. Understanding this relationship is crucial for problems where the activity of the radioactive isotope is calculated.

Radioactive Activity

Activity refers to the rate at which a radioactive substance decays, usually measured in disintegrations per second (Becquerels) or curies. It is calculated using the formula A = ?N, where ? is the decay constant and N is the number of radioactive nuclei. This concept links the microscopic decay properties of the isotope to a measurable macroscopic quantity.

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