Problem 79 - Chemistry: The Central Science in SI Units, Global Edition

Question

According to current regulations, the maximum permissible dose of strontium-90 in the body of an adult is \(1 \mu \mathrm{Ci}\left(1 \times 10^{-6} \mathrm{Ci}\right)\). Using the relationship rate \(=k N\), calculate the number of atoms of strontium-90 to which this dose corresponds. To what mass of strontium-90 does this correspond? The half-life for strontium-90 is \(28.8 \mathrm{yr}\)

Step-by-Step Solution

Answer

The number of strontium-90 atoms corresponding to the maximum permissible dose of 1 µCi for an adult is approximately \(1.54 \times 10^{6}\) atoms. The mass of strontium-90 corresponding to this number of atoms is approximately \(2.3 \times 10^{-16}\) grams.

Step 1: Calculate the decay constant k

We will use the formula for decay constant based on half-life, which is given by: \(k = \frac{ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of strontium-90. Substitute the given half-life value for strontium-90 (28.8 years) and calculate the decay constant k: \(k = \frac{ln(2)}{28.8}\) \(k = 0.0241 \, \mathrm{yr}^{-1}\)

Step 2: Calculate the number of strontium-90 atoms, N

We will use the relationship between activity (rate) and the number of radioactive atoms, given by: \(A = kN\), where A is the activity, k is the decay constant, and N is the number of strontium-90 atoms. The maximum permissible activity (rate) for an adult is given as 1 µCi = \(1 \times 10^{-6}\) Ci. We must convert this to decays per year. Knowing that 1 Ci = \(3.7 \times 10^{10} \, \mathrm{decays/s}\), we can convert to decays per year: \(\mathrm{1 \, µCi \times 10^{-6} \, Ci/µCi \times 3.7 \times 10^{10} \, decays/s/Ci \times (365 \times 24 \times 3600) \, s/yr}\) \(\mathrm{A = 3.7 \times 10^4 \, decays/yr}\) Now, we can solve for the number of strontium-90 atoms, N: \(N = \frac{A}{k}\) \(\mathrm{N = \frac{3.7 \times 10^4 \, decays/yr}{0.0241 \, yr^{-1}}}\) \(N = 1.54 \times 10^{6} \, \mathrm{atoms}\)

Step 3: Calculate the mass of strontium-90

To calculate the mass of strontium-90, we will use the number of atoms found in step 2, the molar mass of strontium-90 (89.907 g/mol), and Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol). First, find the number of moles of strontium-90: \(\mathrm{moles = \frac{1.54 \times 10^{6} \, atoms}{6.022 \times 10^{23} \, atoms/mol}}\) \(\mathrm{moles = 2.56 \times 10^{-18} \, mol}\) Now, calculate the mass of strontium-90 using the molar mass: \(\mathrm{mass = 2.56 \times 10^{-18} \, mol \times 89.907 \, g/mol}\) \(\mathrm{mass = 2.3 \times 10^{-16} \, g}\) The mass of strontium-90 corresponding to the maximum permissible dose of 1 µCi for an adult is approximately \(2.3 \times 10^{-16}\) grams.

Key Concepts

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This happens when the forces within an atom's nucleus become unbalanced, transforming that atom into a more stable form. Understanding this concept is fundamental in nuclear chemistry, as it explains why some elements, like strontium-90, are hazardous to humans. A few key points about radioactive decay:

• It involves the transformation of unstable isotopes into stable ones.
• The process can result in the emission of alpha, beta, or gamma radiation.
• It follows a first-order kinetic process, meaning the rate of decay is directly proportional to the number of radioactive nuclei present.

For strontium-90, knowing its decay characteristics is crucial, especially when determining safe exposure levels. The decay constant, derived from the half-life formula, helps calculate the rate at which an isotope will decay, ensuring safety standards are not surpassed.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a crucial concept when studying radioactive materials. Each isotope has a unique half-life, giving us insight into how quickly a substance will become less radioactive. Here’s why half-life is important:

• Provides a measure of an isotope's stability.
• Aids in calculating the time over which a radioactive sample remains hazardous.
• Essential for dating ancient artifacts and understanding nuclear processes.

In the case of strontium-90, its half-life is 28.8 years, meaning it takes this long for half of any given amount of strontium-90 to decay. This factor is instrumental in calculations like determining the decay constant and predicting future radioactivity levels.

Molar Mass

Molar mass is the mass of one mole of a substance, measured in grams per mole (g/mol). This concept is key in converting between the mass of a substance and its number of atoms or molecules. Knowing the molar mass of strontium-90, which is 89.907 g/mol, allows for the conversion from the number of atoms to actual mass, an essential step in many chemistry calculations. Why molar mass matters:

• Facilitates the conversion between mass and moles in chemical reactions.
• Essential in determining the proportions of elements in compounds.
• Helps calculate the amount of a substance needed or produced in a reaction.

Especially in nuclear chemistry, calculating the molar mass helps in understanding the potential impact of radioactive substances based on their atomic structure and mass.

Avogadro's Number

Avogadro's number is a constant used in chemistry to quantify the number of particles in a mole, equal to approximately \(6.022 imes 10^{23}\) atoms/mol. It bridges the gap between the microscopic and macroscopic understanding of matter. This number is pivotal when converting between moles of a substance and its count of atoms or molecules.A couple of reasons Avogadro's number is significant:

• Allows chemists to count particles by weighing them.
• Helps in performing precise chemical calculations and reactions.
• Enables the scaling of laboratory reactions to industrial processes.

Using Avogadro’s number simplifies the conversion from the number of radioactive atoms, like those of strontium-90, into moles, assisting in the calculation of their corresponding mass and ensuring accurate chemical analysis.