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 \(=\mathrm{kN}\), 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 \(28.8 \mathrm{yr}\).

Short answer

The mass of strontium-90 corresponding to the maximum permissible dose is approximately \( 5.63 \times 10^{-10} \) g.

Step-by-step solution

Find the decay constant

To find the decay constant (k), we'll use the formula: k = \( \frac{\ln{2}}{T_{1/2}} \), where \( T_{1/2} \) is the half-life of the radioactive isotope. Substitute the given half-life for strontium-90: k = \( \frac{\ln{2}}{28.8} \) yr\( ^{-1} \)

Calculate the decay rate

The maximum permissible dose of strontium-90 in the body is given as 1 µCi. We need to convert the given disintegration rate into disintegrations per second (Bq). 1 µCi = 1 x \( 10^{-6} \) Ci = \( 1 \times 10^{-6} \cdot 3.7 \times 10^{10} \) Bq = 37 Bq So, the decay rate (R) is 37 disintegrations per second.

Use the decay rate formula to find the number of atoms

The relationship between the decay rate and the number of radioactive atoms is given as R = kN. We can rearrange this formula to solve for N: N = \( \frac{R}{k} \) Now substitute the calculated decay rate (R) and the decay constant (k) into the formula: N = \( \frac{37}{\frac{\ln{2}}{28.8}} \)

Calculate the mass of strontium-90

To find the mass of strontium-90 corresponding to the calculated number of atoms, we'll use the formula: mass = \( \frac{N \cdot m_r}{N_A} \) where N is the number of atoms, m_r is the molar mass of strontium-90 (90 g/mol), and \( N_A \) is Avogadro's number (6.022 x \( 10^{23} \) mol\( ^{-1} \)). Substitute the calculated number of atoms (N) into the formula: mass = \( \frac{N \cdot 90}{6.022 \times 10^{23}} \) g Calculate the mass using the values obtained in previous steps.

Key concepts

Radioactive Decay Rate

Radioactive decay rate is a measure of how quickly unstable atoms lose their energy by emitting radiation.
For isotopes like strontium-90, this is crucial to determine the safety levels for human exposure. The decay rate typically has units of disintegrations per second, which is measured in Becquerel (Bq).
In practical terms, the rate of decay helps us understand the amount of radioactive material that will diminish over time, influencing both environmental considerations and medical applications where radioactive isotopes are used.
The exercise provided expected students to convert the permissible dose of strontium-90 to a decay rate. Improving this understanding requires a solid grasp on how the decay rate connects with the isotope's half-life and the amount of material present.

Half-life of Isotopes

The half-life of an isotope is the time it takes for half the atoms in a sample to decay.
For strontium-90, with a half-life of 28.8 years, it means that after that period, only half of the original strontium-90 atoms would remain unchanged. Understanding this concept is key to appreciating the long-term effects of radioactive materials and their management in the environment or when used in treatments. To make it easier for students, it helps to provide a real-world context, such as explaining the implications of strontium-90's half-life in nuclear waste management. Since the half-life is a constant, it's directly used to calculate the decay constant (\( k \)), which, combined with the number of atoms (\( N \)), provides the decay rate.

Avogadro's Number

Whenever we talk about atoms, molecules, or ions on a microscopic scale, Avogadro's number is our bridge to the macroscopic world.
This constant, approximately 6.022 x \( 10^{23} \) entities per mole, allows us to relate a substance's microscopic properties to its measurable, bulk amount.
In our exercise, finding the mass of strontium-90 required the use of Avogadro’s number. It helped to translate the abstract concept of individual atoms into a tangible mass that can be understood in a lab or real-world setting. For students, using Avogadro's number in calculations can reveal how incredibly tiny atoms are and how many of them can be found in even the smallest sample of material.