Problem 79 - Chemistry: The Central Science

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 \(1 \times 10^{-6} \mathrm{Ci}\) is approximately \(1.539 \times 10^{6}\) atoms. The mass of strontium-90 this corresponds to is approximately \(2.32 \times 10^{-16}\) grams.

Step 1: Calculate decay constant k

To find the decay constant k, we can use the formula: \[k = \frac{\ln{2}}{t_{1/2}},\] where \(t_{1/2}\) is the half-life of the radioactive material. Given the half-life of strontium-90, t_{1/2} = 28.8 years, we can plug this into the formula and calculate k: \[k = \frac{\ln{2}}{28.8 \mathrm{yr}} \approx 2.406 \times 10^{-2} \mathrm{yr}^{-1}.\]

Step 2: Calculate the number of atoms N

The relationship between activity rate, decay constant, and number of atoms is given as: \[ \text{rate} = kN. \] Now we can plug in the maximum permissible dose (\(1 \times 10^{-6} \mathrm{Ci}\)), the calculated decay constant, and solve for the number of atoms N. Notice that we need to convert the activity from curie to Becquerel (Bq), as 1 Ci = \(3.7 \times 10^{10}\) Bq. \[1 \times 10^{-6} \mathrm{Ci} = 3.7 \times 10^4 \mathrm{Bq}.\] \[3.7 \times 10^4 \mathrm{Bq} = (2.406 \times 10^{-2} \mathrm{yr}^{-1})N.\] Now solve for N: \[N \approx 1.539 \times 10^{6} \text{ atoms}.\]

Step 3: Calculate the mass of strontium-90

To find the mass of strontium-90, we can use the formula: \[ m = \frac{N \times M}{N_A}, \] where m is the mass of strontium-90, N is the number of atoms, M is the molar mass of strontium-90, and \(N_A\) is Avogadro's number (\(6.022 \times 10^{23} \mathrm{mol}^{-1}\)). The molar mass of strontium-90 is 89.907 g/mol. Plugging in the values, we get: \[ m = \frac{(1.539 \times 10^{6} \text{ atoms}) \times (89.907 \mathrm{g/mol})}{(6.022 \times 10^{23} \mathrm{mol}^{-1})} \approx 2.32 \times 10^{-16} \mathrm{g}. \] Therefore, the mass of strontium-90 corresponding to the maximum permissible dose is approximately \(2.32 \times 10^{-16}\) grams.

Key Concepts

Decay Constant

When trying to understand the rate at which a radioactive substance decreases in time, we turn to a key concept known as the decay constant. This value, usually denoted by the symbol k, represents the probability of a single atom decaying per unit time. In simpler terms, it is a measure of the speed of the radioactive decay process.

As shown in the provided exercise, to calculate the decay constant, one commonly used formula involves the half-life of the substance, which is the time required for half of the substance to decay. The equation \[k = \frac{\ln{2}}{t_{1/2}},\] connects the decay constant to the half-life. Practically, finding k gives us a value that is crucial for further calculations regarding the substance's behavior over time.

Half-Life

The term half-life is a central concept in the field of nuclear physics and radiochemistry. It is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. Half-life is denoted as \(t_{1/2}\) and it is a specific characteristic of each radioactive isotope.

In our context, knowing the half-life of strontium-90 is essential to understand how long the substance remains active and to calculate the decay constant. The longer the half-life, the slower the rate of decay. The half-life can also offer insights into the appropriate safety measures needed when handling radioactive materials, as it informs us about the time scale on which radioactive exposure decreases.

Strontium-90

Strontium-90 is a radioactive isotope that is generated as a byproduct of nuclear fission, found in nuclear fallout. It has a half-life of approximately 28.8 years, which makes it a considerable hazard due to its potential longevity in the environment.

The isotope is of particular concern in nuclear safety and medicine. In the body, strontium can mimic calcium, being incorporated into bone structure, and may lead to various bone disorders and diseases. Therefore, understanding and calculating the permissible levels of strontium-90, like in our exercise, is critical for safeguarding human health.

Activity Rate in Becquerel

The activity rate of a radioactive material is a measure of the decay events that occur per second. It is communicated in units of Becquerel (Bq), where one Becquerel is equal to one decay per second. This unit of measurement is named after Henri Becquerel, one of the discoverers of radioactivity.

Using the activity rate, scientists and engineers can calculate the number of atoms disintegrating each second, which is vital for assessing the level of radiation exposure and for calculating the dose of radioactive isotopes in medical applications. Converting other units such as curie (Ci) to Becquerel is crucial, as shown in the steps of the solution, for standardizing measurements across different applications and regions.

Avogadro's Number

In the realm of chemistry and physics, Avogadro's number is a fundamental constant. It is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Avogadro's number, denoted as \(N_A\), is approximately \(6.022 \times 10^{23}\) entities per mole.

This number plays a crucial role when dealing with substances at the atomic or molecular scale, as it allows for the translation between microscopic measurements and macroscopic amounts of material. For instance, in the exercise provided, Avogadro's number is used to convert the number of strontium-90 atoms to grams, thereby bridging the gap between atomic scale calculations and tangible, real-world quantities.