Question

Why is strontium-90 a particularly dangerous isotope for humans? The half-life of strontium-90 is 29.1 years. Calculate the radioactivity in millicuries of $15.6\mathrm{mg}$ of ${}^{90}\mathrm{Sr}$

Short answer

2140 mCi of radioactivity.

Step-by-step solution

Understanding the Problem

Strontium-90 is dangerous because it can mimic calcium and get incorporated into bones, where it continues to release radioactive particles. We are asked to calculate the radioactivity of 15.6 mg of Strontium-90 in millicuries.

Calculate Moles of Strontium-90

First, find the molar mass of strontium-90, which is approximately 89.91 g/mol. Convert 15.6 mg to grams, then use the molar mass to find moles: $$15.6extmg=0.0156extg$$$$extMolesof{}^{90}extSr=\frac{0.0156extg}{89.91extg/mol}\approx 1.735\times {10}^{-4}extmoles$$

Calculate the Number of Atoms

Use Avogadro's number, which is approximately $6.022\times {10}^{23}$ atoms/mol, to find the number of atoms: $$extNumberofatoms=1.735\times {10}^{-4}extmoles\times 6.022\times {10}^{23}extatoms/mol\approx 1.045\times {10}^{20}extatoms$$

Determine the Activity in Becquerels

The activity $A$ in becquerels (Bq) can be calculated using the number of atoms $N$ and the decay constant $\lambda$, where $\lambda =\frac{\ln (2)}{\text{half-life}}$:$$\lambda =\frac{\ln (2)}{29.1extyears}=\frac{0.693}{29.1\times 365\times 24\times 3600extseconds}\approx 7.56\times {10}^{-10}{\text{ s}}^{-1}$$$$A=\lambda N=7.56\times {10}^{-10}{\text{ s}}^{-1}\times 1.045\times {10}^{20}\approx 7.9\times {10}^{10}\text{ Bq}$$

Convert Activity to Millicuries

To convert becquerels to curies, use the conversion $1extCi=3.7\times {10}^{10}extBq$. Then convert curies to millicuries (1 Ci = 1000 mCi):$$A=\frac{7.9\times {10}^{10}extBq}{3.7\times {10}^{10}extBq/Ci}\approx 2.14extCi$$$$A=2.14extCi\times 1000extmCi/Ci=2140extmCi$$

Key concepts

Strontium-90

Strontium-90 is a radioactive isotope of the element strontium. It is produced as a byproduct of nuclear fission in nuclear reactors and during nuclear weapons explosions. Due to its radioactive nature, strontium-90 is hazardous, especially when it enters the human body. Strontium-90 specifically mimics calcium and is readily absorbed into the bones and teeth. This is concerning because, once in the bones, it can emit beta particles that damage bone marrow and potentially lead to bone cancer and leukemia. When considering its environmental and biological impact, understanding its behavior and pathways is essential for scientific and medical reasons.

Half-Life

The half-life of a radioactive isotope like strontium-90 is the time it takes for half of the isotope to decay. The half-life of strontium-90 is 29.1 years. This means, starting with a given amount, after 29.1 years, only half will remain while the rest will have decayed into other elements.
Radioactive decay follows an exponential pattern, meaning that the decrease in the number of radioactive atoms is proportional to the amount present. This gradual process makes strontium-90 a persistent environmental and health hazard due to its relatively long half-life, leading to prolonged exposure risks.

Decay Constant

The decay constant ($\lambda$) is a useful measure in understanding the rate of radioactive decay. It is defined as the fraction of the amount of a radioactive substance that decays per unit time. For strontium-90, the decay constant is calculated using its half-life with the formula:$$\lambda =\frac{\ln (2)}{\text{half-life}}$$Substituting the known half-life of strontium-90, 29.1 years, it can be computed as follows:$$\lambda =\frac{0.693}{29.1\times 365\times 24\times 3600\text{ seconds}}\approx 7.56\times {10}^{-10}{\text{ s}}^{-1}$$The decay constant is vital for determining the radioactive activity because it quantifies how quickly a given isotope undergoes radioactive decay.

Molar Mass

The molar mass of an element or compound is the mass of one mole of that substance. For strontium-90, an isotope of strontium, the molar mass is approximately 89.91 g/mol. This value is essential for converting a measured mass of strontium-90, like the 15.6 mg used in calculations, into moles, which is a key step in determining the total radioactivity.
To find the number of moles, convert mass from milligrams to grams and divide by the molar mass: $$15.6\text{ mg}=0.0156\text{ g}$$$$\text{Moles of }{}^{90}\text{Sr}=\frac{0.0156\text{ g}}{89.91\text{ g/mol}}\approx 1.735\times {10}^{-4}\text{ moles}$$.
Using moles, you can then determine the number of atoms and, ultimately, the activity of the strontium-90 sample, which serves as a basis for calculating its radioactivity.