Problem 91 - Chemistry The Molecular Nature of Matter

Question

Iodine- 131 is a radioisotope present in radioactive fallout that targets the thyroid gland. If \(1.00 \mathrm{mg}\) of \({ }^{131} \mathrm{I}\) has an activity of \(4.6 \times 10^{12} \mathrm{~Bq}\), what is the decay constant for 131 I? What is the half-life, in seconds?

Step-by-Step Solution

Answer

The decay constant for \(^{131}I\) is approximately \(\lambda = 3.1 \times 10^{-6} s^{-1}\) and the half-life is about \(t_{1/2} = 2.24 \times 10^{5} s\).

Step 1: Understanding Activity

The activity (A) of a radioactive isotope is defined by the equation A = \(\lambda N\), where \(\lambda\) is the decay constant and N is the number of atoms. Since we know the activity (A = \(4.6 \times 10^{12}\) Bq) and the mass of iodine-131 (m = 1.00 mg), we can find the decay constant \(\lambda\) by first determining the number of atoms (N).

Step 2: Calculating the Number of Atoms (N)

To calculate N, we use the mass (m) and the molar mass (M) of \(^{131}I\). Convert the mass from milligrams to grams, then use Avogadro's number (\(N_A = 6.022 \times 10^{23}\) atoms/mol) to find the number of atoms: N = \(\frac{m}{M} \times N_A\). The molar mass of iodine-131 is approximately 131 g/mol.

Step 3: Finding the Decay Constant (\(\lambda\))

Once we know N, we can rearrange the activity equation to solve for the decay constant: \(\lambda = \frac{A}{N}\). Insert the values of A and N to calculate \(\lambda\).

Step 4: Calculating the Half-Life (\(t_{1/2}\))

The half-life of a radioactive isotope is related to the decay constant by the equation \(t_{1/2} = \frac{\ln(2)}{\lambda}\). Use the value of \(\lambda\) obtained in the previous step to calculate the half-life in seconds.

Key Concepts

Half-Life Calculation

The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei in a sample to decay. This concept is critical in fields ranging from archaeology to medicine, as it helps to determine how long a substance will remain radioactive.

To calculate half-life, we can use the decay constant \(\lambda\). The relationship between half-life \(t_{1/2}\) and decay constant is given by the equation \(t_{1/2} = \frac{\ln(2)}{\lambda}\). Here, \(\ln(2)\) is the natural logarithm of 2, which is approximately 0.693. Knowing this relationship allows us to predict how a sample's activity will change over time, which is vital for safety in handling radioactive materials.

For example, if a scientist measures the decay constant of a radioisotope, they could use the above equation to determine its half-life, providing insight into the duration of its potential effects on living organisms or the environment. This calculation is essential when dealing with isotopes like Iodine-131, which can pose health risks.

Activity of a Radioactive Isotope

The activity of a radioactive isotope refers to the number of decay events that occur per unit of time. Measured in becquerels (Bq), one becquerel equates to one decay per second. The activity \(A\) is determined by the equation \(A = \lambda N\), where \(\lambda\) is the decay constant and \(N\) is the number of undecayed atoms present in the sample.

This formula is significant because it directly relates the microscopic properties of atoms — their decay rate — to a macroscopic observable: the activity we can measure. In practical situations, this helps in establishing the amount of a radioactive material that is safe to handle or is needed for a medical diagnosis.

When provided with the activity and the mass of a radioactive substance, we can backtrack to find its decay constant, assuming we know the number of atoms present, often calculated using Avogadro’s number. This process enables us to understand the dynamic behavior of the radioactive decay process and make predictions about future activity levels.

Avogadro's Number

Avogadro's number, denoted \(N_A\), is a fundamental constant in chemistry that represents the number of atoms or molecules in one mole of a substance. It is approximately \(6.022 \times 10^{23}\) entities per mole.

Understanding Avogadro's number is essential when dealing with substances at the atomic or molecular level, as it allows for the conversion between the number of atoms and the mass of a substance. For instance, if we have a 1.00 mg sample of Iodine-131, we can determine the number of atoms by using the molar mass of Iodine-131 (131 g/mol) in the equation \(N = \frac{m}{M} \times N_A\), where \(m\) is mass and \(M\) is molar mass.

Practical Use of Avogadro's Number

In the case of radioactive isotopes, this conversion is significant because it links the macroscopic world that we perceive to the microscopic, allowing us to calculate the activity of a radioactive sample. Therefore, Avogadro's number is not just an abstract concept; it’s a bridge that helps us to apply our knowledge of atomic physics in real-world situations.