Question

Iodine-131 is used to treat thyroid cancer. It decays by beta emission and has a half-life of \(8.1\) days. (a) Write a balanced nuclear reaction for the decay of iodine-131. (b) What is the activity (in Ci) of a 2.50-mg sample of the isotope?

Short answer

Question: Write a balanced nuclear decay equation for Iodine-131 and calculate the activity of a 2.50-mg sample of the isotope in curies (Ci). Answer: The balanced nuclear decay equation for Iodine-131 is: \(\text{Iodine-131}_{53}^{131} \rightarrow \text{Xenon-131}_{54}^{131} + \text{Electron or beta particle}_{-1}^{0}\) The activity of the 2.50-mg sample of Iodine-131 is 30.8 Ci.

Step-by-step solution

Write the balanced nuclear decay equation for Iodine-131

In beta emission, a neutron in the nucleus transforms into a proton, releasing an electron with high energy, called a beta particle. The decay of iodine-131 can be represented as: \(\text{Iodine-131} \rightarrow \text{Product nucleus} + \text{Electron or beta particle}\) To balance the decay equation, we will use the atomic mass and atomic number balances: \(\text{Iodine-131}_{53}^{131} \rightarrow \text{Product nucleus}_{Z}^{A} + \text{Electron or beta particle}_{-1}^{0}\) Since a neutron becomes a proton in beta decay, the atomic mass number stays the same, and the atomic number increases by 1. The product nucleus is Xenon-131: \(\text{Iodine-131}_{53}^{131} \rightarrow \text{Xenon-131}_{54}^{131} + \text{Electron or beta particle}_{-1}^{0}\)

Calculate the activity of the sample

The half-life of Iodine-131 is given as 8.1 days. We will use the decay constant (\(\lambda\)) to calculate the activity. The relation between decay constant and half-life is: \(\lambda = \frac{\ln 2}{t_{1/2}}\) We will then use the decay constant (\(\lambda\)) and the number of nuclei in the sample (N) to find the activity. The activity (A) is given by: \(A=\lambda \cdot N\) First, find the decay constant: \(\lambda = \frac{\ln 2}{8.1\, \text{days}} = 0.0855\, \text{day}^{-1}\) Now, we need to find the number of nuclei (N) in the 2.50-mg sample of Iodine-131. Given that the atomic mass of Iodine-131 is approximately 131 g/mol, we can find the number of moles and convert it to nuclei using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol): \(N = \frac{2.50\, \text{mg}}{131\, \frac{\text{g}}{\text{mol}}} \times \frac{1\, \text{g}}{1000\, \text{mg}} \times 6.022 \times 10^{23}\, \frac{\text{atoms}}{\text{mol}} = 1.15 \times 10^{18} \, \text{nuclei}\) Finally, calculate the activity: \(A=\lambda \cdot N = 0.0855\, \text{day}^{-1} \times 1.15 \times 10^{18} \, \text{nuclei} = 9.83 \times 10^{16} \, \text{nuclei}\, \text{day}^{-1}\) We must then convert this activity to curies (Ci). One curie is defined as \(3.7 \times 10^{10}\) disintegrations per second(DPS). To convert to DPS, we will multiply the activity by the conversion factor of seconds in a day: \(A = 9.83 \times 10^{16} \, \text{nuclei}\, \text{day}^{-1} \times \frac{1\, \text{day}}{86400\, \text{s}} = 1.14 \times 10^{12} \, \text{nuclei}\, \text{s}^{-1}\) Now, convert the activity in nuclei/s to curies: \(A_\text{Ci} = \frac{1.14 \times 10^{12} \, \text{nuclei}\, \text{s}^{-1}}{3.7 \times 10^{10}\, \frac{\text{nuclei}}{\text{s}\,\text{Ci}}} = 30.8\, \text{Ci}\) The activity of the 2.50-mg Iodine-131 sample is 30.8 curies.

Key concepts

Beta Emission

Beta emission is a type of radioactive decay in which an unstable atomic nucleus releases a beta particle to stabilize itself. A beta particle can be either an electron or a positron, but in the case of iodine-131, we are dealing with the emission of an electron. During this process, a neutron in the nucleus is transformed into a proton, increasing the element's atomic number by one while keeping its mass number the same.

As a result of beta emission, the parent atom changes into a new element with a different number of protons. This change is necessary for atoms with an imbalance between protons and neutrons to reach a more stable configuration. In medical applications like treating thyroid cancer, iodine-131's radiation can be beneficial for destroying cancerous cells due to its beta-emitting properties.

Half-Life Calculation

The concept of half-life is central to understanding radioactive decay. It represents the time required for half of the radioactive atoms in a sample to decay. For iodine-131, the half-life is 8.1 days, which means that after this period, the activity of the sample will reduce to half of its initial value.

Using the formula \( t_{1/2} = \frac{\ln 2}{\lambda} \), where \( t_{1/2} \) is the half-life and \( \lambda \) is the decay constant, one can determine the rate at which the decay process happens. It's important to understand that the half-life is a statistical measure and does not change regardless of the amount of substance or its activity. This predictability is what makes the half-life a useful tool to determine how long a radioactive substance will remain active or hazardous.

Activity of Radioactive Sample

The activity of a radioactive sample refers to the number of decays occurring per unit time. It is directly proportional to the number of unstable nuclei present in the sample. To calculate activity, we use the formula \( A = \lambda \cdot N \), where \( A \) is the activity, \( \lambda \) is the decay constant, and \( N \) is the number of radioactive nuclei.

Activity is commonly measured in units like becquerels (Bq) or curies (Ci), with one becquerel representing one decay per second, and one curie being equal to 37 billion decays per second. Knowing the activity of a substance like iodine-131 is critical, especially in medical settings, to ensure that the correct dose is administered to patients for effective treatment.

Nuclear Reaction Equation

To capture the process of radioactive decay, we use nuclear reaction equations. These equations show the transformation of one element to another and the emission of radioactive particles. For iodine-131, the equation is \( \text{Iodine-131}_{53}^{131} \rightarrow \text{Xenon-131}_{54}^{131} + \text{Electron}_{-1}^{0} \).

This notation indicates that an iodine-131 atom transforms into a xenon-131 atom (increasing proton number) and releases a beta particle (the electron). The equation must be balanced, showing that the mass number on both sides remains unchanged and the charge is conserved. Writing and balancing these equations correctly is fundamental to understanding the changes occurring during the decay process and predicting the new substances formed.