Problem 34 - Chemistry and Chemical Reactivity

Question

Iodine $$-131\left(t_{4}=8.04 \text { days }\right)$$ a \(\beta\) emitter, is used to treat thyroid cancer. (a) Write an equation for the decomposition of \(^{131}\) I. (b) If you ingest a sample of NaI containing \(^{131} 1,\) how much time is required for the activity to decrease to \(35.0 \%\) of its original value?

Step-by-Step Solution

Answer

13.69 days are needed for the activity to decrease to 35%.

Step 1: Identify the Decay Equation

The decomposition of a radioactive isotope can be represented by a nuclear decay equation. For iodine-131, it undergoes beta decay, changing into xenon-131. The equation is: \[ _{53}^{131} \text{I} \rightarrow _{54}^{131}\text{Xe} + \beta^- + \overline{u}_e \]where \(\beta^-\) is the beta particle (electron) and \(\overline{u}_e\) is the antineutrino.

Step 2: Understand the Half-Life Concept

The half-life \( t_{1/2} \) of a radioactive isotope is the time it takes for half of the isotope to decay. For iodine-131, this is given as \( t_{1/2} = 8.04 \) days. This will help us determine how much of the sample remains after a certain time.

Step 3: Use the Decay Formula to Find Time

To find the time required for the radioactivity to reduce to 35% of its original value, we use the decay formula:\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]where \(N(t)\) is the quantity remaining after time \(t\), \(N_0\) is the initial quantity, and \(t_{1/2}\) is the half-life.

Step 4: Solve the Equation for Time

We need to solve for \(t\) when \(N(t) = 0.35N_0\):\[ 0.35 = \left( \frac{1}{2} \right)^{\frac{t}{8.04}} \]Take the natural logarithm of both sides to solve for \(t\):\[ \ln{0.35} = \frac{t}{8.04} \cdot \ln{0.5} \]Rearranging gives:\[ t = \frac{8.04 \cdot \ln{0.35}}{\ln{0.5}} \]

Step 5: Calculate the Time

Now compute the value:\[ \ln{0.35} \approx -1.0498 \]\[ \ln{0.5} \approx -0.6931 \]\[ t \approx \frac{8.04 \cdot -1.0498}{-0.6931} \approx 13.69 \text{ days} \]

Key Concepts

Half-Life

Half-life is a fundamental concept in understanding radioactive decay. It describes the time period necessary for half of a radioactive substance to transform or decay into another state. This means that after one half-life, only half of the original radioactive atoms remain.
For example, if a substance has a half-life of 8 days, after 8 days only 50% of it would be left active. After another 8 days (or two half-lives in total), only 25% would be remaining—continuing this pattern.

• Half-life is intrinsic to the substance and does not depend on the initial amount of material.
• It helps calculate how much of a radioactive material remains after a certain time.
• Useful in medicine, archeology, and other fields to determine the decay rate of substances.

Understanding half-life allows predicting how long a radioactive material will remain active or how quickly it will decay to a level considered safe.

Beta Decay

Beta decay is a type of radioactive decay where a neutron in the nucleus of an atom is transformed into a proton. During this process, the atom emits a beta particle, which is a fast-moving electron, and an antineutrino.
This changes the element into another one on the periodic table by increasing the atomic number by one while maintaining the mass number.
In the case of Iodine-131, which undergoes beta decay:

• The neutron in iodine converts to a proton.
• A beta particle and an antineutrino are emitted.
• The decay results in the formation of a xenon-131 atom.

This transformation allows iodine-131 to be used in medicine, especially in treatments targeting thyroid cancer, as it helps to alter the cellular structure of cancerous tissues.

Iodine-131

Iodine-131 is a radioactive isotope of iodine, commonly used in medical treatments and diagnostics. It is particularly important in managing thyroid conditions, like cancer and hyperthyroidism.
This isotope emits beta particles during its decay, which makes it effective in destroying cancerous cells in the thyroid gland.

• Its half-life of approximately 8 days allows a balance between effective treatment and minimized exposure to radiation.
• Once it decays through beta decay, iodine-131 becomes xenon-131, a stable element.
• The emitted radiation is used to both detect (via scanning) and treat disease at the cellular level within the thyroid.

Understanding how iodine-131 decays and its interactions in the body is crucial for safe and effective medical application. Its calculated administration ensures targeted therapy, potentially reducing risks compared to more invasive procedures.