Problem 32 - Chemistry and Chemical Reactivity

Question

Iodine-131 is used to treat thyroid cancer. (a) The isotope decays by \(\beta\) particle emission. Write a balanced equation for this process. (b) Iodine-131 has a half-life of 8.04 days. If you begin with \(2.4 \mu \mathrm{g}\) of radioactive \(^{131} \mathrm{I},\) what mass remains after 40.2 days?

Step-by-Step Solution

Answer

(a) \(^{131}_{53}\text{I} \rightarrow \,^{131}_{54}\text{Xe} + \beta^{-}\) (b) 0.075 µg remains.

Step 1: Understanding Beta Decay

Beta decay involves the emission of a beta particle, which means a neutron in the nucleus of an atom converts into a proton, and a beta particle (electron) is emitted. In a balanced equation for iodine-131 decaying by beta emission, iodine-131 (\(^{131}_{53}\text{I}\)) will transform into an isotope of xenon.

Step 2: Writing the Balanced Equation

In the balanced equation for beta decay, the atomic number increases by 1 (since a neutron changes into a proton), while the mass number remains the same. This gives us:\[^{131}_{53}\text{I} \rightarrow \,^{131}_{54}\text{Xe} + \beta^{-}\]

Step 3: Calculate the Number of Half-Lives

First, determine how many half-lives 40.2 days represent. We use the formula:\[\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life}}\]Substituting the given values, we get:\[\frac{40.2 \text{ days}}{8.04 \text{ days}} = 5\]

Step 4: Apply the Half-Life Formula

Use the formula for decay based on half-lives. The formula for the remaining mass is:\[\text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}\]Given an initial mass of 2.4 µg and 5 half-lives, calculate:\[\text{Remaining mass} = 2.4 \times \left(\frac{1}{2}\right)^5\]

Step 5: Perform the Calculation

Calculate the remaining mass:\[2.4 \times \left(\frac{1}{2}\right)^5 = 2.4 \times \frac{1}{32} = 0.075 \mu \text{g}\]So, the remaining mass of iodine-131 after 40.2 days is 0.075 µg.

Key Concepts

Beta Decay

Beta decay is a type of radioactive decay where a beta particle, which is essentially an electron, is emitted from an atom's nucleus. This process occurs when a neutron in the nucleus transforms into a proton. As a result:

• The atomic number of the element increases by one, because a new proton is added.
• The mass number remains unchanged since the total number of protons and neutrons does not change.

In the case of iodine-131, when it undergoes beta decay, it transforms into xenon-131. The atomic number increases from 53 (iodine) to 54 (xenon), indicating the element has changed, but the mass number, 131, remains the same.
To visualize this as a nuclear equation, iodine-131 is written as:\[^{131}_{53}\text{I} \rightarrow \,^{131}_{54}\text{Xe} + \beta^{-}\]Here, the beta particle is represented as \(\beta^{-}\), which indicates a negatively charged electron being emitted.

Half-Life Calculation

The concept of half-life is crucial for understanding the rate at which a radioactive isotope decays. A half-life is the period required for half of the radioactive atoms in a sample to decay.
For iodine-131, the half-life is 8.04 days. Therefore, after each period of 8.04 days, only half of the initial iodine-131 remains. To determine how much of the iodine-131 remains after a specific time, such as 40.2 days, we calculate the number of half-lives that have passed:

• We use the formula: \[\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life}}\]
• With 40.2 days elapsed: \[\frac{40.2 \text{ days}}{8.04 \text{ days}} = 5\]

In this case, 5 half-lives have elapsed, meaning the iodine-131 sample has gone through five cycles of halving its mass.

Radioactive Isotopes

Radioactive isotopes are variants of chemical elements with unstable nuclei that release radiation as they decay into stable forms. These isotopes may have too many or too few neutrons, which causes the instability.
Iodine-131 is one such radioactive isotope. It is widely used in medical applications, particularly in treating thyroid cancer, due to its ability to deliver targeted radiation to thyroid tissues. This therapeutic use arises from iodine's natural affinity for the thyroid gland.

• As iodine-131 decays, it emits beta particles, creating a successor element, such as xenon-131.
• Its half-life of 8.04 days is ideal for medical purposes, providing enough time for treatment while not lingering excessively in the body.

Understanding isotopes like iodine-131 makes it easier to appreciate their applications in nuclear medicine.

Nuclear Equations

Nuclear equations are used to represent the changes occurring in the nucleus during radioactive decay. Unlike chemical equations, nuclear equations focus on the atomic number and mass number.
In beta decay, the nucleus changes as follows:

• A neutron converts into a proton, emitting an electron (beta particle) in the process.
• The new element has an atomic number increased by one, while its mass number stays unchanged.

For iodine-131, the nuclear equation is balanced by ensuring mass numbers and atomic numbers align on both sides of the equation:\[^{131}_{53}\text{I} \rightarrow \,^{131}_{54}\text{Xe} + \beta^{-}\]Through this balance, the atomic and mass number explain how matter and energy conservation occur during the decay.