Problem 35 - Chemistry and Chemical Reactivity

Question

Iodine- \(131(t_{1 / 2}=8.04 \text { days), a } \beta\) 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 \(^{131}\) I, how much time is required for the activity to decrease to \(35.0 \%\) of its original value?

Step-by-Step Solution

Answer

11.89 days

Step 1: Write the Nuclear Decay Equation

The decay of iodine-131 involves the emission of a beta particle. The nuclear equation for this decay is: \[ ^{131}_{53}\text{I} \rightarrow \ ^{131}_{54}\text{Xe} + \beta^- + \bar{u}_e \] Here, iodine-131 transforms into xenon-131 with the emission of a beta particle \(\beta^-\) and an antineutrino \(\bar{u}_e\).

Step 2: Understand the Decay Formula

The decay formula is given by: \[ N(t) = N_0 \times e^{-\lambda t} \] or can be expressed as: \[ \frac{N(t)}{N_0} = e^{-\lambda t} \] Here, \(N(t)\) is the remaining quantity, \(N_0\) is the original quantity, \(\lambda\) is the decay constant, and \(t\) is time.

Step 3: Calculate the Decay Constant

The decay constant \(\lambda\) is related to the half-life \(t_{1/2}\) by the formula: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \] Substitute \(t_{1/2} = 8.04\) days: \[ \lambda = \frac{\ln(2)}{8.04} \approx 0.0862 \text{ days}^{-1} \]

Step 4: Set Up the Activity Equation

We need the time when the activity decreases to 35.0% of its original value: \[ \frac{N(t)}{N_0} = 0.35 \] Using the decay equation: \[ 0.35 = e^{-0.0862 \times t} \]

Step 5: Solve for Time \(t\)

Take the natural logarithm of both sides to solve for \(t\): \[ \ln(0.35) = -0.0862 \times t \] \[ t = \frac{\ln(0.35)}{-0.0862} \approx 11.89 \text{ days} \]

Step 6: Conclusion

The time required for the activity to decrease to 35.0% of its original value is approximately 11.89 days.

Key Concepts

Nuclear Decay Equation

When dealing with radioactive materials like iodine-131, the nuclear decay equation is essential to understand their transformation process. In iodine-131 decay, a beta particle is emitted, converting it into a different element. The nuclear equation for iodine-131 decay is:

• \[ ^{131}_{53}\text{I} \rightarrow \ ^{131}_{54}\text{Xe} + \beta^- + \bar{u}_e \]

This equation shows iodine-131 (\(^{131}_{53}\text{I}\)) decaying into xenon-131 (\(^{131}_{54}\text{Xe}\)) by emitting a beta particle (\(\beta^-\)) and an antineutrino (\(\bar{u}_e\)).
Understanding this equation helps in predicting the products of radioactive decay, an essential aspect in nuclear chemistry and related fields.

Decay Constant

The decay constant (\(\lambda\)) is a crucial parameter in radioactive decay, which helps determine how quickly a substance undergoes decay. It is specific to each radioactive isotope and is related to the isotope's half-life. The formula connecting decay constant to half-life is:

• \[ \lambda = \frac{\ln(2)}{t_{1/2}} \]

In simple terms, the decay constant expresses the probability per unit time that a given nucleus will decay. For iodine-131, with a half-life of 8.04 days, the decay constant is calculated as follows:

• \[ \lambda = \frac{\ln(2)}{8.04} \approx 0.0862 \text{ days}^{-1} \]

Knowing \(\lambda\) allows us to make predictions about how quickly the activity of a radioactive sample decreases over time.

Half-Life Calculation

Half-life (\(t_{1/2}\)) is a vital concept in understanding radioactive decay. It is the time required for half of the radioactive nuclei in a sample to decay. For iodine-131, the half-life is 8.04 days, meaning every 8.04 days, the amount of iodine-131 reduces by half.
This concept is crucial because it helps predict how long a radioactive material will remain active. If you know the half-life, you can estimate how much time it will take for the activity of a substance to decrease to any specified level. This is done using the decay formula, involving calculations like those done for time when activity decreases to 35% of its original value.

Beta Decay

Beta decay is a common type of radioactive decay where a beta particle is emitted. In beta decay, there are two types — beta-minus (\(\beta^-\)) and beta-plus (\(\beta^+\)). Iodine-131 undergoes beta-minus decay.
During beta-minus decay, a neutron is converted into a proton, emitting a beta particle which is similar to an electron. A neutrino or antineutrino is also emitted, depending on the type of beta decay:

• Beta-minus decay emits an antineutrino (\(\bar{u}_e\)).

This process increases the atomic number of the nucleus by one, transforming iodine (element 53) into xenon (element 54). Understanding beta decay helps in appreciating how elements transmute into others through radioactive processes.

Radioactive Decay Formula

The radioactive decay formula is a mathematical representation of how the quantity of a radioactive substance decreases over time. It's given by:

• \[ N(t) = N_0 \times e^{-\lambda t} \]

In this formula:

• \(N(t)\) is the remaining quantity at time \(t\).
• \(N_0\) is the initial quantity.
• \(\lambda\) is the decay constant.

This exponential equation helps calculate remaining amounts of radioactive material and predict activities at future times. For example, to find what time it takes for a sample to decay to 35% of its original quantity, we use:

• \[ \frac{N(t)}{N_0} = 0.35 = e^{-\lambda t} \]

Solving this equation gives us the time taken for that particular activity level.