Problem 56 - Physics

Question

Both gold \(\frac{198}{79} \mathrm{Au}\left(T_{1 / 2}=2.69\right.\) days \()\) and iodine \(\frac{131}{53} \mathrm{I}\left(T_{1 / 2}=8.04\right.\) days \()\) are used in diagnostic medicine related to the liver. At the time laboratory supplies are monitored, the activity of the gold is observed to be five times greater than the activity of the iodine. How many days later will the two activities be equal?

Step-by-Step Solution

Answer

The activities will be equal after approximately 9.46 days.

Step 1: Write the Decay Formula

The decay of a radioactive substance can be described using the formula \( A = A_0 e^{-\lambda t} \), where \( A_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is time. To find \( \lambda \), use \( \lambda = \frac{\ln 2}{T_{1/2}} \).

Step 2: Calculate Decay Constants

For gold, the decay constant \( \lambda_{Au} \) is \( \lambda_{Au} = \frac{\ln 2}{2.69} \approx 0.2576 \) per day. For iodine, the decay constant \( \lambda_I \) is \( \lambda_I = \frac{\ln 2}{8.04} \approx 0.0862 \) per day.

Step 3: Use Initial Activity Relationship

Let \( A_{0,Au} \) and \( A_{0,I} \) be the initial activities of gold and iodine. Given that gold's activity is initially five times that of iodine, \( A_{0,Au} = 5A_{0,I} \).

Step 4: Equate Activities Using Decay Formula

Set their activities equal at time \( t \):\[ A_{0,Au} e^{-\lambda_{Au} t} = A_{0,I} e^{-\lambda_I t} \]Substitute \( A_{0,Au} = 5A_{0,I} \) into the equation:\[ 5A_{0,I} e^{-0.2576 t} = A_{0,I} e^{-0.0862 t} \]

Step 5: Simplify and Solve for Time \( t \)

Cancel \( A_{0,I} \) from both sides and simplify:\[ 5 e^{-0.2576t} = e^{-0.0862t} \]Take the natural logarithm of both sides:\[ \ln 5 - 0.2576t = -0.0862t \]Rearrange to solve for \( t \):\[ \ln 5 = (0.2576 - 0.0862)t \]\[ t = \frac{\ln 5}{0.2576 - 0.0862} \]

Step 6: Calculate the Time \( t \)

Calculate \( t \):\[ t = \frac{\ln 5}{0.1714} \approx 9.46 \] days.

Key Concepts

Half-Life

The half-life of a radioactive isotope is the period it takes for half of the original quantity of the substance to decay. Understanding half-life helps predict how long a radioactive sample will remain active.
The concept is crucial when working with radioactive materials like gold-198 and iodine-131, as used in the given problem. Half-life is measured as a time unit— for instance, gold-198 has a half-life of 2.69 days, meaning every 2.69 days, half of the gold-198 atoms have decayed into another element.
In practical terms:

• Half-life helps determine the dosage of radioactive material in medical treatments or tests.
• It indicates how frequently a material needs replenishment to maintain effectiveness in applications, like in diagnostic medicine.

Understanding half-life allows scientists and medical professionals to manage and utilize radioactive materials safely and effectively.

Decay Constant

The decay constant is a fundamental parameter in the study of radioactive decay. It symbolizes how quickly a radioactive substance undergoes decay. Mathematically, it is represented by the symbol \( \lambda \) and is closely related to the half-life of the isotope.
The decay constant can be calculated using the formula \( \lambda = \frac{\ln 2}{T_{1/2}} \), where \( T_{1/2} \) is the half-life of the substance. In the provided problem:

• Gold-198 has a decay constant \( \lambda_{Au} = 0.2576 \) per day.
• Iodine-131 has a decay constant \( \lambda_I = 0.0862 \) per day.

The decay constant allows us to quantitatively describe how quickly a sample loses its radioactivity. It is an essential factor in calculations predicting the activity of a sample over time, influencing decisions in areas such as medicine and nuclear energy.

Diagnostic Medicine

Diagnostic medicine is a critical field in healthcare that uses various techniques to detect and monitor diseases. Radioactive isotopes, like gold-198 and iodine-131, play a pivotal role in this realm.
These isotopes are chosen based on their specific properties such as their half-life and decay constant, which make them suitable for certain diagnostic procedures:

• Gold-198: Often used in imaging procedures due to its relatively short half-life.
• Iodine-131: Widely used in thyroid function tests and treatment due to its longer half-life.

Radioactive isotopes are instrumental in non-invasive medical imaging, helping professionals to observe internal organs and detect anomalies without surgical interventions. This usage underscores the importance of understanding radioactive decay and decay constants in the medical field.

Gold-198

Gold-198 is a radioactive isotope extensively used in medical diagnostics, especially in the liver. Its relatively short half-life of 2.69 days makes it ideal for temporary treatments and imaging, as its radioactivity declines rapidly, reducing long-term radiation exposure to patients.
Some key points about gold-198 include:

• It emits beta and gamma radiation, aiding in imaging and therapeutic applications.
• The short half-life means it quickly loses radioactivity, minimizing lingering effects.

Gold-198's properties align well with the needs of diagnostic medicine, proving vital in scenarios where quick imaging is essential, and prolonged exposure is undesirable.

Iodine-131

Iodine-131 is another prominent isotope used in diagnostic and treatment purposes, most notably for thyroid-related conditions. It has a longer half-life of 8.04 days, making it suitable for applications that require prolonged radiation effects.
Consider these points about iodine-131:

• It is effective in treating hyperthyroidism and thyroid cancer, delivering radiation directly to the thyroid.
• Iodine-131's half-life strikes a balance, being long enough to be effective yet short enough to minimize long-term radiation exposure.

This isotope's radiaoactivity ensures diagnostics are accurate and treatments are effective, making it a crucial tool in therapeutic applications related to the thyroid gland.