Problem 17 - College Physics

Question

Radioactive tracers. Radioactive isotopes are often intro- duced into the body through the bloodstream. Their spread through the body can then be monitored by detecting the appearance of radiation in different organs. \(^{131} \mathrm{I}, \mathrm{a} \beta^{-}\) emitter with a half-life of 8.0 \(\mathrm{d}\) is one such tracer. Suppose a scientist introduces a sample with an activity of 375 \(\mathrm{Bq}\) and watches it spread to the organs. (a) Assuming that the sample all went to the thyroid gland, what will be the decay rate in that gland 24 \(\mathrm{d}\) (about 2\(\frac{1}{2}\) weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0 \(\mathrm{Bq}\) , what percent of the tracer went to that gland? (c) What isotope remains after the I-131 decays?

Step-by-Step Solution

Answer

(a) 47.1 Bq; (b) 36.1% tracer in the gland; (c) Isotope remaining is \( ^{131}\mathrm{Xe} \).

Step 1: Understanding Radioactive Decay

To solve this problem, we need to understand that radioactive decay is described by the equation \( A(t) = A_0 \times e^{-\lambda t} \), where \( A(t) \) is the activity at time \( t \), \( A_0 \) is the initial activity, and \( \lambda \) is the decay constant. The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by \( \lambda = \frac{\ln(2)}{T_{1/2}} \).

Step 2: Calculating the Decay Constant

First, we need to calculate the decay constant \( \lambda \) using the half-life of I-131, which is 8.0 days. The formula is \( \lambda = \frac{\ln(2)}{T_{1/2}} \). Plugging in the half-life, we get: \( \lambda = \frac{0.693}{8.0 \text{ days}} = 0.0866 \text{ days}^{-1} \).

Step 3: Calculate Activity After 24 Days

Using the radioactive decay equation \( A(t) = A_0 \times e^{-\lambda t} \), calculate the activity after 24 days. Here, \( t = 24 \) days and \( A_0 = 375 \) Bq. Therefore: \( A(24) = 375 \times e^{-0.0866 \times 24} \). Calculating this gives \( A(24) \approx 47.1 \text{ Bq} \).

Step 4: Determine Percent of Tracer in Gland

After 24 days, the observed decay rate is 17.0 Bq. To find what percent of the tracer went to the gland, we use the formula: \( \text{percent} = \frac{A_{observed}}{A(24)} \times 100 \% \), where \( A(24) = 47.1 \) Bq. This gives \( \text{percent} = \frac{17.0}{47.1} \times 100 \% \approx 36.1\% \).

Step 5: Identify the Isotope After Decay

After \( ^{131}\mathrm{I} \) decays through beta decay, it becomes \( ^{131}\mathrm{Xe} \) (Xenon-131). This is the product of beta decay for iodine-131.

Key Concepts

Radioactive isotopes

Radioactive isotopes, often used in medical and scientific applications, are atoms that contain an unstable combination of neutrons and protons. These isotopes will undergo radioactive decay over time. This means that they will release particles or electromagnetic waves to reach a more stable state. This property makes them very useful.

• In medicine, they can serve as tracers, highlighting certain areas of the body without invasive procedures.
• In industry, they are used in applications such as detecting leaks or measuring material thickness.

In our example, Iodine-131 ( ^{131} I) is a radioactive isotope commonly used as a tracer to study the spread of substances in the body, given its emission of radiation that can be easily detected. It helps doctors monitor where and how effectively iodine is absorbed in organs like the thyroid gland, which can be crucial for diagnosing and treating diseases.

Half-life

The half-life of a radioactive isotope is the time it takes for half the quantity of the isotope to decay. During this time, half of the original radioactive atoms transmute into a more stable form. This concept is fundamental in radiochemistry and nuclear physics.

• It helps scientists determine how long a substance remains active and detectable.
• In medical treatments, understanding half-life is essential for safety and effectiveness.

For Iodine-131, the half-life is 8 days. Knowing this, we can predict how much ^{131}I will remain or disintegrate over time. For example, after 8 days, 50% of a sample will have decayed, leaving the other half intact, and this % loss continues in a predictable pattern thereafter. This reliable decay makes using isotopes like ^{131} I for tracing studies possible.

Beta decay

Beta decay is a type of radioactive decay where a beta particle, which can be an electron or positron, is emitted from an atomic nucleus. This process results in the transformation of a neutron into a proton or vice versa. Beta decay is crucial for changing the identity of an isotope.

• It can result in a change in the atomic number, creating a new element or isotope.
• It is one of the three main types of radioactive decay, alongside alpha and gamma decay.

In the case of Iodine-131 ( ^{131} I), it undergoes beta minus ( β^{-} ) decay. This results in the conversion of a neutron in the iodine nucleus into a proton, accompanied by the emission of an electron (beta particle) and an antineutrino. This process transforms ^{131}I into Xenon-131 ( ^{131}Xe), a non-radioactive and stable isotope. Understanding beta decay is pivotal for grasping how radioactive tracers change once administered.

Decay constant

The decay constant ( λ ) is a value that describes the rate of decay of a radioactive isotope. It directly ties to the half-life of the substance and helps calculate the decay over time. Essentially, it's a probability factor in equations calculating the remainder of active isotopes.

• The larger the decay constant, the faster the isotope decays.
• It is critical for calculating how much activity remains at any given time.

In mathematical terms, the decay constant is given by λ = rac{ln(2)}{T_{1/2}} , where T_{1/2} is the half-life. For Iodine-131, the decay constant is approximately 0.0866 per day, derived from its 8-day half-life. With this, we can calculate the remaining activity at any time using the equation A(t) = A_0 imes e^{-λt} , crucial for precise medical diagnostics and nuclear physics applications.