Question

\({ }^{131}\) I is an isotope of Iodine that \(\beta\) decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with \({ }^{131} \mathrm{I}\) is injected into the blood of a person. The activity of the amount of \({ }^{131} \mathrm{I}\) injected was \(2.4 \times 10^{5}\) Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After \(11.5\) hours, \(2.5 \mathrm{ml}\) of blood is drawn from the person's body, and gives an activity of \(115 \mathrm{~Bq}\). The total volume of blood in the person's body, in liters is approximately (you may use \(e^{x} \approx 1+x\) for \(|x| \ll 1\) and \(\left.\ln 2 \approx 0.7\right)\)

Short answer

The total volume of blood in the person's body is approximately \( 2.1 \) liters.

Step-by-step solution

Determine the time elapsed in terms of half-lives

First, find out how many half-lives have passed from the time the serum was injected until the blood sample was taken. To do this, convert the 11.5 hours to days by dividing by 24, and then divide the result by the half-life period of 8 days to find the number of half-lives elapsed.

Calculate the decay factor

To find the decay factor, use the relationship between the number of half-lives passed and the decay factor, which is given by the formula: decay factor = \( (1/2)^n \) where n is the number of half-lives. Since the decay factor is approximate for a small number of half-lives, we can use the approximation \( e^{-\text{decay constant} \times time} \) where the decay constant is \( \text{ln}(2) \) divided by half-life. The elapsed time in days, multiplied by the natural log of 2, and divided by the half-life will give us an approximation for the decay constant times time. Using the approximation \( e^x \) for small x, we can also state \( e^{\text{-decay constant} \times \text{time}} \) as \( 1 - \text{decay constant} \times \text{time} \) for the decay factor.

Compute the initial and final activities

The initial activity is given as \( 2.4 \times 10^5 \) Bq. Compute the final activity of the \( { }^{131} \text{I} \) after 11.5 hours using the decay factor from step 2. The final activity will be the initial activity multiplied by the decay factor.

Find the ratio of final to initial activity per volume

The activity measured from the 2.5 ml of blood is 115 Bq. To compare this with the initial activity, we need to express this in terms of activity per liter (the initial activity is already per liter since it's the total in the body). Convert the activity of the sample from Bq per 2.5 ml to Bq per liter by multiplying by 400 (since there are 1000 ml in a liter and we want to find out the activity for 1000 ml while we have the activity for 2.5 ml).

Calculate the total blood volume

Finally, to find the total blood volume in the person's body, set up a proportion of the initial total activity to the final activity per volume. This proportion, together with the decay factor, is used to solve for the total volume. The equation will be \( \text{initial activity} \times \text{decay factor} = \text{final activity per liter} \times \text{total blood volume in liters} \). Solve for the total blood volume in liters.

Key concepts

Half-Life of Isotopes

Understanding the concept of half-life is essential when dealing with radioactive isotopes such as Iodine-131. The half-life of an isotope is the time it takes for half of the radioactive atoms present in a sample to decay to a more stable form. In the case of Iodine-131, its half-life is 8 days.

This means that every 8 days, the amount of Iodine-131 in a given sample will reduce to half of its initial quantity due to radioactive decay. For practical applications, knowing the half-life allows us to predict how long a radioactive substance will retain a certain level of activity, which is crucial in medical treatments, radiometric dating, and nuclear power generation.

For example, if a serum labeled with Iodine-131 is injected into a person's bloodstream, the understanding of half-life helps in calculating how long the isotope will be active in the body. This calculation involves determining how many half-lives have elapsed since the time of injection and comparing this with the activity level observed at a particular time post-injection.

Exponential Decay

Radioactive decay follows a pattern known as exponential decay. This pattern is characterized by a consistent decrease in the amount of the radioactive substance over time. In mathematical terms, the quantity of the substance decreases by the same proportion in each successive time period.

The formula for exponential decay, especially as it pertains to half-life, is often expressed as: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \] where \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, \( \frac{1}{2} \) represents the fraction remaining after one half-life, and \( T \) is the half-life period of the isotope.

In terms of the aforementioned serum and blood sample problem, the exponential decay law allows us to calculate the predicted activity of the injected Iodine-131 isotope after any given time interval by using the time elapsed and the known half-life. For small elapsed times, we may use simplifying approximations to ease the computation, but the fundamental concept remains grounded in the exponential nature of the decay process.

Becquerel Activity Unit

The Becquerel (Bq) is the SI unit for measuring the activity of a radioactive substance, and it is defined as one decay per second. High-level activity means that many decays are occurring each second, while low-level activity indicates fewer decays per second.

When we discuss the activity of Iodine-131 in the example, we're referring to how many atoms of Iodine-131 are decaying each second. Initially, the activity of the injected serum was \(2.4 \times 10^{5}\) Bq, indicating a substantial number of decay events per second. As time progresses, this activity decreases due to the natural decay process, measured by drawing a blood sample and assessing its activity.

By relating the initial and final activity levels, taking into account the elapsed time and the decay process, we're able to gauge the effectiveness of the substance for its intended use, such as diagnostic imaging or treatment, and make necessary adjustments for safe and effective application.