Problem 33
Question
Effective Half-life Radioactive substances are removed from living organisms by two processes: natural physical decay and biological metabolism. Each process contributes to an effective half-life \(E\) that is defined by $$ 1 / E=1 / P+1 / B $$ where \(P\) is the physical half-life of the radioactive substance and \(B\) is the biological half-life. (a) Radioactive iodine, \(I-131\), is used to treat hyperthyroidism (overactive thyroid). It is known that for human thyroids, \(P=8\) days and \(B=24\) days. Find the effective half-life of I-131 in the thyroid. (b) Suppose the amount of I-131 in the human thyroid after \(t\) days is modeled by \(A(t)=A_{0} e^{k t}\), \(k<0 .\) Use the effective half-life found in part (a) to determine the percentage of radioactive iodine remaining in the human thyroid gland two weeks after its ingestion.
Step-by-Step Solution
Verified Answer
(a) Effective half-life is 6 days. (b) About 19.8% of I-131 remains after two weeks.
1Step 1: Identify Given Values
From the problem, we know the physical half-life of iodine (\( P \)) is 8 days, and the biological half-life (\( B \)) is 24 days. We will use these values to find the effective half-life \( E \).
2Step 2: Substitute into the Effective Half-life Formula
The formula for effective half-life is given by \( 1/E = 1/P + 1/B \). Substitute \( P = 8 \) and \( B = 24 \) into this equation. \[ \frac{1}{E} = \frac{1}{8} + \frac{1}{24} \]
3Step 3: Calculate the Sum
To find \( \frac{1}{E} \), you need to add \( \frac{1}{8} + \frac{1}{24} \). First, find a common denominator:\[ \frac{1}{8} = \frac{3}{24} \] Now add: \[ \frac{3}{24} + \frac{1}{24} = \frac{4}{24} \] This simplifies to \( \frac{1}{6} \).
4Step 4: Solve for Effective Half-life E
We have \( \frac{1}{E} = \frac{1}{6} \), therefore, to find \( E \), take the reciprocal: \[ E = 6 \] So, the effective half-life is 6 days.
5Step 5: Model the Decay Using Effective Half-life
Given that \( A(t) = A_{0} e^{kt} \) and we know \( E = 6 \) days, determine \( k \) using the relation \( E = \frac{ln(2)}{-k} \). Solve for \( k \):\[ 6 = \frac{ln(2)}{-k} \]\[ k = \frac{ln(2)}{6} \]
6Step 6: Substitute t and Calculate Remaining Percentage
Two weeks is 14 days, so substitute \( t = 14 \) and \( k = \frac{ln(2)}{6} \) into the decay model:\[ A(14) = A_{0} e^{\left(-\frac{ln(2)}{6}\cdot 14\right)} \] This results in:\[ A(14) = A_{0} e^{-\left(\frac{14 \cdot ln(2)}{6}\right)} \] Calculating:\[ A(14) = A_{0} e^{-1.617} \]\[ A(14) \approx A_{0} \times 0.198 \]Hence, approximately 19.8% of the radioactive iodine remains.
Key Concepts
Radioactive decayBiological half-lifePhysical half-lifeExponential decayHalf-life calculations
Radioactive decay
Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This results in the transformation of the original unstable substance into a different, more stable element. Radioactive decay occurs naturally and follows a predictable pattern. The rate at which this decay happens is specific to each radioactive substance, characterized by its half-life.
As time progresses, the number of radioactive atoms decreases exponentially, which means the rate of decay diminishes with time. Key forms of decay include:
As time progresses, the number of radioactive atoms decreases exponentially, which means the rate of decay diminishes with time. Key forms of decay include:
- Alpha decay, which emits an alpha particle consisting of 2 protons and 2 neutrons.
- Beta decay, involving the conversion of a neutron to a proton (or vice versa) with the emission of a beta particle (an electron or positron).
- Gamma decay, in which the nucleus releases excess energy as gamma rays, often following another type of decay.
Biological half-life
The biological half-life is a measure of the time it takes for a substance (such as a radioisotope) to lose half of its activity due to biological processes. This concept is different from the physical half-life, which is based on nuclear decay and does not account for how the body metabolizes or eliminates a substance.
For substances like Iodine-131, which are used in medical treatments, the biological half-life is determined by how quickly the body processes and removes the iodine. Factors affecting the biological half-life include:
For substances like Iodine-131, which are used in medical treatments, the biological half-life is determined by how quickly the body processes and removes the iodine. Factors affecting the biological half-life include:
- Metabolic rate, which can vary significantly between individuals.
- Age and overall health of the person.
- Organ function, particularly the liver and kidneys, which play key roles in processing substances.
Physical half-life
The physical half-life refers to the time required for half of the radioactive atoms in a sample to decay. It is an inherent property of radioactive isotopes, independent of external conditions such as temperature or pressure. Each isotope has its own unique physical half-life, influencing how long it remains active in applications.
For instance, Iodine-131 has a physical half-life of 8 days. This means that in 8 days, half of the Iodine-131 atoms will have decayed into a stable state or another isotope. This concept is crucial for:
For instance, Iodine-131 has a physical half-life of 8 days. This means that in 8 days, half of the Iodine-131 atoms will have decayed into a stable state or another isotope. This concept is crucial for:
- Determining the lifespan of radioactive materials used in medicine and industry.
- Calculating exposure risks in nuclear facilities.
- Managing radioactive waste and its potential environmental impact.
Exponential decay
Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This type of decay is characteristic of radioactive substances, reflected in the way they lose mass or intensity over time.
The mathematical model is given by the formula:
\[ A(t) = A_0 e^{kt} \] where \( A(t) \) is the amount remaining at time \( t \), \( A_0 \) is the initial amount, and \( k \) is the decay constant, typically negative for decay processes. The decay constant \( k \) is related to the half-life of the substance by the formula:
\[ k = \frac{ln(2)}{\text{half-life}} \] Understanding exponential decay allows scientists to model how quickly a substance will lose its radioactivity over time. This is crucial in fields like medicine, where knowing the decay rate helps predict how long a treatment remains effective.
The mathematical model is given by the formula:
\[ A(t) = A_0 e^{kt} \] where \( A(t) \) is the amount remaining at time \( t \), \( A_0 \) is the initial amount, and \( k \) is the decay constant, typically negative for decay processes. The decay constant \( k \) is related to the half-life of the substance by the formula:
\[ k = \frac{ln(2)}{\text{half-life}} \] Understanding exponential decay allows scientists to model how quickly a substance will lose its radioactivity over time. This is crucial in fields like medicine, where knowing the decay rate helps predict how long a treatment remains effective.
Half-life calculations
Half-life calculations are essential to understanding how quickly a substance will reduce to half its initial quantity, whether through radioactive decay or biological processes. These calculations enable scientists and healthcare professionals to predict timelines for decay and the effective life of a substance in various environments.
When calculating effective half-life, which combines both biological and physical half-lives, the following formula is used:
\[ \frac{1}{E} = \frac{1}{P} + \frac{1}{B} \] where \( E \) is the effective half-life, \( P \) is the physical half-life, and \( B \) is the biological half-life. This formula accounts for both how quickly the substance decays naturally and how it is processed by the body.
Understanding these calculations allows for:
When calculating effective half-life, which combines both biological and physical half-lives, the following formula is used:
\[ \frac{1}{E} = \frac{1}{P} + \frac{1}{B} \] where \( E \) is the effective half-life, \( P \) is the physical half-life, and \( B \) is the biological half-life. This formula accounts for both how quickly the substance decays naturally and how it is processed by the body.
Understanding these calculations allows for:
- Determining safe and effective dosing in medical treatments.
- Planning disposal timelines for radioactive waste.
- Assessing the environmental impact and safety of radioactive materials.
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