Radioactive Half-Life: Videos & Practice Problems

Radioactive half-life is the time required for half of a radioisotope to decay, and it's a constant value that does not depend on the initial concentration of the substance. A radioisotope, or nuclide, is an unstable isotope that emits radiation during its decay, which can include alpha decay, beta decay, electron capture, or positron emission. The half-life can be calculated using the equation:

$t_{\frac{1}{2}}=\frac{\ln \left(2\right)}{k}$

where ln(2) is approximately 0.693 and k is the decay constant. This equation highlights that half-life is inversely proportional to the decay constant and remains constant throughout the reaction, as demonstrated by a flat line when plotted against time. Understanding half-life is crucial for applications such as radiometric dating and nuclear medicine.

Intro to Radioactive Half-Life

Radioactive half-life, denoted as \( T_{\text{half}} \), is the time required for half of a given quantity of a radioisotope to decay. A radioisotope is characterized by an unstable nucleus that emits radiation during its decay process. Common types of decay include alpha decay, beta decay, electron capture, and positron emission.

To determine the half-life of a radioisotope, one can analyze the remaining percentage of the substance over time. For instance, if we start with 100% of a radioisotope at day 0, the decay process can be tracked as follows: after one half-life, 50% remains; after another half-life, 25% remains; and after yet another half-life, only 12.5% remains. In this example, if it takes 3 days to reduce the quantity from 100% to 50%, and the same duration applies for each subsequent halving, we can conclude that the half-life of this particular radioisotope is 3 days.

Method 1 of Radioactive Half-Life

In the study of radioactive decay, understanding the concept of half-life is crucial. The half-life is defined as the time required for half of a sample of a radioactive substance to decay. The relationship between half-life and the decay constant, denoted as \( k \), is expressed through the formula:

\[\text{Half-life} = \frac{\ln(2)}{k}\]

Here, \( \ln(2) \) is a constant approximately equal to 0.693. The decay constant \( k \) is measured in time inverse (e.g., days^-1), and it should not be confused with time \( t \). A key characteristic of half-life is that it remains constant throughout the decay process, regardless of the initial concentration of the substance. This means that if a sample loses half of its material every three days, this interval does not change; it consistently takes three days to reduce the quantity by half.

When visualizing this concept on a graph, the half-life appears as a flat, horizontal line, indicating that it does not vary over time. This constancy is a fundamental aspect of first-order reactions, where the rate of decay is directly proportional to the amount of substance present. Thus, the half-life serves as a reliable measure in understanding the kinetics of radioactive decay, reinforcing the idea that it is a constant characteristic of the decay process.

Method 1 of Radioactive Half-Life Example

To determine the half-life of Plutonium-244 given its decay constant, we can use the relationship defined by the radioactive decay formula. The half-life (\(t_{1/2}\)) is calculated using the equation:

\(t_{1/2} = \frac{\ln(2)}{k}\)

In this equation, \(k\) represents the decay constant. For Plutonium-244, the decay constant is provided as \(8.66 \times 10^{-9} \, \text{years}^{-1}\). To find the half-life, we substitute this value into the equation:

\(t_{1/2} = \frac{\ln(2)}{8.66 \times 10^{-9}}\)

Calculating this gives:

\(t_{1/2} \approx 8.00 \times 10^{7} \, \text{years}\)

This result indicates that it takes approximately \(8.00 \times 10^{7}\) years for Plutonium-244 to decay to half of its original amount. Understanding this concept is crucial in fields such as nuclear physics and radiometric dating, where the stability and longevity of isotopes are significant.

Method 2 of Radioactive Half-Life Example

In the study of radioactive decay, understanding the concentrations of radioactive nuclei is essential. This involves using the radioactive half-life equation and the integrated rate law to analyze changes in activity over time. The half-life equation is expressed as:

$$ t_{1/2} = \frac{\ln(2)}{k} $$

where \( t_{1/2} \) is the half-life and \( k \) is the decay constant. The integrated rate law for radioactive decay can be represented as:

$$ \ln(N_f) = -kt + \ln(N_0) $$

In this equation, \( N_f \) is the final concentration, \( N_0 \) is the initial concentration, and \( t \) is the time elapsed. To illustrate these concepts, consider a sample of Radon-222 with an initial activity of \( 8.5 \times 10^4 \) disintegrations per second, which decreases to \( 3.7 \times 10^4 \) disintegrations per second after 7.3 days. The goal is to determine the half-life of Radon-222.

First, we can rearrange the integrated rate law to isolate the decay constant \( k \):

$$ \ln(3.7 \times 10^4) = -k(7.3) + \ln(8.5 \times 10^4) $$

By simplifying this equation, we can find \( k \). After calculating, we find:

$$ k \approx 0.113936 \, \text{days}^{-1} $$

Next, we substitute \( k \) back into the half-life equation:

$$ t_{1/2} = \frac{\ln(2)}{0.113936} $$

Calculating this gives us a half-life of approximately 6.1 days. This process highlights the relationship between initial and final concentrations, time, and the decay constant in determining the half-life of a radioactive substance.

Method 3 of Radioactive Half-Life Example

In the context of radioactive decay, method 3 focuses on calculating the fraction or percentage of a radioisotope remaining after a certain period, utilizing both the radioactive half-life equation and the integrated rate law. The fraction of a radioisotope can be expressed as the ratio of the final concentration to the initial concentration. To convert this fraction into a percentage, one can multiply by 100.

For instance, consider iodine-131, which has a half-life of 8.021 days and is used in thyrotherapy. If a sample is estimated to be 6.25 months old, we first need to determine the decay constant (k). The relationship between half-life (t_1/2) and the decay constant is given by the equation:

$$ t_{1/2} = \frac{\ln(2)}{k} $$

Rearranging this equation allows us to solve for k:

$$ k = \frac{\ln(2)}{t_{1/2}} $$

Substituting the half-life value, we find:

$$ k = \frac{\ln(2)}{8.021 \text{ days}} \approx 0.08642 \text{ days}^{-1} $$

Next, we need to convert the age of the sample from months to days. Assuming an average month is about 30 days, 6.25 months translates to:

$$ 6.25 \text{ months} \times 30 \text{ days/month} = 187.5 \text{ days} $$

Now, we can apply the integrated rate law, which can be rearranged to find the fraction remaining:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) = -kt $$

Substituting the known values:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) = - (0.08642 \text{ days}^{-1})(187.5 \text{ days}) $$

This results in:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) \approx -16.20375 $$

To isolate the fraction, we exponentiate both sides:

$$ \frac{[A_f]}{[A_i]} = e^{-16.20375} $$

Calculating this gives a fraction of approximately:

$$ \frac{[A_f]}{[A_i]} \approx 9.18 \times 10^{-8} $$

This value represents the fraction of iodine-131 remaining in the sample after 6.25 months, demonstrating the application of radioactive decay principles in practical scenarios.

The half-life of arsenic-74 is about 18 days. If a sample initially contains 5.13 x 10⁴ mg arsenic-74, what mass (in mg) would be left after 80 days?

• A. 2.36 x 10³ mg
• B. 7.02 x 10² mg
• C. 1.43 x 10³ mg
• D. 1.14 x 10⁸ mg

Answer: (A)

What percentage of carbon – 14 ( t_1/2 = 5715 years) remains in a sample estimated to be 18,315 years old? 

• A. 11.09%
• B. 27.96%
• C. 18.72%
• D. 10.85%

Answer: (D)

Here’s what students ask on this topic:

What is the half-life of a radioactive isotope?

The half-life of a radioactive isotope, also known as a radionuclide, is the time required for half of the atoms in a sample to undergo radioactive decay. During decay, the unstable isotope transforms into a more stable form, which may be another element or a different isotope of the same element. This process emits radiation in the form of particles or electromagnetic waves.

The half-life is a characteristic property of each radioactive isotope and can vary widely, from fractions of a second to millions of years, depending on the isotope in question. For example, the half-life of carbon-14, which is used in radiocarbon dating, is about 5,730 years, whereas the half-life of uranium-238 is about 4.5 billion years. Knowing the half-life allows scientists to determine the age of materials or to understand the timing of processes in nuclear physics and medicine.

How is the half-life of a radioactive parent isotope defined?

The half-life of a radioactive parent isotope is defined as the amount of time it takes for half of the atoms in a sample of the isotope to decay into its daughter isotopes. This decay process happens at a constant rate, which is characteristic of each radioactive isotope. During each half-life, 50% of the remaining radioactive atoms will transform, resulting in a predictable decrease in the amount of the parent isotope over time. It's important to note that the half-life is not affected by external factors such as temperature, pressure, or chemical state; it is a fixed property of the isotope. For example, if the half-life of a particular isotope is 5 years, then after 5 years, only half of the original sample would remain, and after another 5 years (10 years in total), only a quarter of the original sample would be left.

Among the following radioactive parent isotopes, which one has the shortest half-life?

To determine which radioactive parent isotope has the shortest half-life, you would need to compare the specific half-lives of the isotopes in question. The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms present to decay into their daughter isotopes. This can range from fractions of a second to millions of years, depending on the isotope.

However, as the list of isotopes isn't provided in your question, I can't give a specific answer. For example, Polonium-214 has a very short half-life of 164 microseconds, while Uranium-238 has a half-life of about 4.5 billion years. To find out which isotope has the shortest half-life, you would need to look up the half-lives of each isotope mentioned and compare them. The one with the smallest numerical value for its half-life is the one that decays the fastest, meaning it has the shortest half-life.