Question

Define "third-life" in a similar way to "half-life," and determine the "third- life" for a nuclide that has a half-life of 31.4 years.

Short answer

The third-life for a nuclide, defined similarly to half-life as the time taken for a substance to reduce to one-third of its initial amount, can be calculated using the relationship \( T_{1/3} = \frac{\ln{\frac{1}{3}}}{\ln{\frac{1}{2}}} \times T_{1/2} \). For a nuclide with a half-life of 31.4 years, its third-life is approximately 20.0 years.

Step-by-step solution

Defining Half-life and Third-life

Half-life is defined as the time taken for a substance to reduce to half its initial amount. Similarly, we can define "third-life" as the time taken for a substance to reduce to one-third of its initial amount.

Relationship between Third-life and Half-life

Let's consider a substance that has a half-life of T years. After this time, the remaining amount of the substance will be half of the initial amount. To obtain the third-life, we need to find the time it takes for the substance to reduce to one third of its initial amount. To do this, let's look at the general equation that governs this kind of decay: \( N_t = N_0 \times \frac{1}{2}^{\frac{t}{T_{1/2}}} \) Where: - \(N_t\) is the remaining amount of the substance after t years. - \(N_0\) is the initial amount of the substance. - \(T_{1/2}\) is the half-life of the substance. To determine the third-life (\(T_{1/3}\)), we need to modify the equation so that it represents the remaining amount of the substance equal to one-third of the initial amount. Therefore, the equation becomes: \( N_t = N_0 \times \frac{1}{3}^{\frac{t}{T_{1/3}}} \) Now, we have to find a relationship between \(T_{1/2}\) and \(T_{1/3}\) to solve the problem.

Deriving the Relationship between Third-life and Half-life

First, let's find the ratio of the initial amount (\(N_0\)) to the remaining amount (\(N_t\)) at the half-life: \( \frac{N_0}{N_t} = \frac{1}{2} \) Similarly, let's find the ratio of the initial amount (\(N_0\)) to the remaining amount (\(N_t\)) at the third-life: \( \frac{N_0}{N_t} = \frac{1}{3} \) Now let's equate both expressions in terms of the half-life and third-life: \( \frac{1}{2}^{\frac{t}{T_{1/2}}} = \frac{1}{3}^{\frac{t}{T_{1/3}}} \) To achieve a relationship between \(T_{1/2}\) and \(T_{1/3}\), let's take the natural logarithm of both sides: \( \frac{t}{T_{1/2}} \times \ln{\frac{1}{2}} = \frac{t}{T_{1/3}} \times \ln{\frac{1}{3}} \) Divide both sides by t: \( \frac{\ln{\frac{1}{2}}}{T_{1/2}} = \frac{\ln{\frac{1}{3}}}{T_{1/3}} \) Now, we can obtain the relationship: \( T_{1/3} = \frac{\ln{\frac{1}{3}}}{\ln{\frac{1}{2}}} \times T_{1/2} \)

Calculating Third-life

Using the given half-life of the nuclide, T = 31.4 years, let's find the third-life (T_{1/3}): \( T_{1/3} = \frac{\ln{\frac{1}{3}}}{\ln{\frac{1}{2}}} \times 31.4 \approx 20.0 \) years So, the third-life for the nuclide with a half-life of 31.4 years is approximately 20.0 years.

Key concepts

Third-life

In the world of radioactive decay, we frequently hear about half-life—the time it takes for half of a radioactive substance to decay. However, there's another less commonly discussed concept: third-life. Third-life is the time duration required for a radioactive substance to reduce to one-third of its initial amount. It's an alternative way to measure radioactive decay rates and offers a broader understanding of this process.

To calculate the third-life, you use a similar approach to half-life calculations. By using specific equations related to decay, you can determine how long it will take for only one-third of the original nuclide to remain. This concept, like half-life, is critical in fields such as nuclear physics and radioactive dating, as it can help estimate timelines concerning nuclear reactions and material dating.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This decay process allows the nuclei to transform into a new state, often forming different elements or isotopes.

There are several types of radioactive decay, such as alpha decay, beta decay, and gamma decay. Each type involves the emission of different particles or electromagnetic radiation, contributing to the transformation of a nuclide.

• Alpha decay emits alpha particles, consisting of two protons and two neutrons.
• Beta decay involves the conversion of neutrons into protons with the release of electrons or positrons.
• Gamma decay emits high-energy photons without changing the number of protons or neutrons.

This decay is essential in various applications, from carbon dating archaeological artifacts to medical treatments and power generation in nuclear reactors. Understanding it is crucial for any applications involving radioactive materials.

Nuclides

Nuclides are distinct species of atoms characterized by the number of protons and neutrons within their nuclei. Each nuclide corresponds to a unique atomic configuration and has specific properties, such as mass and stability.

Every element on the periodic table can have multiple nuclides, known as isotopes, differing only in the number of neutrons. These isotopes may be stable or radioactive, the latter undergoing decay to reach a more stable state. Some nuclides are used in industry and medicine due to their radioactive nature.

The consistency in atomic and mass numbers helps in identifying nuclides, making it possible to predict their behavior and energy levels during radioactive decay. This understanding assists physicists and chemists in various applications such as tracing chemical pathways or understanding stellar processes.

Exponential Decay

Exponential decay describes how the quantity of something decreases at a rate proportional to its current value. In the context of radioactive decay, exponential decay models the process through which a radioactive substance diminishes over time.

The mathematical representation for this process is given by:\[ N(t) = N_0 \times e^{-\lambda t} \]Where:

• \( N(t) \) is the quantity at time \( t \).
• \( N_0 \) is the initial quantity.
• \( \lambda \) is the decay constant.

Exponential decay is a universal concept appearing also in areas like finance and biology. Its characteristic feature is a steady ratio of decrease, making predictions and calculations straightforward once the decay constant is known. Observing how substances decline exponentially helps develop more accurate models for predicting changes over time, critical in fields utilizing radioactive materials.

Natural Logarithm

The natural logarithm is a mathematical function denoted as \( \ln(x) \). It is the logarithm to the base \( e \), where \( e \) (approximately 2.718) is the constant of natural growth.

In the context of nuclear physics and radioactive decay, natural logarithms are crucial for calculating decay over time, such as half-life and third-life calculations. They transform multiplicative relationships into additive ones, simplifying the process of determining decay timelines.

For instance, when determining the time for half or one-third of a substance to remain, taking the natural logarithm allows you to rearrange and solve exponential decay equations easily. The property of turning multiplication into addition (through logarithms) greatly facilitates complex calculations involving rates and exponential functions, making them indispensable in sciences involving continuous growth or decay.