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Chapter 5: Problem 4

Explain the difference between the half-life of an isotope and the mean lifetime of a radioactive nucleus.

Short Answer

Expert verified
Half-life is the time for half of a sample to decay, while mean lifetime is the average time for a single nucleus to decay. Both are related by the formula \(t_{1/2} = \tau \ln(2)\).

Step by step solution

01

Understanding Half-life

The half-life of an isotope is the time required for half of the radioactive nuclei in a sample to decay. It is a measure of the rate at which an isotope undergoes radioactive decay. For example, if an isotope has a half-life of 10 years, after 10 years, half of the original amount will have decayed into another element or isotope.
02

Exploring Mean Lifetime

The mean lifetime of a radioactive nucleus is the average time that a single nucleus will exist before decaying. It is mathematically related to the half-life but provides a different perspective on how long a nucleus can be expected to last, based on statistical averages of decay processes.
03

Relating Half-life and Mean Lifetime

The mean lifetime (\(\tau\)) and half-life (\(t_{1/2}\)) are related through the formula:\[t_{1/2} = \tau \ln(2).\]This equation shows that while both terms describe how radioactive decay progresses over time, the mean lifetime is longer than the half-life since it refers to average lifetimes rather than the specific moment when half of the sample has decayed.
04

Practical Implications

Half-life is typically used when discussing the decay of large quantities of material because it is easier to measure half of a large sample decaying. Meanwhile, mean lifetime is more useful in theoretical calculations involving single nuclei and statistical averages. Both are valuable but suited for slightly different contexts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life
The half-life of a radioactive isotope is a fundamental concept in understanding radioactive decay. It refers to the time it takes for half of a given quantity of radioactive nuclei in a sample to decay into another element or isotope. During this process, the original substance becomes less radiologically active, transforming into a different substance over time.

Some key points about half-life include:
  • It is a consistent property of a radioactive isotope, meaning it does not change over time or with varying conditions like temperature or pressure.
  • The half-life provides a "clock" to measure how fast a substance undergoes radioactive decay.
  • In real-world applications, half-life helps in dating archaeological finds, managing nuclear waste, and even in medical treatments with radioactive isotopes.
Understanding half-life is crucial because it allows us to predict how long a radioactive substance will remain active, thereby informing safety protocols and practical applications.
Mean Lifetime
The mean lifetime of a radioactive nucleus is another vital yet distinct concept used to understand radioactive decay. Unlike half-life, mean lifetime focuses on the average time a single radioactive nucleus will take before it decays.

Here are some important details about mean lifetime:
  • It is a statistical measure, describing how long a nucleus will last on average in a sample.
  • The mean lifetime provides a deeper insight into the behavior of individual radioactive atoms.
  • Mathematically, it is related to the half-life by the equation \[t_{1/2} = \tau \ln(2)\], revealing a mathematical connection between these two concepts.
Mean lifetime is invaluable in theoretical physics and in situations where understanding the decay of individual atoms is pertinent, such as in experimental nuclear physics and certain medical applications.
Radioactive Isotopes
Radioactive isotopes, or radioisotopes, are variants of a chemical element that have an unstable nucleus. This instability causes them to emit radiation as they decay into a more stable form. This process can involve the release of alpha, beta, or gamma radiation, each with its own characteristics.

Important things to know about radioactive isotopes:
  • They occur naturally, such as carbon-14, or can be artificially produced in laboratories and reactors.
  • Each isotope has its unique half-life and decay pattern, influencing how we handle and use them.
  • Radioisotopes are widely used in medicine, industry, and research 鈥� from cancer treatments to radiocarbon dating.
Understanding radioactive isotopes is essential due to their widespread application and potential hazards. Knowing their half-lives and decay characteristics helps in safely utilizing these powerful tools across multiple fields.

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Most popular questions from this chapter

Could carbon- 14 dating be used to determine the age of fibres estimated to be a million years old?

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