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Chapter 4: Problem 16

How does the decay rate of an isotope relate to its half-lifc?

Short Answer

Expert verified
Decay rate is inversely proportional to half-life, through the decay constant.

Step by step solution

01

Understanding the Relationship

The decay rate of an isotope is directly related to its half-life, which is the time it takes for half of a given quantity of an isotope to decay. The relationship involves exponential decay, where a constant decay rate reduces the quantity over time.
02

Introducing the Decay Constant (位)

The decay constant (位) is a unique value for each radioactive isotope that indicates its decay rate. It is related to the half-life (T鈧�/鈧�) through the formula: \[位 = \frac{0.693}{T_{1/2}}\] This expression shows that the decay constant is the natural logarithm of 2 (approximately 0.693) divided by the half-life of the isotope.
03

Connecting Decay Rate and Half-Life

The decay rate refers to how rapidly the number of radioactive atoms decreases, which is governed by the decay constant. Since the half-life is inversely proportional to the decay constant, a shorter half-life indicates a higher decay rate and vice versa.

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Key Concepts

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

half-life
Half-life is a crucial concept in understanding radioactive decay. It refers to the duration required for half of the radioactive nuclei in a sample to decay into another element or a more stable form. This time period is unique to each isotope. To visualize this, imagine a container filled with popcorn kernels, each representing a radioactive atom. Half-life is like the time it takes for half of those kernels to pop.

Here's why the half-life is important:
  • Predictable decay: Knowing the half-life helps predict how long an isotope will remain active or dangerous.
  • Age dating: Scientists use half-lives to date ancient objects by measuring isotope ratios.
  • Medical uses: In medicine, isotopes with short half-lives are used in diagnostic imaging.
decay rate
The decay rate of an isotope refers to the speed at which the isotope loses its radioactive particles over time. It's about how fast the atom transforms into another element or a more stable state. Picture a sand timer where the sand grains represent radioactive particles. The decay rate tells us how quickly the sand flows from one compartment to another.

Two essential points about decay rate:
  • Expressed mathematically: The decay rate is often represented as a negative exponential function, highlighting the decrease of nuclei over time.
  • Variable dependency: The decay rate directly depends on both the decay constant and the half-life of the isotope.
decay constant
The decay constant, denoted by the Greek letter 位 (lambda), is a fixed value for each radioactive isotope that quantifies the probability of decay per unit time. You can think of it as a unique identifier for the isotope's tendency to transform.

Here are some key features of the decay constant:
  • Calculated relationship: The decay constant relates to the half-life with the formula \[位 = \frac{0.693}{T_{1/2}}\] This formula uses the natural logarithm of 2 because, in an experimental setting, half of the isotopes decay during the half-life period.
  • Proportionality: A larger decay constant indicates a more rapid decay process.
  • Measurement unit: It is often measured in reciprocal time units like seconds鈦宦� or years鈦宦�, emphasizing the rate of decay progression.

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