Problem 71
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.
Step-by-Step Solution
Verified Answer
The third-life, similar to half-life, is the time required for a quantity to reduce to one-third of its initial value. To calculate the third-life for a nuclide with a half-life of 31.4 years, first determine its decay constant, \(λ\), using the half-life equation: \(λ = \frac{ln(2)}{t_{1/2}} ≈ 0.0221\text{ year}^{-1}\). Then, calculate the third-life using the equation \(t_{1/3} = \frac{ln(3)}{λ} ≈ 49.85\text{ years}\).
1Step 1: Define third-life
Third-life can be defined as the time required for a quantity (in this case, the number of radioactive atoms in a sample of a nuclide) to reduce to one-third its initial value.
2Step 2: Understand the concept of half-life
Half-life is the time required for a quantity to reduce to half its initial value. In nuclear decay, it is often used to describe the time required for half of the radioactive atoms in a sample to decay. We are given the half-life of a nuclide as 31.4 years.
3Step 3: Calculate the decay constant
To find the third-life, we'll first need to calculate the decay constant of the nuclide. The decay constant is represented as λ (lambda). It can be calculated using the half-life equation:
\[λ = \frac{ln(2)}{t_{1/2}}\]
where
\(t_{1/2}\) is the half-life
\(ln(2)\) is the natural logarithm of 2
Now, plug in the given half-life value:
\[λ = \frac{ln(2)}{31.4}\]
\[λ ≈ 0.0221\text{ year}^{-1}\]
4Step 4: Calculate the third-life
Next, we'll use the decay constant to find the third-life. We can define third-life (\(t_{1/3}\)) using the decay constant in a similar equation as the half-life equation:
\[t_{1/3} = \frac{ln(3)}{λ}\]
where
\(ln(3)\) is the natural logarithm of 3
Now, plug in the previously calculated decay constant:
\[t_{1/3} = \frac{ln(3)}{0.0221}\]
\[t_{1/3} ≈ 49.85\text{ years}\]
So, the third-life for the given nuclide is approximately 49.85 years.
Key Concepts
Half-LifeDecay ConstantNatural Logarithm
Half-Life
In the realm of nuclear physics and radioactive decay, "half-life" carries a specific meaning explored through nuclear decay.*.
It is the period required for half of the radioactive atoms in a sample to decay.** This means that if you start with 100 atoms of a radioactive substance, after one half-life, only 50 will remain as the rest would have decayed into another element or isotope.
- Half-life is a constant for each radioactive element, representing its rate of decay.*
- Every radioactive substance has its unique half-life duration, dictating how quickly or slowly it decays.
The concept is crucial for understanding and predicting the behavior of radioactive substances over time. It's a key measurement when dealing with nuclear materials and understanding radioactivity's impact in environmental or medical scenarios.
It is the period required for half of the radioactive atoms in a sample to decay.** This means that if you start with 100 atoms of a radioactive substance, after one half-life, only 50 will remain as the rest would have decayed into another element or isotope.
- Half-life is a constant for each radioactive element, representing its rate of decay.*
- Every radioactive substance has its unique half-life duration, dictating how quickly or slowly it decays.
The concept is crucial for understanding and predicting the behavior of radioactive substances over time. It's a key measurement when dealing with nuclear materials and understanding radioactivity's impact in environmental or medical scenarios.
Decay Constant
The decay constant, symbolized by the Greek letter λ (lambda), is a fundamental concept in the study of radioactive decay. It represents the probability of decay of a single atom in a unit of time.*
To understand this deeply, consider a substance with a large number of radioactive atoms. Each atom has a specific probability of decaying at any given moment, quantified by the decay constant.*
- The decay constant links with half-life through the relationship: \[ λ = \frac{ln(2)}{t_{1/2}} \].
*- Here, \(ln(2)\) is the natural logarithm of 2, and \(t_{1/2}\) is the half-life.
The smaller the decay constant, the longer the half-life, indicating a slower rate of radioactive decay. This relationship helps scientists understand how quickly a radioactive sample loses its activity or decays into another form.
To understand this deeply, consider a substance with a large number of radioactive atoms. Each atom has a specific probability of decaying at any given moment, quantified by the decay constant.*
- The decay constant links with half-life through the relationship: \[ λ = \frac{ln(2)}{t_{1/2}} \].
*- Here, \(ln(2)\) is the natural logarithm of 2, and \(t_{1/2}\) is the half-life.
The smaller the decay constant, the longer the half-life, indicating a slower rate of radioactive decay. This relationship helps scientists understand how quickly a radioactive sample loses its activity or decays into another form.
Natural Logarithm
Natural logarithms are an essential mathematical tool often used in exponential decay calculations, such as those involving radioactive decay.*
The natural logarithm is the logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It’s often denoted as \(ln\).
- In nuclear physics, natural logarithms aid in converting exponential decay equations into linear forms, making them easier to solve.
- They are particularly useful when calculating periods like half-life and third-life.*
For instance, the half-life equation uses \(ln(2)\), and the calculation of third-life involves \(ln(3)\), which are the natural logarithms of 2 and 3, respectively. These mathematical computations become more straightforward using natural logs, helping to break down complex exponential relationships in nuclear decay analyses.
The natural logarithm is the logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It’s often denoted as \(ln\).
- In nuclear physics, natural logarithms aid in converting exponential decay equations into linear forms, making them easier to solve.
- They are particularly useful when calculating periods like half-life and third-life.*
For instance, the half-life equation uses \(ln(2)\), and the calculation of third-life involves \(ln(3)\), which are the natural logarithms of 2 and 3, respectively. These mathematical computations become more straightforward using natural logs, helping to break down complex exponential relationships in nuclear decay analyses.
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