Chapter 19: Problem 89
Determine the activity of \(10.0 \mathrm{mg}\) of \({ }^{226} \mathrm{Ra}\) in units of \(\mathrm{Bq}\) and \(\mathrm{Ci}\). The half-life of \({ }^{226} \mathrm{Ra}\) is 1600 years.
Short Answer
Expert verified
Activity: \(3.64 \times 10^{8} \text{ Bq} \), \(9.84 \times 10^{-3} \text{ Ci}\).
Step by step solution
01
Understanding Activity
Activity refers to the rate at which a radioactive material decays. It is calculated using the formula: \[ A = \lambda N \]where \( A \) is the activity in becquerels (Bq), \( \lambda \) is the decay constant, and \( N \) is the number of radioactive nuclei.
02
Calculating the Decay Constant
The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by the formula:\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]For \(^{226} \text{Ra}\), the half-life \( T_{1/2} \) is 1600 years, which needs to be converted to seconds. Thus:\[ \lambda = \frac{0.693}{1600 \times 365 \times 24 \times 3600} \approx 1.37 \times 10^{-11} \text{ s}^{-1} \]
03
Calculating the Number of Nuclei
To find \( N \), use the mass of the sample and the molar mass of \(^{226} \text{Ra}\), which is 226 g/mol. The number of nuclei \( N \) is computed as:\[ N = \frac{10.0 \times 10^{-3} \text{ g} \times 6.022 \times 10^{23}}{226 \text{ g/mol}} \approx 2.66 \times 10^{19} \text{ nuclei} \]
04
Calculating the Activity in Becquerels
Using the values of \( \lambda \) and \( N \), calculate the activity \( A \) in Bq as follows:\[ A = \lambda N = 1.37 \times 10^{-11} \text{ s}^{-1} \times 2.66 \times 10^{19} \text{ nuclei} \approx 3.64 \times 10^{8} \text{ Bq} \]
05
Converting Activity to Curies
To convert from becquerels to curies (Ci), use the conversion:\[ 1 ext{ Ci} = 3.7 \times 10^{10} ext{ Bq} \]Thus:\[ A = \frac{3.64 \times 10^{8} \text{ Bq}}{3.7 \times 10^{10} \text{ Bq/Ci}} \approx 9.84 \times 10^{-3} \text{ Ci} \]
06
Final Step: Results Summary
The activity of 10.0 mg of \(^{226} \text{Ra}\) is approximately \(3.64 \times 10^{8} \text{ Bq} \) or \(9.84 \times 10^{-3} \text{ Ci} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Half-life calculation
The concept of half-life is central to understanding how quickly a radioactive substance decreases in activity. The half-life is the time it takes for half of the radioactive nuclei in a sample to decay.
Think of it like cutting a loaf of bread in half repeatedly; after each cut, you have half as much as before. In the context of radioactive decay, each half-life period reduces the number of active radioactive atoms by half.
To calculate the half-life mathematically, you use the formula:
Think of it like cutting a loaf of bread in half repeatedly; after each cut, you have half as much as before. In the context of radioactive decay, each half-life period reduces the number of active radioactive atoms by half.
To calculate the half-life mathematically, you use the formula:
- \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/T_{1/2}} \]
- Where \( N(t) \) is the number of radioactive atoms remaining after time \( t \),
- \( N_0 \) is the initial number of atoms, and \( T_{1/2} \) is the half-life.
Decay constant
The decay constant is a unique property of each radioactive isotope. It represents the probability of a given nucleus decaying per unit time.
In essence, it tells us how quickly a substance will lose its radioactivity.
To calculate this constant, we use the relation between the half-life \( T_{1/2} \) and the decay constant \( \lambda \) as given by:
Understanding the decay constant helps in predicting the behavior of a radioactive element over time, which is critical in fields such as nuclear medicine, radiometric dating, and nuclear power.
In essence, it tells us how quickly a substance will lose its radioactivity.
To calculate this constant, we use the relation between the half-life \( T_{1/2} \) and the decay constant \( \lambda \) as given by:
- \[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
Understanding the decay constant helps in predicting the behavior of a radioactive element over time, which is critical in fields such as nuclear medicine, radiometric dating, and nuclear power.
Nuclear chemistry
Nuclear chemistry focuses on reactions and changes happening at the atomic nucleus level. Unlike traditional chemistry which deals mainly with electron shell interactions, nuclear chemistry involves changes in the core of the atom that affect atomic mass, atomic number, and energy levels.
This field encompasses various sub-disciplines like:
Understanding nuclear chemistry allows scientists to harness radioactive materials effectively and safely for electricity production, medical diagnostics and treatment, and deep space exploration.
It also aids in addressing safety and environmental concerns associated with radioactive waste.
This field encompasses various sub-disciplines like:
- Radioactive decay.
- Nuclear fission - Splitting of heavy atomic nuclei like in nuclear reactors.
- Nuclear fusion - Joining of light nuclei like those occurring in the sun.
Understanding nuclear chemistry allows scientists to harness radioactive materials effectively and safely for electricity production, medical diagnostics and treatment, and deep space exploration.
It also aids in addressing safety and environmental concerns associated with radioactive waste.
Radioactivity units conversion
Measuring radioactivity requires an understanding of different units of measurement. The two most important units for measuring radioactivity are Becquerels (Bq) and Curies (Ci).
Understanding these conversions is essential for communicating radioactive levels in a comprehensible manner, especially in fields like environmental monitoring, nuclear medicine, and radiation safety.
- Becquerels denote one decay per second, providing a direct measure of activity that is part of the International System of Units (SI).
- Curies are an older unit based on the activity of 1 gram of radium-226, which is equal to \(3.7 \times 10^{10}\) decays per second.
- \[ 1 \, \text{Ci} = 3.7 \times 10^{10} \, \text{Bq} \]
Understanding these conversions is essential for communicating radioactive levels in a comprehensible manner, especially in fields like environmental monitoring, nuclear medicine, and radiation safety.