Problem 36 - Chemistry and Chemical Reactivity

Question

Strontium- 90 is a hazardous radioactive isotope that resulted from atmospheric testing of nuclear weapons. A sample of strontium carbonate containing "Sr is found to have an activity of \(1.0 \times 10^{3} \mathrm{dpm} .\) One year later, the activity of this sample is 975 dpm. (a) Calculate the half-life of strontium-90 from this information. (b) How long will it take for the activity of this sample to drop to \(1.0 \%\) of the initial value?

Step-by-Step Solution

Answer

(a) The half-life of strontium-90 is approximately 28.8 years. (b) It will take about 288 years for the activity to drop to 1% of the initial value.

Step 1: Understanding the Decay Formula

The decay of radioactive isotopes follows the exponential equation \( A(t) = A_0 e^{-kt} \), where \( A(t) \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( k \) is the decay constant, and \( e \) is the base of the natural logarithm. We are given \( A_0 = 1.0 \times 10^3 \text{ dpm} \) and \( A(1) = 975 \text{ dpm} \).

Step 2: Determining the Decay Constant

Using the formula \( A(t) = A_0 e^{-kt} \), substitute \( A(1) = 975 \), \( A_0 = 1000 \), and \( t = 1 \text{ year} \). This gives us the equation \( 975 = 1000 e^{-k} \). Solving for \( k \), we get \( e^{-k} = \frac{975}{1000} = 0.975 \). Taking the natural logarithm on both sides, we find \( k = -\ln(0.975) \).

Step 3: Calculating the Half-Life

The half-life \( T_{1/2} \) can be found using the relation \( T_{1/2} = \frac{\ln(2)}{k} \). Substitute \( k = -\ln(0.975) \) from Step 2 to find \( T_{1/2} \). Calculate to find \( T_{1/2} \approx 28.8 \text{ years} \).

Step 4: Determining Time for Activity to Drop to 1%

To find when the activity will be 1% of its initial value, use \( A(t) = 0.01 A_0 = 0.01 \times 1000 = 10 \text{ dpm} \). Substituting into the decay formula gives \( 10 = 1000 e^{-kt} \) and solving for \( t \) gives \( e^{-kt} = 0.01 \). Use \( k \) from Step 2: \( t = \frac{-\ln(0.01)}{k} \).

Step 5: Final Calculation and Answer

Substitute the value of \( k = -\ln(0.975) \) and \( \ln(0.01) = -4.605 \) into the equation derived in Step 4 to calculate \( t \). This gives \( t \approx 288 \text{ years} \).

Key Concepts

Strontium-90

Strontium-90 is a radioactive isotope that is often a byproduct of nuclear fission. It was notably released into the atmosphere during past atomic weapon testing.
This isotope is a concern due to its ability to replace calcium in bones, thereby posing health risks.

• Strontium-90 decays by emitting beta radiation.
• It has no stable isotopes, meaning it is inherently unstable and always subject to radioactive decay.
• When incorporated into biological organisms, it can persist and radiate for long periods, as it substitutes calcium in bone tissues.

Understanding Strontium-90 is crucial when studying radioactive fallout and its long-term environmental persistence.

Half-Life Calculation

Calculating the half-life of a radioactive substance is key in understanding its long-term behavior. The half-life is the time it takes for half of a radioactive sample to decay.

• For Strontium-90, determining its half-life requires examining changes in activity over a specified period.
• This involves the exponential decay formula: \[ A(t) = A_0 e^{-kt} \] where:
  • \( A(t) \) is the activity at time \( t \).
  • \( A_0 \) is the initial activity.
  • \( k \) is the decay constant.
• By calculating the decay constant \( k \), the half-life \( T_{1/2} \) is derived using:\[ T_{1/2} = \frac{\ln(2)}{k} \]

This calculation helps predict how quickly the substance's radioactivity will decrease to safer levels.

Nuclear Physics

Nuclear physics is the branch of physics that deals with the components and behavior of atomic nuclei. Radioactive decay, such as that of Strontium-90, falls within this realm.

• Nuclear physics explores the forces and interactions that cause unstable isotopes to emit radiation.
• The understanding of decay processes, such as beta decay, informs the analysis of various isotopes' stability and safety.
• This field also encompasses the study of nuclear reactions, such as fission and fusion, which are foundational to nuclear energy and weapons technology.
• Beta decay, like in Strontium-90, involves a neutron transforming into a proton, emitting an electron (beta particle), and an antineutrino.

Through nuclear physics, scientists develop safety protocols for handling and storing radioactive materials, as well as predict and mitigate potential environmental impacts.