Problem 39 - Chemistry and Chemical Reactivity

Question

Strontium-90 is a hazardous radioactive isotope that resulted from atmospheric nuclear testing. A sample of strontium carbonate containing \(^{90} \mathrm{Sr}\) 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) 27.4 years; (b) 182 years.

Step 1: Identify key information

From the problem, we know the initial activity, \(A_0 = 1.0 \times 10^3\) dpm, and the activity after one year, \(A = 975\) dpm.

Step 2: Use the decay formula

The decay of radioactive substances can be described by the formula \(A = A_0 e^{-\lambda t}\), where \(A\) is the activity after time \(t\), \(A_0\) is the initial activity, and \(\lambda\) is the decay constant.

Step 3: Calculate the decay constant

We have \(A = 975\) dpm, \(A_0 = 1.0 \times 10^3\) dpm, and \(t = 1\) year. Substitute these into the formula to find \(\lambda\):\[975 = 1.0 \times 10^3 \times e^{-\lambda \times 1}\]Solve for \(\lambda\):\[\lambda = - \ln\left(\frac{975}{1000}\right) = 0.0253 \, \text{per year}\]

Step 4: Calculate the half-life

The half-life \(T_{1/2}\) is related to the decay constant by \(T_{1/2} = \frac{\ln 2}{\lambda}\). Substitute the calculated \(\lambda\):\[\T_{1/2} = \frac{\ln 2}{0.0253} = 27.4 \, \text{years}\]

Step 5: Determine time for 1.0% activity

We want the activity \(A\) to be \(1.0\%\) of \(A_0\), which is \(0.01 \times 1.0 \times 10^3 = 10\) dpm. Use the decay formula:\[10 = 1.0 \times 10^3 \times e^{-0.0253t}\]Solve for \(t\):\[t = -\frac{\ln(0.01)}{0.0253} = 182 \, \text{years}\]

Key Concepts

Strontium-90

Strontium-90 is a radioactive isotope which gained notoriety due to nuclear testing and accidents. It is one of the fission products of uranium and plutonium, which are found in the fallout from nuclear explosions. This element poses significant health risks because it mimics calcium and can be absorbed into bones.

Once in the body, Strontium-90 can irradiate bone tissue for years, increasing the risk of bone cancer and leukemia. It releases energy in the form of beta particles during its radioactive decay. Due to its long-lived nature in the environment and its biological impact, it is monitored carefully in nuclear safety programs. Keeping informed about Strontium-90 helps us understand both the benefits and risks of nuclear activities.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the material to decay. Calculating the half-life is important in predicting how long a substance will stay radioactive and assist in managing nuclear waste.

To calculate the half-life, we use the formula: \[ T_{1/2} = \frac{\ln 2}{\lambda} \]where \( \lambda \) is the decay constant, and \( \ln 2 \) is the natural logarithm of 2. In the case of Strontium-90, we determined the half-life to be approximately 27.4 years. This means that, over this period, the activity of a sample will decrease to half of its initial amount.

Understanding half-life helps not only in safe handling and storage of radioactive materials but also in capitalizing on the usages of radiation in medical and industrial applications.

Decay Constant

The decay constant \( \lambda \) represents the probability per unit time that a nucleus will decay. It is a crucial parameter in understanding and predicting radioactive decay processes. In the formula for radioactive decay, \[ A = A_0 e^{-\lambda t} \]where \( A \) is the activity at time \( t \), and \( A_0 \) is the initial activity, \( \lambda \) helps calculate not just the rate but also the kinetics of decay.

For Strontium-90, we calculated \( \lambda \) to be 0.0253 per year. A higher decay constant means a substance will decay more rapidly. This concept allows for the planning of activities related to radioactive materials, ensuring safety and proper timing in applications ranging from medical treatments to archaeological dating.

Nuclear Chemistry

Nuclear chemistry involves the study of radioactive elements and the changes they undergo during decay, fission, or fusion. This branch of chemistry is key to understanding how elements like Strontium-90 and others behave under different conditions.

Key aspects include:

• Radioactive decay processes, which explain how unstable nuclei lose energy.
• Measurements of activity, which help determine the number of decays per unit time and are vital for safety assessments.
• Applications of nuclear reactions in medicine, such as in cancer treatment or diagnostic imaging.

Understanding nuclear chemistry not only enhances our knowledge about the universe and its origins but also aids in the development of technologies that benefit society, like nuclear power and radioactive tracing in biological research.