Question

Strontium-90 is one of the products of the fission of uranium-235. This strontium isotope is radioactive, with a half-life of 28.1 years. Calculate how long (in years) it will take for \(1.00 \mathrm{~g}\) of the isotope to be reduced to \(0.200 \mathrm{~g}\) by decay.

Short answer

It takes approximately 65.9 years for 1.00 g of Strontium-90 to decay to 0.200 g.

Step-by-step solution

Understand the Half-life Formula

The half-life formula is used to determine the time it takes for a radioactive substance to decay to a certain amount. The formula is: \[N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where \(N(t)\) is the remaining amount of substance at time \(t\), \(N_0\) is the initial amount, \(T_{1/2}\) is the half-life, and \(t\) is the time elapsed.

Identify Given Values

We need to identify the values from the problem:- \(N_0 = 1.00\, \text{g}\) (initial amount of substance)- \(N(t) = 0.200\, \text{g}\) (remaining amount of substance)- \(T_{1/2} = 28.1\, \text{years}\) (half-life of Strontium-90)

Rearrange the Formula to Solve for Time

Rearrange the half-life formula to solve for \(t\):\[0.200 = 1.00 \left(\frac{1}{2}\right)^{\frac{t}{28.1}}\]Divide both sides by 1.00 to simplify the equation:\[0.200 = \left(\frac{1}{2}\right)^{\frac{t}{28.1}}\]

Take the Logarithm of Both Sides

Apply the logarithm to both sides to solve for \(t\). This will require the use of the natural logarithm (ln) or common logarithm (log):\[\log(0.200) = \frac{t}{28.1} \log\left(\frac{1}{2}\right)\]

Solve for t

Continue to solve for \(t\):\[ t = \frac{\log(0.200)}{\log\left(\frac{1}{2}\right)} \times 28.1\]Compute the values:\(\log(0.200) \approx -0.69897\)\(\log\left(\frac{1}{2}\right) \approx -0.30103\)Now substitute these values back into the formula:\[t = \frac{-0.69897}{-0.30103} \times 28.1 \approx 65.9\]

Key concepts

Strontium-90

Strontium-90 is a significant byproduct of nuclear reactions, particularly the fission of uranium-235. It gains its importance and notoriety due to its radioactivity and the implications on human health and the environment. Strontium-90 is chemically similar to calcium, which means that when it enters living organisms, it can be incorporated into bones and teeth.
This property makes it especially hazardous, particularly because it emits beta radiation during decay. This radiation can damage living tissue, leading to health issues such as cancer.
Understanding its behavior in the environment, including its decay process, is vital for safely managing nuclear materials and mitigating its impact.

Half-life Calculation

The concept of half-life is central to understanding how radioactive materials like strontium-90 decay over time. The half-life is the time required for half of the radioactive atoms in a sample to decay.
For strontium-90, this time period is 28.1 years. To calculate how long it takes for a radioactive sample to decrease from 1.00 g to 0.200 g, we can use the half-life formula:

• Start with the formula: \(N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\)
• You know \(N_0 = 1.00 \, \text{g}\), \(N(t) = 0.200 \, \text{g}\), and \(T_{1/2} = 28.1 \, \text{years}\).
• Rearrange the formula to solve for \(t\): \(\frac{N(t)}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\)

This derivation involves understanding logarithms, as you'll need to take the log of both sides to isolate \(t\).
Once calculated, we find that it takes approximately 65.9 years for the mass to reduce to 0.200 g.

Nuclear Fission

Nuclear fission is the process that splits a heavier atomic nucleus into two or more lighter nuclei, releasing a significant amount of energy during the process. It's the principle behind nuclear reactors and atomic bombs.
When uranium-235 undergoes fission, it not only produces other elements like strontium-90 but also releases energy as heat and radiation. This released energy can be used to generate electricity in power plants or can result in a powerful explosion in nuclear weapons.
Understanding nuclear fission is crucial for harnessing its power safely and effectively. It also helps us to comprehend the origin of certain isotopes, like strontium-90, and the potential hazards they pose.

• Energy Release: Massive amounts of energy are freed during fission.
• Byproducts: Elements like strontium-90 are produced.
• Applications: Can be used in both energy generation and weaponry.

Recognizing the implications of fission helps in developing safety measures for its application and managing its environmental impact.