Question

The first atomic explosion was detonated in the desert north of Alamogordo, New Mexico, on July 16, 1945. What fraction of the strontium- \(90\left(t_{1 / 2}=28.9\right.\) years) originally produced by that explosion still remains as of July \(16,2009 ?\)

Short answer

As of July 16, 2009, approximately \(6.749\%\) of the strontium-90 originally produced by the first atomic explosion still remains.

Step-by-step solution

Calculate the time elapsed

To find the elapsed time, subtract the initial year from the final year: Elapsed time (t) = 2009 - 1945 t = 64 years

Apply the half-life formula

Now, we will use the half-life formula to find the remaining fraction of strontium-90: \(N_t = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\) We are interested in finding the remaining fraction of strontium-90 (\(\frac{N_t}{N_0}\)), so we can rearrange the equation: \(\frac{N_t}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\) Plugging in the values we have: \(\frac{N_t}{N_0} = \left(\frac{1}{2}\right)^{\frac{64}{28.9}}\)

Calculate the remaining fraction

Using a calculator, compute the remaining fraction of strontium-90: \(\frac{N_t}{N_0} \approx 0.06749\)

Express the fraction as a percentage

To express the remaining fraction as a percentage, multiply the fraction by 100: Percentage remaining = 0.06749 × 100 = 6.749% As of July 16, 2009, approximately 6.749% of the strontium-90 originally produced by the first atomic explosion still remains.

Key concepts

Half-Life

The concept of half-life is crucial in understanding radioactive decay, particularly in fields like nuclear chemistry.
Half-life refers to the time it takes for half of a radioactive substance to decay.
This decay follows an exponential pattern, meaning the quantity of substance decreases by half after each half-life period.

• For example, if a substance has a half-life of 10 years, after 10 years, only 50% of the original substance remains.
• After another 10 years (or two half-lives), this amount further reduces to 25% of the initial quantity.

Understanding half-life allows scientists to predict how long a radioactive substance will remain active.
This is especially important when dealing with nuclear materials, as it helps in managing their use and disposal safely.

Strontium-90

Strontium-90 is a radioactive isotope of the element strontium, notable for its relevance in nuclear chemistry.
It's produced as a byproduct in nuclear fission found in nuclear reactors and nuclear explosions.
This isotope has a half-life of approximately 28.9 years, making it one of the more persistent radioactive materials.

• Strontium-90 can find its way into bones due to its chemical similarity to calcium.
• This poses health risks as it can affect bone marrow and potentially cause leukemia.

Because of its prolonged presence and hazards, strontium-90 is closely monitored in environments contaminated by nuclear activities.
The ability to calculate how much strontium-90 remains over time helps manage these risks and plan environmental clean-up efforts.

Nuclear Chemistry

Nuclear chemistry is a branch of chemistry that focuses on the reactions and properties of atomic nuclei.
It's integral in understanding processes like radioactive decay, nuclear fission, and nuclear fusion.

• Radioactive decay involves the transformation of unstable nuclei into more stable ones, releasing energy in the process.
• Nuclear fission is the splitting of a large nucleus into smaller ones, releasing a significant amount of energy, as seen in nuclear reactors and bombs.

Nuclear chemistry has vast applications that range from energy production in nuclear reactors to the development of medical treatments through radiation therapy.
Safety is a critical concern in this field, which is why understanding concepts such as half-life and isotopes like strontium-90 is essential.
This knowledge helps in managing nuclear waste and reducing the environmental impact of nuclear activities.