Problem 69 - University Physics with Modern Physics

Question

Measurements indicate that 27.83\(\%\) of all rubidium atoms currently on the earth are the radioactive \(^{87} \mathrm{Rb}\) isotope. The rest are the stable \(^{87} \mathrm{Rb}\) isotope. The half-life of \(^{87} \mathrm{Rb}\) is \(4.75 \times 10^{10} \mathrm{y}\) . Assuming that no rubidium atoms have been formed since, what percentage of rubidium atoms were \(^{87} \mathrm{Rb}\) when our solar system was formed \(4.6 \times 10^{9} \mathrm{y}\) ago?

Step-by-Step Solution

Answer

Approximately 31.97\% of rubidium atoms were \(^ {87} \mathrm{Rb} \) when the solar system formed.

Step 1: Understand the Given and Required

We know that 27.83% of the rubidium atoms currently are the radioactive isotope \(^ {87} \mathrm{Rb} \). We need to find the percentage of \(^ {87} \mathrm{Rb} \) atoms when the solar system formed 4.6 billion years ago. The half-life of \(^ {87} \mathrm{Rb} \) is given as \(4.75 \times 10^{10} \) years.

Step 2: Write the Decay Formula

Use the decay formula \( N = N_0 \times \left( \frac{1}{2} \right)^{t / t_{1/2}} \), where \( N \) is the current number of \(^{87} \mathrm{Rb} \) atoms, \( N_0 \) is the initial number, \( t \) is the time, and \( t_{1/2} \) is the half-life.

Step 3: Express Current and Initial Percentages

Let the original percentage of \(^{87} \mathrm{Rb} \) be \( P_0 \) and the current percentage be \( P \). We have \( P = P_0 \times \left( \frac{1}{2} \right)^{4.6 \times 10^9 / 4.75 \times 10^{10}} \).

Step 4: Calculate Current Ratio of Decay

Compute the decay factor for the 4.6 billion years. \(\left( \frac{1}{2} \right)^{4.6 \times 10^9 / 4.75 \times 10^{10}} \approx 0.8707\).

Step 5: Solve for Initial Percentage

Since \( P = 27.83\% \), solve the equation \( 27.83 = P_0 \times 0.8707 \) to find \( P_0 \). Calculate \( P_0 = \frac{27.83}{0.8707} \approx 31.97\%\).

Step 6: Conclusion

The percentage of \(^{87} \mathrm{Rb} \) when the solar system was formed was approximately \(31.97\%\).

Key Concepts

Half-life calculation

The concept of half-life is crucial in understanding radioactive decay. It refers to the amount of time it takes for half of a given amount of a radioactive isotope to decay. For example, in the case of the radioactive isotope Rubidium-87, its half-life is given as \(4.75 \times 10^{10}\) years. This means that every \(4.75 \times 10^{10}\) years, half of the Rubidium-87 atoms will have decayed into another element.

When calculating how much of a radioactive isotope remains after a certain period, we use the formula:

• \( N = N_0 \times \left( \frac{1}{2} \right)^{t / t_{1/2}} \)

Where:

• \(N\) is the current amount of the isotope.
• \(N_0\) is the initial amount of the isotope.
• \(t\) is the elapsed time.
• \(t_{1/2}\) is the half-life of the isotope.

This formula helps in determining how much of a radioactive substance remains after a specific amount of time, given its half-life. In the provided problem, we calculated the decay over \(4.6 \times 10^9\) years to find out how much Rubidium-87 remained from the initial composition.

Isotope composition

Isotopes are different forms of an element's atoms, which have the same number of protons but a different number of neutrons. This difference in neutron number leads to variations in the atomic mass and can alter the stability of the atom.

In naturally occurring elements, isotopes are often found in specific and stable ratios, but some isotopes are radioactive. Radioactive isotopes, like Rubidium-87, decay over time at a predictable rate, quantified by their half-lives. Knowing the isotope composition, including both stable and radioactive isotopes, enables scientists to date materials and understand historical geological and astronomical events.

• The original problem involves understanding that only \(27.83\%\) of present rubidium atoms are Rubidium-87.
• Back in the solar system's formation, a higher percentage of natural Rubidium would have been Rubidium-87, due to the lapse of radioactive decay over billions of years.

Using the decay calculations, the initial isotope composition is estimated to determine how the ratios have shifted over time, giving insights into the processes that have occurred since the solar system's formation.

Rubidium

Rubidium is a soft, silvery-white metallic element in the alkali metal group, with the symbol \(Rb\) and atomic number 37. It's naturally occurring, mostly as the stable isotope Rubidium-85, which accounts for 72.17\% of naturally occurring rubidium. However, Earth also contains a minor percentage of the radioactive isotope Rubidium-87, as discussed in our exercise.

Rubidium-87 has been useful in dating geological materials due to its long half-life. The isotopic composition of rubidium is crucial for scientists using radiometric dating techniques, particularly the rubidium-strontium method. This method relies on the decay of Rubidium-87 into Strontium-87 over time, a decay process that can help date ancient rocks and minerals.

• Rubidium was not synthesized but is found naturally as the isotopes \(^ {85} \text{Rb}\) and \(^ {87} \text{Rb}\).
• The decay process and isotope ratios help study the Earth's history.
• Major uses of rubidium include deep scientific research and various industrial applications like specialty glasses.

Understanding its isotopic nature aids in interpreting geological data and advances knowledge in fields that study the Earth's processes and history.