Problem 47 - Chemistry and Chemical Reactivity

Question

A technique to date geological samples uses rubidium-87, a long-lived radioactive isotope of rubidium $$\left(t_{1}=4.8 \times 10^{10} \text { years }\right)$$. Rubidium-87 decays by \(\beta\) emission to strontium-87. If rubidium-87 is part of a rock or mineral, then strontium- 87 will remain trapped within the crystalline structure of the rock. The age of the rock dates back to the time when the rock solidified. Chemical anal. ysis of the rock gives the amounts of \(^{\pi \pi} \mathrm{Rb}\) and \(^{\mathrm{X} 7} \mathrm{Sr}\). From these data, the fraction of \(^{\mathrm{s} 7} \mathrm{Rb}\) that remains can be calculated. Suppose analysis of a stony meteorite determined that 1.8 mmol of \(^{\text {* }}\) Rb and 1.6 mmol of \(^{\text {mige }}\) Sr (the portion of "Sr formed by decomposition of "Rb) were present. Estimate the age of the meteorite. (Hint: The amount of \(^{\mathrm{s} 2} \mathrm{Rb}\) at \(t_{0}\) is moles \(^{\mathrm{s} 7} \mathrm{Rb}+\) moles \(^{\mathrm{s} 7} \mathrm{Sr}\).)

Step-by-Step Solution

Answer

The age of the meteorite is approximately 1.9 billion years.

Step 1: Understand the Given Data

We have 1.8 mmol of \(^{87} \mathrm{Rb}\) and 1.6 mmol of \(^{87} \mathrm{Sr}\). The strontium is formed from the decay of rubidium, so the initial quantity of \(^{87} \mathrm{Rb}\) at \(t_0\) is the sum of both, which is 3.4 mmol.

Step 2: Calculate the Fraction Remaining

The fraction of \(^{87} \mathrm{Rb}\) remaining can be found by dividing the current amount by the initial amount: \(\frac{1.8}{3.4}\).

Step 3: Apply the Decay Formula

Use the decay equation \(N = N_0 e^{-\lambda t}\), where \(N_0\) is the initial amount, \(N\) is the remaining amount, \(\lambda\) is the decay constant, and \(t\) is time. Simplify the fraction to \(\frac{1.8}{3.4} = e^{-\lambda t}\).

Step 4: Calculate the Decay Constant

Use half-life \(t_{1/2} = 4.8 \times 10^{10}\) years to find \(\lambda = \frac{0.693}{t_{1/2}}\). Substitute \(t_{1/2}\) to get \(\lambda = \frac{0.693}{4.8 \times 10^{10}}\).

Step 5: Solve for Time \(t\)

Take the natural log of the fraction \(\frac{1.8}{3.4}\) and solve for \(t\) using \(-\lambda t = \ln \left(\frac{1.8}{3.4}\right)\). Calculate \(t = -\frac{\ln \left(\frac{1.8}{3.4}\right)}{\lambda}\).

Step 6: Calculate Final Age of Meteorite

Substitute the values to find \(t\). Calculate \(-\frac{\ln \left(\frac{1.8}{3.4}\right)}{\left(\frac{0.693}{4.8 \times 10^{10}}\right)}\) to estimate \(t\).

Key Concepts

Rubidium-Strontium Dating

Rubidium-Strontium dating is a method used to determine the age of rocks and minerals by examining the ratio of rubidium-87 and its decay product, strontium-87. This process relies on the radioactive decay of rubidium-87, which transforms into strontium-87 over time. Rubidium is found naturally in many minerals, and when those minerals form, they start with a certain amount of rubidium-87. As time passes, rubidium-87 decays into strontium-87, and by measuring the current ratio of these isotopes, scientists can back-calculate to determine how long this decay has been occurring.
- **Initial and Present Quantities**: When applying this dating method, it's important to determine both the current amount of rubidium-87 present and the initial amount (including what has decayed to strontium-87). - **Decay Process**: The process through which rubidium-87 decays involves beta emission, which changes it into strontium-87 while releasing energy. Rubidium-Strontium dating is an invaluable tool in understanding the geological history of our planet, allowing scientists to date materials that are billions of years old.

Decay Constant

The decay constant (\(\lambda\)) is a crucial part of understanding radioactive decay, representing the probability per unit time that a given atom will decay. Its value is unique for each radioactive isotope, and knowing the decay constant allows us to determine the rate at which a radioactive substance decreases over time.
To calculate the decay constant from the half-life of a substance, use the relationship: \[\lambda = \frac{0.693}{t_{1/2}}\]where \(t_{1/2}\) is the half-life. The constant 0.693 is derived from the natural logarithm of 2, because half-life is a measure of time it takes for half of the substance to decay. Understanding the decay constant helps in determining the age of geological samples because it allows for calculations that reveal how long decay has taken place.- **Importance**: Knowing \(\lambda\) enables precise predictions about the decay patterns and age determination in radiometric dating methods.- **Physical Interpretation**: The smaller the decay constant, the slower the decay process, reflecting longer stability for the radioactive isotope.

Half-life Calculation

Half-life is the time required for half of the radioactive atoms in a sample to decay. This concept is fundamental in radiometric dating methods like the Rubidium-Strontium dating.
The formula to relate half-life with decay constant is:\[t_{1/2} = \frac{0.693}{\lambda}\]This indicates that the half-life is inversely proportional to the decay constant. By knowing the half-life of rubidium-87, which is about \(4.8 \times 10^{10}\) years, scientists can use it to measure geological time scales accurately.- **Utility in Dating**: Through half-life, the age of rocks and fossils can be determined by measuring the remaining radioactive isotopes.- **Comparison with Other Elements**: Different isotopes have different half-lives, which can vary from fractions of a second to billions of years, making some isotopes suitable for dating very old samples while others are suitable for relatively recent events.

Geological Samples Dating

Dating geological samples involves using radiometric techniques to estimate the age of rocks, minerals, and even meteorites. These dating methods, including Rubidium-Strontium dating, rely on the natural radioactive decay of certain isotopes over time.
- **Sample Analysis**: When analyzing a geological sample, scientists measure the current ratios of parent and daughter isotopes to estimate how long the decay process has been occurring. - **Importance of Accurate Measurements**: Precise measurement of isotopic ratios is vital for accurate age prediction. Errors in measurement can lead to significant inaccuracies in age estimation. By understanding geological samples dating, scientists can reconstruct the history of the Earth and even other celestial bodies, providing insights into processes that occurred billions of years ago. This method not only enhances our geological knowledge but also our understanding of the timeline of events that shaped our planet and solar system.