Problem 55 - Chemistry and Chemical Reactivity

Question

The oldest-known fossil found in South Africa has been dated based on the decay of Rb-87. $$^{87} \mathrm{Rb} \longrightarrow^{87} \mathrm{Sr}+_{-1}^{0} \beta \quad t_{1 / 2}=4.8 \times 10^{10} \text { years }$$ If the ratio of the present quantity of \(^{87} \mathrm{Rb}\) to the original quantity is \(0.951,\) calculate the age of the fossil.

Step-by-Step Solution

Answer

The age of the fossil is approximately 3.43 billion years.

Step 1: Understand the Half-life Formula

The half-life formula is used to calculate the age of a sample using the current amount of a substance and its original amount. The formula is given by:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/t_{1/2}} \]where \( N(t) \) is the remaining amount after time \( t \), \( N_0 \) is the initial amount, and \( t_{1/2} \) is the half-life of the substance.

Step 2: Rearrange the Formula for Time

To find the time \( t \) since the fossil was formed, we need to rearrange the half-life formula. Start by taking the ratio of the current amount \( N(t) \) to the original amount \( N_0 \):\[ 0.951 = \left( \frac{1}{2} \right)^{t/t_{1/2}} \]Take the natural logarithm on both sides to solve for \( t \):\[ \ln(0.951) = \frac{t}{t_{1/2}} \times \ln\left(\frac{1}{2}\right) \]

Step 3: Solve for the Age of the Fossil

With the formula rearranged, solve for \( t \):\[ t = \frac{\ln(0.951)}{\ln(0.5)} \times t_{1/2} \]Substitute the half-life value \( t_{1/2} = 4.8 \times 10^{10} \text{ years} \) into the equation:\[ t = \frac{\ln(0.951)}{\ln(0.5)} \times 4.8 \times 10^{10} \]Calculate the values:- \( \ln(0.951) \approx -0.05098 \)- \( \ln(0.5) \approx -0.69315 \)Then calculate:\[ t \approx \frac{-0.05098}{-0.69315} \times 4.8 \times 10^{10} \approx 3.43 \times 10^9 \text{ years} \]

Step 4: Conclude the Calculation

The age of the fossil in years is calculated as \( 3.43 \times 10^9 \) years. This means that the fossil is approximately 3.43 billion years old.

Key Concepts

Rb-Sr decay

In the context of dating ancient materials such as fossils, Rubidium-87 (\(^{87} \mathrm{Rb}\)) plays a crucial role due to its radioactive decay process. Rubidium-87 decays into Strontium-87 (\(^{87} \mathrm{Sr}\)) by emitting a beta particle (\(_{-1}^{0} \beta\)). This decay process is vital in radiometric dating because it provides a reliable clock to measure the time elapsed since the formation of a fossil.
- **What is Rb-Sr Decay?** It's a transformation in which \(^{87} \mathrm{Rb}\) loses a beta particle to become \(^{87} \mathrm{Sr}\). This happens at a constant rate over millions of years, making it ideal for dating fossils.
- **Why use Rb-Sr decay for dating?** The long half-life of \(^{87} \mathrm{Rb}\) at approximately 4.8 billion years allows scientists to date very ancient rocks and fossils, helping reconstruct Earth's early history.
In summary, Rb-Sr decay is a method that provides important insights into the age of geological formations. It helps us understand how old certain materials are, by tracking the natural conversion of \(^{87} \mathrm{Rb}\) to \(^{87} \mathrm{Sr}\).

Half-life calculation

Half-life is a fundamental concept in radiometric dating. It is defined as the amount of time it takes for half of a sample of a radioactive isotope to decay. This measure remains constant over time and provides a convenient way to learn about the age of materials.
- **Understanding the Half-life Formula:** The formula \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/t_{1/2}} \] is used to calculate how much of a radioactive substance remains after a certain period. Here, \(N(t)\) is the remaining amount, \(N_0\) is the original quantity, \(t\) is the time past since the decay began, and \(t_{1/2}\) represents the half-life.
- **Rearranging for Time:** In determining the age of a sample, like a fossil, you may need to solve for \(t\) using known values of \(N(t)\) and \(N_0\). This is done by rearranging and solving the equation \[ t = \frac{\ln(N(t) / N_0)}{\ln(0.5)} \times t_{1/2} \].
This calculation provides the time elapsed since the sample's formation.
Understanding how to calculate half-life using these formulas makes it easier to apply these concepts practically, such as estimating the age of ancient fossils.

Age of fossils

Dating fossils accurately is a significant part of understanding Earth's history and evolutionary processes. The age of a fossil can be determined using radiometric dating methods like the Rb-Sr decay method.
- **Importance of Determining Fossil Age:** Knowing the age of fossils helps paleontologists construct a timeline of life on Earth. This timeline provides insights into how different species evolved over time and how environmental changes have influenced these processes.
- **Calculation in Practice:** For example, in the given fossil problem,scientists used the decay of \(^{87} \mathrm{Rb}\) to \(^{87} \mathrm{Sr}\) and calculated the ratio \(0.951\) of the present to original amount. Plugging into the rearranged formula, they found the fossil's age to be approximately 3.43 billion years.
With this data, scientists don't just learn about the fossil itself but also gain a broader understanding of the Earth's ancient environments. Each fossil dated contributes to our knowledge of Earth's biological heritage and the history encapsulated within its strata.