Problem 39 - Chemistry: Atoms First

Question

A sample of rock was found to contain \(8.23 \mathrm{mg}\) of rubidium- 87 and \(0.47 \mathrm{mg}\) of strontium- 87 . (a) Calculate the age of the rock if the half-life of the decay of rubidium by \(\beta\) emission is \(4.7 \times 10^{10} \mathrm{y}\). (b) If some 87 \(38 \mathrm{Sr}\) was initially present in the rock, would the rock be younger, older, or the same age as the age calculated in (a)? Explain your answer.

Step-by-Step Solution

Answer

The age of the rock is the number of half-lives multiplied by the half-life of Rb-87. If initial Sr-87 was present, the rock would be younger than the age calculated in (a).

Step 1: Understanding the Problem

Determine the current amounts of rubidium-87 and strontium-87 in the rock sample. Use these amounts to calculate the number of half-lives passed, which will allow calculation of the age of the rock. The half-life of rubidium-87 is given.

Step 2: Calculate the Number of Half-lives

Using the decay formula \( N = N_0 (1/2)^{t/T} \) where \( N \) is the remaining amount, \( N_0 \) is the initial amount, \( t \) is the time elapsed, and \( T \) is the half-life. Here \( N = 0.47 \text{mg} \) (of Sr-87), \( N_0 = N_0 (\text{of Rb-87}) + 0.47 \text{mg} \) and \( T = 4.7 \times 10^{10} \) years. The initial amount of Rb-87 before decay, \( N_0 \), is unknown.

Step 3: Express the Number of Rb-87 Half-lives

Use the ratio of Sr-87 to Rb-87 to find the number of half-lives: \( \frac{0.47 \text{mg}}{8.23 \text{mg}} = (1/2)^n \), where \( n \) is the number of half-lives.

Step 4: Solve the Half-life Equation

Take the natural logarithm of both sides to solve for \( n \): \( \ln(\frac{0.47}{8.23}) = n \ln(1/2) \). Solve for \( n \) to find the number of half-lives that have passed.

Step 5: Calculate the Age of the Rock

Multiply the number of half-lives by the half-life duration: \( \text{Age of the rock} = n \times T \), where \( T = 4.7 \times 10^{10} \) years.

Step 6: Discuss the Effect of Initial Sr-87

If initial Sr-87 was present, the calculated age in part (a) would be an overestimate. This is because the Sr-87 to Rb-87 ratio would be higher than it would be if all of the Sr-87 were produced from Rb-87 decay alone.

Key Concepts

Rubidium-Strontium Dating

Rubidium-strontium dating is a radiometric dating technique used by scientists to determine the age of rocks and minerals on the basis of the quantities present of certain isotopes. Rubidium-87 (Rb-87) decays into strontium-87 (Sr-87) over time. By measuring the ratio of these two isotopes in a rock sample, and knowing the half-life of Rb-87, we can calculate the time that has passed since the rock formed. This method assumes no initial Sr-87 was present when the rock was formed; otherwise, this would affect the final age calculation.

The process starts by carefully crushing and separating minerals in the rock and then precisely measuring the relative amounts of Rb-87 and Sr-87 using a mass spectrometer. This data, along with the knowledge of the Rb-87 half-life, is used to compute the age of the rock.

Half-Life Calculation

The half-life of a radioactive isotope is the time it takes for half of the isotope to decay. This concept is crucial when it comes to understanding radioactive dating methods. Knowing the half-life allows us to calculate the age of rocks because we can determine how many half-lives have passed based on the current ratio of parent to daughter isotopes.

For example, if you know the amount of a radioactive isotope originally present and how much remains now, you can use the half-life to calculate the number of half-lives that have passed, which directly correlates to the age of the rock sample. This is because the radioactive decay process follows a predictable pattern, allowing for such calculations to be made.

Decay Formula

To determine the age of rocks using rubidium-strontium dating, scientists employ the decay formula:
\[ N = N_0 (1/2)^{t/T} \]
Here, \(N\) is the current quantity of the daughter isotope (Sr-87), \(N_0\) is the initial quantity of the parent isotope (Rb-87), \(t\) is the time that has passed, and \(T\) is the half-life of the parent isotope. By rearranging this formula, we can solve for the time that has passed since the rock formed. The decay rate is constant, so this calculation yields reliable results as long as the system has remained closed, meaning that neither isotope has been added or removed since the rock’s formation.

Furthermore, in cases where some of the daughter isotope was present at the beginning, additional steps must be taken to determine the initial quantities accurately, which can make the calculations more complex.

Geological Timescale

The geological timescale is a system of chronological dating that classifies geological strata in time. It is used by geologists, paleontologists, and other Earth scientists to describe the timing and relationships of events in the history of our planet. By using radioactive dating techniques, including rubidium-strontium dating, scientists are able to place ancient rocks within the detailed framework of the geological timescale.

This timescale spans from the formation of the Earth approximately 4.54 billion years ago to the present, and it is divided into eons, eras, periods, epochs, and ages, reflecting major changes in Earth's conditions and the life forms that existed. Radiometric dating plays a central role in defining the ages of the various divisions of the timescale, contributing to our understanding of Earth's long and complex history.