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Chapter 20: Problem 94

The naturally occurring isotope rubidium-87 decays by beta emission to strontium- 87 . This decay is the basis of a method for determining the ages of rocks. A sample of rock contains \(102.1 \mu \mathrm{g}{ }^{87} \mathrm{R} \mathrm{b}\) and \(5.0 \mu \mathrm{g}{ }^{87} \mathrm{Sr}\). What is the age of the rock? The half-life of rubidium- 87 is \(4.8 \times 10^{10} \mathrm{y}\)

Short Answer

Expert verified
The age of the rock is approximately \\(3.24 \times 10^9\\) years.

Step by step solution

01

Identify Initial and Current Quantities

In this problem, you begin with a mixture of Rb-87 and Sr-87. The initial amount of Rb-87 (when all of it was Rb) is the total of Rb-87 and Sr-87 together. Calculate the initial rubidium content:\[\text{Initial }^{87} \mathrm{Rb} = 102.1 \, \mu \mathrm{g} + 5.0 \, \mu \mathrm{g} = 107.1 \, \mu \mathrm{g}\]
02

Determine the Number of Half-Lives

The number of half-lives the isotope has undergone is determined using the proportion of Rubidium remaining. Given that the current amount of Rb is 102.1 碌g:\[\frac{\text{Current }^{87} \mathrm{Rb}}{\text{Initial }^{87} \mathrm{Rb}} = \frac{102.1}{107.1}\]Calculating this gives:\[\frac{102.1}{107.1} \approx 0.953\]
03

Use the Decay Formula

Use the formula to determine the number of half-lives:\[\left(\frac{1}{2}\right)^n = \frac{\text{Current }^{87} \mathrm{Rb}}{\text{Initial }^{87} \mathrm{Rb}}\]Substitute the fraction found from Step 2:\[\left(\frac{1}{2}\right)^n = 0.953\]Taking the logarithm of both sides:\[n \cdot \log_{10}(0.5) = \log_{10}(0.953)\]
04

Solve for n

Solve for \(n\) by isolating it:\[n = \frac{\log_{10}(0.953)}{\log_{10}(0.5)}\]Notice that:\[\log_{10}(0.953) \approx -0.0203 \,\text{and}\ \, \log_{10}(0.5) \approx -0.3010\]Hence:\[n \approx \frac{-0.0203}{-0.3010} \approx 0.0674\]
05

Calculate the Age of the Rock

Now, use the half-life to calculate the age of the rock:\[\text{Age} = n \cdot t_{1/2} = 0.0674 \times 4.8 \times 10^{10} \, \text{y}\]Calculating that gives us:\[\text{Age} \approx 3.24 \times 10^9 \, \text{years}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rubidium-Strontium Dating
Rubidium-Strontium dating is a form of radioactive dating that establishes the age of geological specimens through the decay process of isotopes. In this method, the key decay process involves rubidium-87, which decays into strontium-87 over time. Most geological samples like rocks contain these isotopes in measurable quantities, making them suitable subjects for dating.
  • Rubidium-87 is present in minerals from the time of their formation, but it decays over long periods.
  • Strontium-87 accumulates as a result of the decay, providing scientists with clues about the rock's age.
  • The current and original amounts of rubidium and strontium are used to estimate how many decay cycles, or half-lives, have elapsed.
  • By understanding this decay cycle, researchers can determine the time since the rock or mineral was formed.
This technique is instrumental in geology, giving insights into the Earth's history and the timeline of events such as volcanic eruptions and the formation of various landforms.
Half-Life Calculation
Understanding the half-life of a radioactive isotope is crucial in calculating how old something is. The half-life is the time required for half of the isotope present in a sample to decay. For rubidium-87, the half-life is quite long, at approximately 48 billion years.
  • This duration means that rubidium-87 is suitable for dating ancient formations because it remains in significant amounts even after billion-year time scales.
  • In calculations, the number of elapsed half-lives is determined using the proportion of the original isotope to its remaining quantity.
  • This method involves mathematical formulas that relate half-life to age.
Understanding the half-life of isotopes like rubidium-87 equips scientists with the ability to measure ages spanning over vast geological timelines, dating back to the formation of the Earth's crust.
Isotope Decay
Isotopes are forms of elements with the same number of protons but different numbers of neutrons. Isotope decay refers to the process by which an unstable isotope loses energy by emitting radiation, transitioning into a more stable form. The decay process for rubidium-87 involves its transformation into strontium-87.
  • In radioactive decay, isotopes lose atomic particles which changes them to different elements or isotopes.
  • Each decay cycle, characterized by a transition from one isotope to another, is a measurable and predictable process.
  • This conversion assists in calculating the time elapsed since the rock was formed, using the balance of isotopes present.
While isotope decay appears complex, it essentially follows predictable patterns that allow scientists to back-calculate the age and history of formations within the Earth's geology.
Beta Emission in Chemistry
Beta emission is a type of radioactive decay where an atomic nucleus releases a beta particle. This happens in isotopes like rubidium-87, which transforms into strontium-87 by undergoing beta decay.
  • A beta particle can be an electron or a positron, which is ejected from the nucleus.
  • This process alters the atomic number while maintaining the atomic mass, effectively transforming the element into a new one.
  • For rubidium-87, beta emission reduces the proton count in favor of strontium, which is one atomic number higher.
In chemistry, understanding beta emissions provides insight into the processes that change one element into another on atomic scales. For dating methods like rubidium-strontium dating, this principle forms the crux of measuring geological time.

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Most popular questions from this chapter

Tritium, or hydrogen- 3 , is formed in the upper atmosphere by cosmic rays, similar to the formation of carbon- 14 . Tritium has been used to determine the age of wines. A certain wine that has been aged in a bottle has a tritium content only \(64 \%\) of that in a similar wine of the same mass that has just been bottled. How long has the aged wine been in the bottle? The half-life of tritium is \(12.3 \mathrm{y}\).

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