Problem 56 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Rubidium-strontium dating. The rubidium isotope \({ }_{37}^{87} \mathrm{Rb}\), a \(\beta\) emitter with a half-life of \(4.75 \times 10^{10} \mathrm{yr}\), is used to determine the age of rocks and fossils. Rocks containing fossils of ancient animals contain a ratio of \({ }_{38}^{87} \mathrm{Sr}\) to \({ }_{37}^{87} \mathrm{Rb}\) of \(0.0260 .\) Assuming that there was no \({ }_{38}^{87} \mathrm{Sr}\) present when the rocks were formed, estimate the age of these fossils.

Step-by-Step Solution

Answer

The age of the fossils is approximately \(1.76 \times 10^9\) years.

Step 1: Understand the Problem

We're using rubidium-strontium dating to estimate the age of rocks. In this process, (\( \text{}_{}^{87} \mathrm{Rb} \)) decays into (\( \text{}_{}^{87} \mathrm{Sr} \)) over time. Given the ratio of these isotopes and the half-life of (\( \text{}_{}^{87} \mathrm{Rb} \)), we can estimate the age of the rock.

Step 2: Formula for Age in Radiometric Dating

We use the formula: \[ t = \frac{1}{\lambda} \ln\left(1 + \frac{N_{s}}{N_{r}} \right) \]where \(t\) is the age, \(\lambda\) is the decay constant, \(N_{s}\) is the number of daughter isotopes (\(\text{}_{}^{87} \mathrm{Sr}\)), and \(N_{r}\) is the number of parent isotopes (\(\text{}_{}^{87} \mathrm{Rb}\)).Also, \(\lambda = \frac{\ln 2}{T_{1/2}}\).

Step 3: Calculate the Decay Constant

The decay constant \(\lambda\) is given by:\[ \lambda = \frac{\ln 2}{T_{1/2}} \]Substituting the half-life of \(\text{}_{}^{87} \mathrm{Rb}\) as \(4.75 \times 10^{10} \text{ yr} \):\[ \lambda = \frac{0.693}{4.75 \times 10^{10}} \approx 1.46 \times 10^{-11} \text{ yr}^{-1} \]

Step 4: Substitute Known Values into the Formula

Given \(N_{s}/N_{r} = 0.0260\), we substitute \(\lambda\) and \(N_{s}/N_{r}\) into the age formula:\[ t = \frac{1}{1.46 \times 10^{-11}} \ln\left(1 + 0.0260 \right) \approx \frac{1}{1.46 \times 10^{-11}} \ln(1.026) \]

Step 5: Calculate the Age

Calculate \(\ln(1.026)\), then find \(t\):\[ \ln(1.026) \approx 0.0257 \]\[ t \approx \frac{0.0257}{1.46 \times 10^{-11}} \approx 1.76 \times 10^{9} \text{ yr} \]Thus, the age of the fossils is about \(1.76 \times 10^{9}\) years.

Key Concepts

Radiometric Dating

Radiometric dating is a fascinating method used to determine the age of objects, such as rocks and fossils, by analyzing the presence and ratio of certain radioactive elements. These elements, known as isotopes, undergo a process of decay over time. During this decay, a parent isotope breaks down to form a daughter isotope at a predictable rate. This predictable decay rate forms the basis of radiometric dating. By measuring the ratio of the parent isotope to the daughter isotope in a given sample, scientists can ascertain how much time has passed since the rock or fossil was formed.

• The age is calculated using the known decay rate of the isotope.
• It is a key tool in both geology and archaeology.
• Provides more precise dating than relative dating methods.

In rubidium-strontium dating, the decay of rubidium-87 ( ext{}_{37}^{87} ext{Rb}) into strontium-87 ( ext{}_{38}^{87} ext{Sr}) is used to date ancient geological formations.

Decay Constant

The decay constant is a crucial parameter in radiometric dating, reflecting how fast a radioactive isotope decays into its daughter isotope. It is denoted by the symbol \( \lambda \), and it's key to calculating the age of an object through the decay formula. Specifically, the decay constant is defined as the fraction of a group of parent isotopes that decay in a unit of time.

Understanding how to calculate the decay constant is essential when working with radiometric dating methods.

• It is expressed in terms of inverse time, typically \( \text{yr}^{-1} \).
• The relationship between the decay constant and the half-life of an isotope is \( \lambda = \frac{\ln 2}{T_{1/2}} \), where \( T_{1/2} \) is the half-life.
• This constant helps us determine the likelihood of decay occurring in a given timeframe.

For rubidium-87, with a half-life of 4.75 x 10^10 years, the decay constant informs us how quickly this isotope turns into strontium-87.

Half-Life

The concept of half-life is fundamental in understanding radiometric dating. It refers to the time required for half of the radioactive parent isotopes in a sample to decay into daughter isotopes. Knowing the half-life enables scientists to calculate the age of materials accurately.

Here's how half-life influences dating:

• Each radioactive isotope has a unique half-life, ranging from fractions of seconds to billions of years.
• For rubidium-87, the half-life is 4.75 x 10^{10} years, making it ideal for dating much older geological materials.
• The concept of half-life helps us relate the amount of original parent isotopes remaining to the amount that has decayed over time.

Essentially, by knowing the half-life and measuring isotopic ratios, the time elapsed since the formation of the rock or fossil can be estimated.

Isotopes

Isotopes are versions of an element that have the same number of protons but different numbers of neutrons. This difference in neutron number gives each isotope a different atomic mass. While most elements are composed of a mixture of isotopes, their radioactive properties vary considerably.

In radiometric dating, isotopes play a pivotal role:

• Rubidium-87 and strontium-87 are isotopes used in rubidium-strontium dating, with rubidium-87 being radioactive.
• The decay of a parent isotope (radioactive) to a daughter isotope (typically stable) is measured to determine age.
• The change in isotopic ratios over time acts like a natural clock, underpinned by physics and chemistry.

Understanding isotopes aids in grasping why certain elements are ideal for dating and how the dating process utilizes their natural decay to estimate age accurately.