Question

Radioactive decay and mass spectrometry are often used to date rocks after they have cooled from a magma. \(^{87} \mathrm{Rb}\) has a half-life of \(4.8 \times 10^{10}\) years and follows the radioactive decay $$^{87} \mathrm{Rb} \longrightarrow^{87} \mathrm{Sr}+\beta^{-}$$ A rock was dated by assaying the product of this decay. The mass spectrum of a homogenized sample of rock showed the \(^{87} \mathrm{Sr} /^{86} \mathrm{Sr}\) ratio to be \(2.25 .\) Assume that the original \(^{87} \mathrm{Sr} /^{86} \mathrm{Sr}\) ratio was 0.700 when the rock cooled. Chemical analysis of the rock gave \(15.5 \mathrm{ppm}\) Sr and 265.4 ppm \(\mathrm{Rb},\) using the average atomic masses from a periodic table. The other isotope ratios were \(^{86} \mathrm{Sr} /^{88} \mathrm{Sr}=\) 0.119 and \(^{84} \mathrm{Sr} /^{88} \mathrm{Sr}=0.007 .\) The isotopic ratio for \(^{87} \mathrm{Rb} /^{85} \mathrm{Rb}\) is 0.330. The isotopic masses are as follows:Calculate the following: (a) the average atomic mass of Sr in the rock (b) the original concentration of \(\mathrm{Rb}\) in the rock in \(\mathrm{ppm}\) (c) the percentage of rubidium- 87 decayed in the rock (d) the time since the rock cooled.

Short answer

The average atomic mass of Sr, original concentration of Rb, percentage decay of Rb and time since rock cooled need to be calculated with the given data using concepts of radioactivity, isotopic ratios and decay constants.

Step-by-step solution

Calculate the average atomic mass of Sr in the rock

Given, the isotopic ratios are \(^{86} \mathrm{Sr} / ^{88} \mathrm{Sr} = 0.119\) and \(^{84} \mathrm{Sr} / ^{88} \mathrm{Sr} = 0.007\). We need to find the ratio \(^{88} \mathrm{Sr} / ^{86} \mathrm{Sr}\) and \(^{88} \mathrm{Sr} / ^{84} \mathrm{Sr}\) by taking reciprocal of the given ratios. Calculate abundance of each isotope and find the average atomic mass using the formula, average atomic mass = sum of (mass of isotope * relative abundance of isotope).

Calculate the original concentration of Rb in the rock in ppm

Given, chemical analysis of the rock gave \(15.5 \mathrm{ppm}\) Sr and \(265.4 \mathrm{ppm}\) Rb and isotopic ratio \(^{87} \mathrm{Rb} / ^{85} \mathrm{Rb} = 0.330\). Calculate isotopic ratios in ppm by multiplying the given ppm with respective isotopic ratios.

Find the percentage of rubidium- 87 decayed in the rock

From the concentration of Rb determined in step 2 and the present concentration, we calculate the difference which will show the decayed Rb. Divide the decayed Rb concentration by the initial Rb concentration and multiply by 100 to get the percentage.

Calculate the time since the rock cooled

The decay of Rb can be calculated using the equation for radioactive decay, \(N=N_{0} e^{-λt}\). Rearrange the formula to find the elapsed time \(t\), where \(λ\), the decay constant, is found using the given half life \(T_{1/2}\) by formula \(λ=0.693/T_{1/2}\). Solve for \(t\).

Key concepts

Mass Spectrometry

Mass spectrometry is a sophisticated analytical technique used to measure the masses of individual atoms or molecules. Scientists use this powerful tool to determine the composition of a sample by ionizing the chemical species of interest and sorting the resulting ions based on their mass-to-charge ratio. When it comes to radioactive decay dating, mass spectrometry can measure the abundances of isotopes, including those of a radioactive parent and its stable daughter. The analysis of these isotopic abundances, particularly in the case of rubidium-strontium (Rb-Sr) dating, is crucial.

For instance, in the exercise, the students are faced with the task of dating a rock by analyzing the ratio of 87Sr to 86Sr isotopes using mass spectrometry. By understanding the isotopic composition and relative abundances, it's possible to calculate the original concentration of rubidium in the rock before decay began. This foundational step is critical in determining the rock's age.

Isotopic Ratio

Understanding Isotopic Ratios in Dating Rocks

Isotopic ratio is the relative amount of one isotope to another within a particular element and is a vital concept in geochronology. In the context of radioactive decay dating, such as Rb-Sr dating in our exercise, the isotopic ratio allows scientists to backtrack and determine a rock's age. The ratio of 87Sr to 86Sr found in a sample is compared to what is presumed to be the initial ratio when the rock formed.

By determining the change in this ratio over time, we can get a snapshot of the radioactive decay process that has occurred. This ratio, coupled with the knowledge of half-life of the parent isotope, enables the calculation of the time elapsed since the rock solidified from magma. It's important to present this to students in an engaging way that shows how a seemingly simple ratio can reveal the history of geological samples.

Half-Life

The concept of half-life is central to understanding radioactive decay and dating rocks. Half-life refers to the amount of time it takes for half of a given quantity of a radioactive isotope to decay into its daughter isotope. In our exercise with Rb-Sr dating, the half-life of 87Rb is stated to be 4.8 billion years.

Understanding the half-life allows us to calculate the decay constant, which is used to determine the age of the rock since it cooled. The longer the half-life, the slower the decay process, and consequently, the older the rock can be for it to still have measurable parent isotopes. In educational content, it's important to articulate that half-life is an exponential decay process, which means that every half-life period that passes, another half of the remaining radioactive material will have decayed. This concept is fundamental in the mathematical calculation of the rock's age in step 4 of the solution.