Radioactive decay and mass spectrometry are often used to...

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.

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.

Key Concepts

Average Atomic Mass Calculation

Calculating the average atomic mass of an element involves taking a weighted average of the masses of its isotopes based on their relative abundances. This calculation is important when converting concentration data into meaningful quantities for radiometric dating calculations and when interpreting mass spectrometry results.

Chemical Analysis and Concentration Measurements (ppm)

Chemical analysis in parts per million (ppm) provides information on the concentration of elements within a rock sample. These measurements are used to quantify the amounts of radioactive parent isotopes and stable daughter isotopes, which are necessary inputs for computing the extent of radioactive decay and determining the age of the rock.

Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the masses and relative concentrations of atoms or molecules. In the context of radiometric dating, it is used to accurately determine isotopic ratios in rock samples. This high-precision measurement is crucial for deducing the original and current abundances of both the parent and daughter isotopes.

Isotopic Ratios

Isotopic ratios refer to the relative abundances of different isotopes of an element. These ratios are essential in radiometric dating as they allow scientists to compare the current amount of a daughter isotope to its original amount. Changes in these ratios, measured using techniques like mass spectrometry, provide the data needed to calculate the time since the rock cooled.

Radiometric Dating

Radiometric dating is a technique used to date materials such as rocks by comparing the relative abundances of radioactive parent isotopes and stable daughter isotopes. By knowing the initial ratio of these isotopes and the decay rate (via half-life), scientists can compute the elapsed time since the formation or cooling of the rock.

Half-Life

The half-life of a radioactive isotope is the time required for half of the original quantity of the isotope to decay. It is a critical factor in radiometric dating because it defines the time scale over which decay occurs. The half-life enables the conversion of measured isotopic ratios into time, providing the age of the rock sample or mineral.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This concept is central to understanding how one isotope transmutes into a different isotope (or element) over time. In radiometric dating, the known decay pattern and decay rate of elements provide a clock by which the age of rocks and minerals can be calculated.

Transcript

00:07 We can estimate the age of a rock, including a meteorite, by knowing the number of moles of the radioactive isotope that we're measuring that we originally start with and knowing how many remain after a set amount of time.

00:28 So for this problem, we know that we have in the meteorite, we have.

00:38 1 .8 millimoles of ribidium 87 and we have 1 .6 millimoles of strontium 87, which is a decay product of rubidium 87.

01:03 So whatever ribidium 87 we started with, some of it has decayed into strontium 87, all the strontium 87 that's in the meteorite came from the original rubidium 87.

01:19 So how does this help us? we can use the formula, the log of the number of moles at the present time divided by the number of moles that we originally started with equals the decay constant, the negative of the decay constant k times the amount of time that is past t.

01:47 So we have the number of moles remaining of ribidium 87.

01:55 That's our 1 .8 millimoles.

01:58 And we have the number of moles of rubidium 87 at time zero at our starting point.

02:04 That is 1 .8 plus 1 .6, which is 3 .4 millimoles.

02:10 Of ribidium 87.

02:13 I'm going to be filling all this in in just a moment for you.

02:18 We're trying to find t, which is the time that is past.

02:25 So t is what we're looking for.

02:28 But we also need the decay constant to solve for t.

02:34 So we get that from the half -life of rabidium.

02:41 87, which we're told is 4 .8 times 10 to the 10 years.

02:56 Now to find the decay constant, we can use the equation, k equals 6 .9, 0 .693 divided by the half -life.