- Last updated

- Save as PDF

- Page ID
- 364128

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vectorC}[1]{\textbf{#1}}\)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

Atoms that have the same number of protons but a different number of neutrons are isotopes. To identify an isotope we use the notation \({}_Z^A E\), where *E *is the element’s atomic symbol, *Z *is the element’s atomic number, and *A *is the element’s atomic mass number. Although an element’s different isotopes have the same chemical properties, their nuclear properties are not identical. The most important difference between isotopes is their stability. The nuclear configuration of a stable isotope remains constant with time. Unstable isotopes, however, disintegrate spontaneously, emitting radioactive decay particles as they transform into a more stable form.

An element’s atomic number, *Z*, is equal to the number of protons and its atomic mass, *A*, is equal to the sum of the number of protons and neutrons. We represent an isotope of carbon-13 as \(_{6}^{13} \text{C}\) because carbon has six protons and seven neutrons. Sometimes we omit *Z *from this notation—identifying the element and the atomic number is repetitive because all isotopes of carbon have six protons and any atom that has six protons is an isotope of carbon. Thus, ^{13}C and C–13 are alternative notations for this isotope of carbon.

## Types of Radioactive Decay Particles

The most important types of radioactive particles are alpha particles, beta particles, gamma rays, and X-rays. An alpha particle, \(\alpha\), is equivalent to a helium nucleus, \({}_2^4 \text{He}\). When an atom emits an alpha particle, the product is a new atom whose atomic number and atomic mass number are, respectively, 2 and 4 less than its unstable parent. The decay of uranium to thorium is one example of alpha emission.

\[_{92}^{238} \text{U} \longrightarrow _{90}^{234} \text{Th}+\alpha \nonumber \]

A beta particle, \(\beta\), comes in one of two forms. A negatron, \(_{-1}^0 \beta\), is produced when a neutron changes into a proton, increasing the atomic number by one, as shown here for lead.

\[_{82}^{214} \mathrm{Pb} \longrightarrow_{83}^{214} \mathrm{Bi} + _{-1}^{0} \beta \nonumber \]

The conversion of a proton to a neutron results in the emission of a positron, \(_{1}^0 \beta\).

\[_{15}^{30} \mathrm{P} \longrightarrow_{14}^{30} \mathrm{Si} + _{1}^{0} \beta \nonumber \]

A negatron, which is the more common type of beta particle, is equivalent to an electron.

The emission of an alpha or a beta particle often produces an isotope in an unstable, high energy state. This excess energy is released as a gamma ray, \(\gamma\), or as an X-ray. Gamma ray and X-ray emission may also occur without the release of an alpha particle or a beta particle.

## Radioactive Decay Rates

A radioactive isotope’s rate of decay, or activity, follows first-order kinetics

\[A-{t} = -\frac{d N}{d t}=\lambda N \label{13.1} \]

where *A *is the isotope’s activity, *N *is the number of radioactive atoms present in the sample at time *t*, and \(\lambda\) is the isotope’s decay constant. Activity is expressed as the number of disintegrations per unit time.

As with any first-order process, we can rewrite Equation \ref{13.1} in an integrated form.

\[N_{t}=N_{0} e^{-\lambda t} \label{13.2} \]

Substituting Equation \ref{13.2} into Equation \ref{13.1} gives

\[A_{t} = \lambda N_{0} e^{-\lambda t}=A_{0} e^{-\lambda t} \label{13.3} \]

If we measure a sample’s activity at time *t *we can determine the sample’s initial activity, *A*_{0}, or the number of radioactive atoms originally present in the sample, *N*_{0}.

An important characteristic property of a radioactive isotope is its half-life, *t*_{1/2}, which is the amount of time required for half of the radioactive atoms to disintegrate. For first-order kinetics the half-life is

\[t_{1 / 2}=\frac{0.693}{\lambda} \label{13.4} \]

Because the half-life is independent of the number of radioactive atoms, it remains constant throughout the decay process. For example, if 50% of the radioactive atoms remain after one half-life, then 25% remain after two half-lives, and 12.5% remain after three half-lives.

Suppose we begin with an *N*_{0} of 1200 atoms During the first half-life, 600 atoms disintegrate and 600 remain. During the second half-life, 300 of the 600 remaining atoms disintegrate, leaving 300 atoms or 25% of the original 1200 atoms. Of the 300 remaining atoms, only 150 remain after the third half-life, or 12.5% of the original 1200 atoms.

Kinetic information about a radioactive isotope usually is given in terms of its half-life because it provides a more intuitive sense of the isotope’s stability. Knowing, for example, that the decay constant for \(_{38}^{90}\text{Sr}\) is 0.0247 yr^{–1} does not give an immediate sense of how fast it disintegrates. On the other hand, knowing that its half-life is 28.1 yr makes it clear that the concentration of \(_{38}^{90}\text{Sr}\) in a sample remains essentially constant over a short period of time.

## Counting Statistics

Radioactivity does not follow a normal distribution because the possible outcomes are not continuous; that is, a sample can emit 1 or 2 or 3 alpha particles (or some other integer value) in a fixed intervale, but it cannot emit 2.59 alpha particles during that same interval. A Poisson distributionprovides the probability that a given number of events will occur in a fixed interval in time or space if the event has a known average rate and if each new event is independent of the preceding event. Mathematically a Poisson distribution is defined by the equation

\[P(X, \lambda) = \frac {e^{-\lambda} \lambda^X} {X !} \nonumber \]

where \(P(X, \lambda)\) is the probability that an event happens *X* times given the event’s average rate, \(\lambda\). The Poisson distribution has a theoretical mean, \(\mu\), and a theoretical variance, \(\sigma^2\), that are each equal to \(\lambda\).

##### Note

For a more detailed discussion of the distribution of data, including normal distributions and Poisson distributions, see Appendix 1.

The accuracy and precision of radiochemical methods generally are within the range of 1–5%. We can improve the precision—which is limited by the random nature of radioactive decay—by counting the emission of radioactive particles for as long a time as is practical. If the number of counts, *M*, is reasonably large (*M *≥ 100), and the counting period is significantly less than the isotope’s half-life, then the percent relative standard deviation for the activity, \((\sigma_A)_{rel}\), is approximately

\[\left(\sigma_{A}\right)_{\mathrm{rel}}=\frac{1}{\sqrt{M}} \times 100 \nonumber \]

For example, if we determine the activity by counting 10 000 radioactive particles, then the relative standard deviation is 1%. A radiochemical method’s sensitivity is inversely proportional to \((\sigma_A)_{rel}\), which means we can improve the sensitivity by counting more particles.

## Analysis of Radioactive Analytes

The concentration of a long-lived radioactive isotope remains essentially constant during the period of analysis. As shown in Example 32.1.1, we can use the sample’s activity to calculate the number of radioactive particles in the sample.

##### Example 32.1.1

The activity in a 10.00-mL sample of wastewater that contains \(_{38}^{90}\text{Sr}\) is \(9.07 \times 10^6\) disintegrations/s. What is the molar concentration of \(_{38}^{90}\text{Sr}\) in the sample? The half-life for \(_{38}^{90}\text{Sr}\) is 28.1 yr.

###### Solution

Solving Equation \ref{13.4} for \(\lambda\), substituting into Equation \ref{13.1}, and solving for *N *gives

\[N=\frac{A \times t_{1 / 2}}{0.693} \nonumber \]

Before we can determine the number of atoms of \(_{38}^{90}\text{Sr}\) in the sample we must express its activity and its half-life using the same units. Converting the half-life to seconds gives *t*_{1/2} as \(8.86 \times 10^8\) s; thus, there are

\[\frac{\left(9.07 \times 10^{6} \text { disintegrations/s }\right)\left(8.86 \times 10^{8} \text{ s}\right)}{0.693} = 1.16 \times 10^{16} \text{ atoms} _{38}^{90}\text{Sr} \nonumber \]

The concentration of \(_{38}^{90}\text{Sr}\) in the sample is

\[\frac{1.16 \times 10^{16} \text { atoms } _{38}^{90} \text{Sr}}{\left(6.022 \times 10^{23} \text { atoms/mol }\right)(0.01000 \mathrm{L})} = 1.93 \times 10^{-6} \text{ M } _{38}^{90}\text{Sr} \nonumber \]

The direct analysis of a short-lived radioactive isotope using the method outlined in Example 32.1.1 is less useful because it provides only a transient measure of the isotope’s concentration. Instead, we can measure its activity after an elapsed time, *t*, and use Equation \ref{13.3} to calculate *N*_{0}.

One example of a characterization application is the determination of a sample’s age based on the decay of a radioactive isotope naturally present in the sample. The most common example is carbon-14 dating, which is used to determine the age of natural organic materials. As cosmic rays pass through the upper atmosphere, some \(_7^{14}\text{N}\) atoms in the atmosphere capture high energy neutrons, converting them into \(_6^{14}\text{C}\). The \(_6^{14}\text{C}\) then migrates into the lower atmosphere where it oxidizes to form C-14 labeled CO_{2}. Animals and plants subsequently incorporate this labeled CO_{2 }into their tissues. Because this is a steady-state process, all plants and animals have the same ratio of \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) in their tissues. When an organism dies, the radioactive decay of \(_6^{14}\text{C}\) to \(_7^{14}\text{N}\) by \(_{-1}^0 \beta\) emission (*t *= 5730 years) leads to predictable reduction in the \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) ratio. We can use the change in this ratio to date samples that are as much as 30000 years old, although the precision of the analysis is best when the sample’s age is less than 7000 years. The accuracy of carbon-14 dating depends upon our assumption that the natural \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) ratio in the atmosphere is constant over time. Some variation in the ratio has occurred as the result of the increased consumption of fossil fuels and the production of \(_6^{14}\text{C}\) during the testing of nuclear weapons. A calibration curve prepared using samples of known age—examples of samples include tree rings, deep ocean sediments, coral samples, and cave deposits—limits this source of uncertainty.

There is no need to prepare a calibration curve for each analysis. Instead, there is a universal calibration curve known as IntCal. The calibration curve from 2013,IntCal13, is described in the following paper: Reimer, P. J., et. al. “IntCal13 and Marine 13 Radiocarbon Age Calibration Curve 0–50,000 Years Cal BP,” *Radiocarbon ***2013**, *55*, 1869–1887. This calibration spans 50 000 years before the present (BP).

##### Example 32.1.2

To determine the age of a fabric sample, the relative ratio of \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) was measured yielding a result of 80.9% of that found in modern fibers. How old is the fabric?

###### Solution

Equation \ref{13.3} and Equation \ref{13.4} provide us with a method to convert a change in the ratio of \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) to the fabric’s age. Letting *A*_{0}* *be the ratio of \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) in modern fibers, we assign it a value of 1.00. The ratio of \(_6^{14}\text{C}\) to \(_6^{12}\text{C}\) in the sample, *A*, is 0.809. Solving gives

\[t=\ln \frac{A_{0}}{A} \times \frac{t_{1 / 2}}{0.693}=\ln \frac{1.00}{0.809} \times \frac{5730 \text { yr }}{0.693}=1750 \text { yr } \nonumber \]

Other isotopes can be used to determine a sample’s age. The age of rocks, for example, has been determined from the ratio of the number of \(_{92}^{238}\text{U}\) to the number of stable \(_{82}^{206}\text{Pb}\) atoms produced by radioactive decay. For rocks that do not contain uranium, dating is accomplished by comparing the ratio of radioactive \(_{19}^{40}\text{K}\) to the stable \(_{18}^{40}\text{Ar}\). Another example is the dating of sediments collected from lakes by measuring the amount of \(_{82}^{210}\text{Pb}\) that is present.