{"id":3667,"date":"2015-05-06T03:51:00","date_gmt":"2015-05-06T03:51:00","guid":{"rendered":"https:\/\/courses.candelalearning.com\/oschemtemp\/?post_type=chapter&#038;p=3667"},"modified":"2016-08-09T18:35:31","modified_gmt":"2016-08-09T18:35:31","slug":"radioactive-decay-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/chapter\/radioactive-decay-2\/","title":{"raw":"Radioactive Decay","rendered":"Radioactive Decay"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>LEARNING OBJECTIVES<\/h3>\r\nBy the end of this module, you will be able to:\r\n<ul>\r\n\t<li>Recognize common modes of radioactive decay<\/li>\r\n\t<li>Identify common particles and energies involved in nuclear decay reactions<\/li>\r\n\t<li>Write and balance nuclear decay equations<\/li>\r\n\t<li>Calculate kinetic parameters for decay processes, including half-life<\/li>\r\n\t<li>Describe common radiometric dating techniques<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-idp102942608\">Following the somewhat serendipitous discovery of radioactivity by Becquerel, many prominent scientists began to investigate this new, intriguing phenomenon. Among them were Marie Curie (the first woman to win a Nobel Prize, and the only person to win two Nobel Prizes in different sciences\u2014chemistry and physics), who was the first to coin the term \u201cradioactivity,\u201d and Ernest Rutherford (of gold foil experiment fame), who investigated and named three of the most common types of radiation. During the beginning of the twentieth century, many radioactive substances were discovered, the properties of radiation were investigated and quantified, and a solid understanding of radiation and nuclear decay was developed.<\/p>\r\nThe spontaneous change of an unstable nuclide into another is <strong>radioactive decay<\/strong>. The unstable nuclide is called the <strong>parent nuclide<\/strong>; the nuclide that results from the decay is known as the <strong>daughter nuclide<\/strong>. The daughter nuclide may be stable, or it may decay itself. The radiation produced during radioactive decay is such that the daughter nuclide lies closer to the band of stability than the parent nuclide, so the location of a nuclide relative to the band of stability can serve as a guide to the kind of decay it will undergo.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214159\/CNX_Chem_21_03_Reaction1.jpg\" alt=\"A diagram shows two spheres composed of many smaller white and green spheres connected by a right-facing arrow with another, down-facing arrow coming off of it. The left sphere, labeled \u201cParent nucleus uranium dash 238\u201d has two white and two green spheres that are near one another and are outlined in red. These two green and two white spheres are shown near the tip of the down-facing arrow and labeled \u201calpha particle.\u201d The right sphere, labeled \u201cDaughter nucleus radon dash 234,\u201d looks the same as the left, but has a space for four smaller spheres outlined with a red dotted line.\" width=\"975\" height=\"347\" data-media-type=\"image\/jpeg\" \/> Figure 1. A nucleus of uranium-238 (the parent nuclide) undergoes \u03b1 decay to form thorium-234 (the daughter nuclide). The alpha particle removes two protons (green) and two neutrons (gray) from the uranium-238 nucleus.[\/caption]\r\n\r\n<section id=\"fs-idp242071680\" data-depth=\"1\">\r\n<div class=\"textbox\" data-type=\"title\">Although the radioactive decay of a nucleus is too small to see with the naked eye, we can indirectly view radioactive decay in an environment called a cloud chamber. Click <a href=\"https:\/\/www.youtube.com\/watch?v=pewTySxfTQk\" target=\"_blank\" rel=\"nofollow\">here<\/a> to learn about cloud chambers and to view an interesting Cloud Chamber Demonstration from the Jefferson Lab.<\/div>\r\n<h2 data-type=\"title\">Types of Radioactive Decay<\/h2>\r\nErnest Rutherford\u2019s experiments involving the interaction of radiation with a magnetic or electric field\u00a0helped him determine that one type of radiation consisted of positively charged and relatively massive \u03b1 particles; a second type was made up of negatively charged and much less massive \u03b2 particles; and a third was uncharged electromagnetic waves, \u03b3 rays. We now know that \u03b1 particles are high-energy helium nuclei, \u03b2 particles are high-energy electrons, and \u03b3 radiation compose high-energy electromagnetic radiation. We classify different types of radioactive decay by the radiation produced.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214200\/CNX_Chem_21_03_Radiation.jpg\" alt=\"A diagram is shown. A gray box on the left side of the diagram labeled \u201cLead block\u201d has a chamber hollowed out in the center in which a sample labeled \u201cRadioactive substance\u201d is placed. A blue beam is coming from the sample, out of the block, and passing through two horizontally placed plates that are labeled \u201cElectrically charged plates.\u201d The top plate is labeled with a positive sign while the bottom plate is labeled with a negative sign. The beam is shown to break into three beams as it passes in between the plates; in order from top to bottom, they are red, labeled \u201cbeta rays,\u201d purple labeled \u201cgamma rays\u201d and green labeled \u201calpha rays.\u201d The beams are shown to hit a vertical plate labeled \u201cPhotographic plate\u201d on the far right side of the diagram.\" width=\"975\" height=\"428\" data-media-type=\"image\/jpeg\" \/> Figure 2. Alpha particles, which are attracted to the negative plate and deflected by a relatively small amount, must be positively charged and relatively massive. Beta particles, which are attracted to the positive plate and deflected a relatively large amount, must be negatively charged and relatively light. Gamma rays, which are unaffected by the electric field, must be uncharged.[\/caption]\r\n<p id=\"fs-idm41800032\"><strong>Alpha (\u03b1) decay<\/strong> is the emission of an \u03b1 particle from the nucleus. For example, polonium-210 undergoes \u03b1 decay:<\/p>\r\n\r\n<div id=\"fs-idp134926448\" data-type=\"equation\">[latex]{}_{\\phantom{1}84}{}^{210}\\text{Po}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{2}{}^{4}\\text{He}+{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}84}{}^{210}\\text{Po}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{2}{}^{4}\\alpha+{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\r\n<p id=\"fs-idp202270256\">Alpha decay occurs primarily in heavy nuclei (A &gt; 200, Z &gt; 83). Because the loss of an \u03b1 particle gives a daughter nuclide with a mass number four units smaller and an atomic number two units smaller than those of the parent nuclide, the daughter nuclide has a larger n:p ratio than the parent nuclide. If the parent nuclide undergoing \u03b1 decay lies below the band of stability, the daughter nuclide will lie closer to the band.<\/p>\r\n<p id=\"fs-idm2875904\"><strong>Beta (\u03b2) decay<\/strong> is the emission of an electron from a nucleus. Iodine-131 is an example of a nuclide that undergoes \u03b2 decay:<\/p>\r\n\r\n<div id=\"fs-idm19827888\" data-type=\"equation\">[latex]{}_{\\phantom{1}53}{}^{131}\\text{I}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}54}{}^{131}\\text{X}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}53}{}^{131}\\text{I}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{-1}{}^{\\phantom{1}0}\\beta_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}54}{}^{131}\\text{Xe}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\r\n<p id=\"fs-idp100287120\">Beta decay, which can be thought of as the conversion of a neutron into a proton and a \u03b2 particle, is observed in nuclides with a large n:p ratio. The beta particle (electron) emitted is from the atomic nucleus and is not one of the electrons surrounding the nucleus. Such nuclei lie above the band of stability. Emission of an electron does not change the mass number of the nuclide but does increase the number of its protons and decrease the number of its neutrons. Consequently, the n:p ratio is decreased, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.<\/p>\r\n<p id=\"fs-idp8043424\"><strong>Gamma emission (\u03b3 emission)<\/strong> is observed when a nuclide is formed in an excited state and then decays to its ground state with the emission of a \u03b3 ray, a quantum of high-energy electromagnetic radiation. The presence of a nucleus in an excited state is often indicated by an asterisk (*). Cobalt-60 emits \u03b3 radiation and is used in many applications including cancer treatment:<\/p>\r\n\r\n<div id=\"fs-idp5264576\" data-type=\"equation\">[latex]{}_{27}{}^{60}\\text{Co*}\\longrightarrow {}_{0}{}^{0}\\gamma+{}_{27}{}^{60}\\text{Co}[\/latex]<\/div>\r\n<p id=\"fs-idp50313008\">There is no change in mass number or atomic number during the emission of a \u03b3 ray unless the \u03b3 emission accompanies one of the other modes of decay.<\/p>\r\n<p id=\"fs-idp18009856\"><strong>Positron emission (\u03b2<sup>+<\/sup> decay<\/strong>) is the emission of a positron from the nucleus. Oxygen-15 is an example of a nuclide that undergoes positron emission:<\/p>\r\n\r\n<div id=\"fs-idm9404976\" data-type=\"equation\">[latex]{}_{\\phantom{1}8}{}^{15}\\text{O}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}7}{}^{15}\\text{N}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}8}{}^{15}\\text{O}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{+1}{}^{\\phantom{1}0}\\beta_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}7}{}^{15}\\text{N}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\r\n<p id=\"fs-idp125268336\">Positron emission is observed for nuclides in which the n:p ratio is low. These nuclides lie below the band of stability. Positron decay is the conversion of a proton into a neutron with the emission of a positron. The n:p ratio increases, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.<\/p>\r\n<p id=\"fs-idp123320528\"><strong>Electron capture<\/strong> occurs when one of the inner electrons in an atom is captured by the atom\u2019s nucleus. For example, potassium-40 undergoes electron capture:<\/p>\r\n\r\n<div id=\"fs-idm22074816\" data-type=\"equation\">[latex]{}_{19}{}^{40}\\text{K}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{18}{}^{40}\\text{Ar}[\/latex]<\/div>\r\n<p id=\"fs-idm42894208\">Electron capture occurs when an inner shell electron combines with a proton and is converted into a neutron. The loss of an inner shell electron leaves a vacancy that will be filled by one of the outer electrons. As the outer electron drops into the vacancy, it will emit energy. In most cases, the energy emitted will be in the form of an X-ray. Like positron emission, electron capture occurs for \u201cproton-rich\u201d nuclei that lie below the band of stability. Electron capture has the same effect on the nucleus as does positron emission: The atomic number is decreased by one and the mass number does not change. This increases the n:p ratio, and the daughter nuclide lies closer to the band of stability than did the parent nuclide. Whether electron capture or positron emission occurs is difficult to predict. The choice is primarily due to kinetic factors, with the one requiring the smaller activation energy being the one more likely to occur.<\/p>\r\n<p id=\"fs-idp132245344\">Figure 3 summarizes these types of decay, along with their equations and changes in atomic and mass numbers.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214202\/CNX_Chem_21_03_RadioDecay.jpg\" alt=\"This table has four columns and six rows. The first row is a header row and it labels each column: \u201cType,\u201d \u201cNuclear equation,\u201d \u201cRepresentation,\u201d and \u201cChange in mass \/ atomic numbers.\u201d Under the \u201cType\u201d column are the following: \u201cAlpha decay,\u201d \u201cBeta decay,\u201d \u201cGamma decay,\u201d \u201cPositron emission,\u201d and \u201cElectron capture.\u201d Under the \u201cNuclear equation\u201d column are several equations. Each begins with superscript A stacked over subscript Z X. There is a large gap of space and then the following equations: \u201csuperscript 4 stacked over subscript 2 He plus superscript A minus 4 stacked over subscript Z minus 2 Y,\u201d \u201csuperscript 0 stacked over subscript negative 1 e plus superscript A stacked over subscript Z plus 1 Y,\u201d \u201csuperscript 0 stacked over subscript 0 lowercase gamma plus superscript A stacked over subscript Z Y,\u201d \u201csuperscript 0 stacked over subscript positive 1 e plus superscript A stacked over subscript Y minus 1 Y,\u201d and \u201csuperscript 0 stacked over subscript negative 1 e plus superscript A stacked over subscript Y minus 1 Y.\u201d Under the \u201cRepresentation\u201d column are the five diagrams. The first shows a cluster of green and white spheres. A section of the cluster containing two white and two green spheres is outlined. There is a right-facing arrow pointing to a similar cluster as previously described, but the outlined section is missing. From the arrow another arrow branches off and points downward. The small cluster to two white spheres and two green spheres appear at the end of the arrow. The next diagram shows the same cluster of white and green spheres. One white sphere is outlined. There is a right-facing arrow to a similar cluster, but the white sphere is missing. Another arrow branches off the main arrow and a red sphere with a negative sign appears at the end. The next diagram shows the same cluster of white and green spheres. The whole sphere is outlined and labeled, \u201cexcited nuclear state.\u201d There is a right-facing arrow that points to the same cluster. No spheres are missing. Off the main arrow is another arrow which points to a purple squiggle arrow which in turn points to a lowercase gamma. The next diagram shows the same cluster of white and green spheres. One green sphere is outlined. There is a right-facing arrow to a similar cluster, but the green sphere is missing. Another arrow branches off the main arrow and a red sphere with a positive sign appears at the end. The next diagram shows the same cluster of white and green spheres. One green sphere is outlined. There is a right-facing arrow to a similar cluster, but the green sphere is missing. Two other arrows branch off the main arrow. The first shows a gold sphere with a negative sign joining with the right-facing arrow. The secon points to a blue squiggle arrow labeled, \u201cX-ray.\u201d Under the \u201cChange in mass \/ atomic numbers\u201d column are the following: \u201cA: decrease by 4, Z: decrease by 2,\u201d \u201cA: unchanged, Z: increased by 1,\u201d \u201cA: unchanged, Z: unchanged,\u201d \u201cA: unchanged, Z: unchanged,\u201d \u201cA: unchanged, Z: decrease by 1,\u201d and \u201cA: unchanged, Z: decrease by 1.\u201d\" width=\"1300\" height=\"865\" data-media-type=\"image\/jpeg\" \/> Figure 3. This table summarizes the type, nuclear equation, representation, and any changes in the mass or atomic numbers for various types of decay.[\/caption]\r\n\r\n<div id=\"fs-idp54313680\" class=\"chemistry everyday-life textbox shaded\" data-type=\"note\">\r\n<h3 data-type=\"title\">PET Scan<\/h3>\r\n<p id=\"fs-idp214075136\">Positron emission tomography (PET) scans use radiation to diagnose and track health conditions and monitor medical treatments by revealing how parts of a patient\u2019s body function (Figure 4). To perform a PET scan, a positron-emitting radioisotope is produced in a cyclotron and then attached to a substance that is used by the part of the body being investigated. This \u201ctagged\u201d compound, or radiotracer, is then put into the patient (injected via IV or breathed in as a gas), and how it is used by the tissue reveals how that organ or other area of the body functions.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214204\/CNX_Chem_21_03_PETScan.jpg\" alt=\"Three pictures are shown and labeled \u201ca,\u201d \u201cb\u201d and \u201cc.\u201d Picture a shows a machine with a round opening connected to an examination table. Picture b is a medical scan of the top of a person\u2019s head and shows large patches of yellow and red and smaller patches of blue, green and purple highlighting. Picture c also shows a medical scan of the top of a person\u2019s head, but this image is mostly colored in blue and purple with very small patches of red and yellow.\" width=\"975\" height=\"332\" data-media-type=\"image\/jpeg\" \/> Figure 4. A PET scanner (a) uses radiation to provide an image of how part of a patient\u2019s body functions. The scans it produces can be used to image a healthy brain (b) or can be used for diagnosing medical conditions such as Alzheimer\u2019s disease (c). (credit a: modification of work by Jens Maus)[\/caption]\r\n<p id=\"fs-idp13254880\">For example, F-18 is produced by proton bombardment of <sup>18<\/sup>O [latex]\\left({}_{\\phantom{1}8}{}^{18}\\text{O}_{\\phantom{}}^{\\phantom{}}+{}_{1}{}^{1}\\text{p}\\longrightarrow {}_{\\phantom{1}9}{}^{18}\\text{F}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\right)[\/latex] and incorporated into a glucose analog called fludeoxyglucose (FDG). How FDG is used by the body provides critical diagnostic information; for example, since cancers use glucose differently than normal tissues, FDG can reveal cancers. The <sup>18<\/sup>F emits positrons that interact with nearby electrons, producing a burst of gamma radiation. This energy is detected by the scanner and converted into a detailed, three-dimensional, color image that shows how that part of the patient\u2019s body functions. Different levels of gamma radiation produce different amounts of brightness and colors in the image, which can then be interpreted by a radiologist to reveal what is going on. PET scans can detect heart damage and heart disease, help diagnose Alzheimer\u2019s disease, indicate the part of a brain that is affected by epilepsy, reveal cancer, show what stage it is, and how much it has spread, and whether treatments are effective. Unlike magnetic resonance imaging and X-rays, which only show how something looks, the big advantage of PET scans is that they show how something functions. PET scans are now usually performed in conjunction with a computed tomography scan.<\/p>\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-idp89067520\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Radioactive Decay Series<\/h2>\r\n<p id=\"fs-idp54304864\">The naturally occurring radioactive isotopes of the heaviest elements fall into chains of successive disintegrations, or decays, and all the species in one chain constitute a radioactive family, or <strong>radioactive decay series<\/strong>. Three of these series include most of the naturally radioactive elements of the periodic table. They are the uranium series, the actinide series, and the thorium series. The neptunium series is a fourth series, which is no longer significant on the earth because of the short half-lives of the species involved. Each series is characterized by a parent (first member) that has a long half-life and a series of daughter nuclides that ultimately lead to a stable end-product\u2014that is, a nuclide on the band of stability (Figure 5). In all three series, the end-product is a stable isotope of lead. The neptunium series, previously thought to terminate with bismuth-209, terminates with thallium-205.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214206\/CNX_Chem_21_03_DecayS.jpg\" alt=\"A graph is shown where the x-axis is labeled \u201cNumber of neutrons, open parenthesis, n, close parenthesis\u201d and has values of 122 to 148 in increments of 2. The y-axis is labeled \u201cAtomic number\u201d and has values of 80 to 92 in increments of 1. Two types of arrows are used in this graph to connect the points. Green arrows are labeled as \u201calpha decay\u201d while red arrows are labeled \u201cbeta decay.\u201d Beginning at the point \u201c92, 146\u201d that is labeled \u201csuperscript 238, U,\u201d a green arrow connects this point to the second point \u201c90, 144\u201d which is labeled \u201csuperscript 234, T h.\u201d A red arrow connect this to the third point \u201c91, 143\u201d which is labeled \u201csuperscript 234, P a\u201d which is connected to the fourth point \u201c92, 142\u201d by a red arrow and which is labeled \u201csuperscript 234, U.\u201d A green arrow leads to the next point, \u201c90, 140\u201d which is labeled \u201csuperscript 230, T h\u201d and is connected by a green arrow to the sixth point, \u201c88, 138\u201d which is labeled \u201csuperscript 226, R a\u201d that is in turn connected by a green arrow to the seventh point \u201c86, 136\u201d which is labeled \u201csuperscript 222, Ra.\u201d The eighth point, at \u201c84, 134\u201d is labeled \u201csuperscript 218, P o\u201d and has green arrows leading to it and away from it to the ninth point \u201c82, 132\u201d which is labeled \u201csuperscript 214, Pb\u201d which is connected by a red arrow to the tenth point, \u201c83, 131\u201d which is labeled \u201csuperscript 214, B i.\u201d A red arrow leads to the eleventh point \u201c84, 130\u201d which is labeled \u201csuperscript 214, P o\u201d and a green arrow leads to the twelvth point \u201c82, 128\u201d which is labeled \u201csuperscript 210, P b.\u201d A red arrow leads to the thirteenth point \u201c83, 127\u201d which is labeled \u201csuperscript 210, B i\u201d and a red arrow leads to the fourteenth point \u201c84, 126\u201d which is labeled \u201csuperscript 210, P o.\u201d The final point is labeled \u201c82, 124\u201d and \u201csuperscript 206, P b.\u201d\" width=\"1300\" height=\"856\" data-media-type=\"image\/jpeg\" \/> Figure 5. Uranium-238 undergoes a radioactive decay series consisting of 14 separate steps before producing stable lead-206. This series consists of eight \u03b1 decays and six \u03b2 decays.[\/caption]\r\n\r\n<\/section><section id=\"fs-idp38328768\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Radioactive Half-Lives<\/h2>\r\n<p id=\"fs-idp135905344\">Radioactive decay follows first-order kinetics. Since first-order reactions have already been covered in detail in the kinetics chapter, we will now apply those concepts to nuclear decay reactions. Each radioactive nuclide has a characteristic, constant <strong>half-life<\/strong> (<em data-effect=\"italics\">t<\/em><sub>1\/2<\/sub>), the time required for half of the atoms in a sample to decay. An isotope\u2019s half-life allows us to determine how long a sample of a useful isotope will be available, and how long a sample of an undesirable or dangerous isotope must be stored before it decays to a low-enough radiation level that is no longer a problem.<\/p>\r\n<p id=\"fs-idm89252176\">For example, cobalt-60, an isotope that emits gamma rays used to treat cancer, has a half-life of 5.27 years (Figure 6). In a given cobalt-60 source, since half of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] nuclei decay every 5.27 years, both the amount of material and the intensity of the radiation emitted is cut in half every 5.27 years. (Note that for a given substance, the intensity of radiation that it produces is directly proportional to the rate of decay of the substance and the amount of the substance.) This is as expected for a process following first-order kinetics. Thus, a cobalt-60 source that is used for cancer treatment must be replaced regularly to continue to be effective.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214207\/CNX_Chem_21_03_HalfLife.jpg\" alt=\"A graph, titled \u201cC o dash 60 Decay,\u201d is shown where the x-axis is labeled \u201cC o dash 60 remaining, open parenthesis, percent sign, close parenthesis\u201d and has values of 0 to 100 in increments of 25. The y-axis is labeled \u201cNumber of half dash lives\u201d and has values of 0 to 5 in increments of 1. The first point, at \u201c0, 100\u201d has a circle filled with tiny dots drawn near it labeled \u201c10 g.\u201d The second point, at \u201c1, 50\u201d has a smaller circle filled with tiny dots drawn near it labeled \u201c5 g.\u201d The third point, at \u201c2, 25\u201d has a small circle filled with tiny dots drawn near it labeled \u201c2.5 g.\u201d The fourth point, at \u201c3, 12.5\u201d has a very small circle filled with tiny dots drawn near it labeled \u201c1.25 g.\u201d The last point, at \u201c4, 6.35\u201d has a tiny circle filled with tiny dots drawn near it labeled.\u201d625 g.\u201d\" width=\"975\" height=\"594\" data-media-type=\"image\/jpeg\" \/> Figure 6. For cobalt-60, which has a half-life of 5.27 years, 50% remains after 5.27 years (one half-life), 25% remains after 10.54 years (two half-lives), 12.5% remains after 15.81 years (three half-lives), and so on.[\/caption]\r\n<p id=\"fs-idm84262544\">Since nuclear decay follows first-order kinetics, we can adapt the mathematical relationships used for first-order chemical reactions. We generally substitute the number of nuclei, <em data-effect=\"italics\">N<\/em>, for the concentration. If the rate is stated in nuclear decays per second, we refer to it as the activity of the radioactive sample. The rate for radioactive decay is:<\/p>\r\n<p id=\"fs-idp16246080\">decay rate = <em data-effect=\"italics\">\u03bbN<\/em> with <em data-effect=\"italics\">\u03bb<\/em> = the decay constant for the particular radioisotope<\/p>\r\n<p id=\"fs-idm22877072\">The decay constant, <em data-effect=\"italics\">\u03bb<\/em>, which is the same as a rate constant discussed in the kinetics chapter. It is possible to express the decay constant in terms of the half-life, <em data-effect=\"italics\">t<\/em><sub>1\/2<\/sub>:<\/p>\r\n\r\n<div id=\"fs-idp219598000\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{{t}_{1\\text{\/}2}}\\text{or}{t}_{1\\text{\/}2}=\\frac{\\text{ln 2}}{\\lambda }=\\frac{0.693}{\\lambda }[\/latex]<\/div>\r\n<p id=\"fs-idp20089360\">The first-order equations relating amount, <em data-effect=\"italics\">N<\/em>, and time are:<\/p>\r\n\r\n<div id=\"fs-idm4205984\" data-type=\"equation\">[latex]{N}_{t}={N}_{0}{e}^{-kt}\\text{or}t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)[\/latex]<\/div>\r\n<p id=\"fs-idp61393104\">where <em data-effect=\"italics\">N<\/em><sub>0<\/sub> is the initial number of nuclei or moles of the isotope, and <em data-effect=\"italics\">N<sub>t<\/sub><\/em> is the number of nuclei\/moles remaining at time <em data-effect=\"italics\">t<\/em>. Example 1 applies these calculations to find the rates of radioactive decay for specific nuclides.<\/p>\r\n\r\n<div id=\"fs-idp79391408\" class=\"textbox shaded\" data-type=\"example\">\r\n<h3>Example 1<\/h3>\r\n<h4 id=\"fs-idm1797296\"><span data-type=\"title\">Rates of Radioactive Decay<\/span><\/h4>\r\n[latex]{}_{27}{}^{60}\\text{Co}[\/latex] decays with a half-life of 5.27 years to produce [latex]{}_{28}{}^{60}\\text{Ni}[\/latex].\r\n<p id=\"fs-idp30533728\">(a) What is the decay constant for the radioactive disintegration of cobalt-60?<\/p>\r\n<p id=\"fs-idp132429920\">(b) Calculate the fraction of a sample of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] isotope that will remain after 15 years.<\/p>\r\n<p id=\"fs-idp176097216\">(c) How long does it take for a sample of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] to disintegrate to the extent that only 2.0% of the original amount remains?<\/p>\r\n\r\n<h4 id=\"fs-idm68606544\"><span data-type=\"title\">Solution<\/span><\/h4>\r\n(a) The value of the rate constant is given by:\r\n<div id=\"fs-idp17616736\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{5.27\\text{y}}=0.132{\\text{y}}^{-1}[\/latex]<\/div>\r\n<p id=\"fs-idp141655056\">(b) The fraction of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] that is left after time <em data-effect=\"italics\">t<\/em> is given by [latex]\\frac{{N}_{t}}{{N}_{0}}[\/latex]. Rearranging the first-order relationship <em data-effect=\"italics\">N<sub>t<\/sub><\/em> = <em data-effect=\"italics\">N<\/em><sub>0<\/sub><em data-effect=\"italics\">e<\/em><sup>\u2013<em data-effect=\"italics\">\u03bbt<\/em><\/sup> to solve for this ratio yields:<\/p>\r\n\r\n<div id=\"fs-idp16374688\" data-type=\"equation\">[latex]\\frac{{N}_{t}}{{N}_{0}}={e}^{-\\lambda t}={e}^{-\\left(0.132\\text{\/y}\\right)\\left(15.0\\text{\/y}\\right)}=0.138[\/latex]<\/div>\r\n<p id=\"fs-idm19598048\">The fraction of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] that will remain after 15.0 years is 0.138. Or put another way, 13.8% of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] originally present will remain after 15 years.<\/p>\r\n<p id=\"fs-idp82234704\">(c) 2.00% of the original amount of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] is equal to 0.0200 [latex]\\times [\/latex] <em data-effect=\"italics\">N<\/em><sub>0<\/sub>. Substituting this into the equation for time for first-order kinetics, we have:<\/p>\r\n\r\n<div id=\"fs-idm21044912\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)=-\\frac{1}{0.132{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{0.0200\\times {N}_{0}}{{N}_{0}}\\right)=29.6\\text{y}[\/latex]<\/div>\r\n<h4 id=\"fs-idp7893440\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\r\nRadon-222, [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex], has a half-life of 3.823 days. How long will it take a sample of radon-222 with a mass of 0.750 g to decay into other elements, leaving only 0.100 g of radon-222?\r\n<div id=\"fs-idp123030208\" data-type=\"note\">\r\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>11.1 days<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-idp76857680\">Because each nuclide has a specific number of nucleons, a particular balance of repulsion and attraction, and its own degree of stability, the half-lives of radioactive nuclides vary widely. For example: the half-life of [latex]{}_{\\phantom{1}83}{}^{209}\\text{Bi}_{\\phantom{}}^{\\phantom{}}[\/latex] is 1.9 [latex]\\times [\/latex] 10<sup>19<\/sup> years; [latex]{}_{\\phantom{1}94}{}^{239}\\text{Ra}_{\\phantom{}}^{\\phantom{}}[\/latex] is 24,000 years; [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex] is 3.82 days; and element-111 (Rg for roentgenium) is 1.5 [latex]\\times [\/latex] 10<sup>\u20133<\/sup> seconds. The half-lives of a number of radioactive isotopes important to medicine are shown in the table below, and others are listed in <a href=\".\/chapter\/half-lives-for-several-radioactive-isotopes-missing-formulas\/\" target=\"_blank\">Half-Lives for Several Radioactive Isotopes<\/a>.<\/p>\r\n\r\n<table id=\"fs-idp14399952\" class=\"span-all\" summary=\"This table has four columns and six rows. The first row is a header row, and it labels each column: \u201cType,\u201d \u201cDecay Mode,\u201d \u201cHalf-life,\u201d and \u201cUses.\u201d Under the \u201cType\u201d column are the following: \u201cF - 18,\u201d \u201cC o - 60,\u201d \u201cT c - 99 m,\u201d \u201cI \u2013 131,\u201d and \u201cT l - 201.\u201d Under the \u201cDecay Mode\u201d column are the following: \u201clowercase beta superscript positive sign decay,\u201d \u201clowercase beta decay, lowercase gamma decay,\u201d \u201clowercase gamma decay,\u201d \u201clowercase beta decay,\u201d and \u201celectron capture.\u201d Under the \u201cHalf-life\u201d column are the following: 110. Minutes, 5.27 years, 8.01 hours, 8.02 days, and 73 hours. Under the \u201cUses\u201d column are the following: PET scans; concern treatment; scans of brain, lung heart bone, etc.; thyroid scans and treatment; heart and arteries scans and cardiac stress tests.\">\r\n<thead>\r\n<tr valign=\"middle\">\r\n<th style=\"text-align: center;\" colspan=\"4\">Half-lives of Radioactive Isotopes Important to Medicine<\/th>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<th style=\"text-align: center;\">Type[footnote]The \u201cm\u201d in Tc-99m stands for \u201cmetastable,\u201d indicating that this is an unstable, high-energy state of Tc-99. Metastable isotopes emit \u03b3 radiation to rid themselves of excess energy and become (more) stable.[\/footnote]<\/th>\r\n<th style=\"text-align: center;\">Decay Mode<\/th>\r\n<th style=\"text-align: center;\">Half-Life<\/th>\r\n<th style=\"text-align: center;\">Uses<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"middle\">\r\n<td>F-18<\/td>\r\n<td>\u03b2<sup>+<\/sup> decay<\/td>\r\n<td>110. minutes<\/td>\r\n<td>PET scans<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td>Co-60<\/td>\r\n<td>\u03b2 decay, \u03b3 decay<\/td>\r\n<td>5.27 years<\/td>\r\n<td>cancer treatment<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td>Tc-99m<\/td>\r\n<td>\u03b3 decay<\/td>\r\n<td>8.01 hours<\/td>\r\n<td>scans of brain, lung, heart, bone<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td>I-131<\/td>\r\n<td>\u03b2 decay<\/td>\r\n<td>8.02 days<\/td>\r\n<td>thyroid scans and treatment<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td>Tl-201<\/td>\r\n<td>electron capture<\/td>\r\n<td>73 hours<\/td>\r\n<td>heart and arteries scans; cardiac stress tests<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-idp25396560\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Radiometric Dating<\/h2>\r\n<p id=\"fs-idm40273488\">Several radioisotopes have half-lives and other properties that make them useful for purposes of \u201cdating\u201d the origin of objects such as archaeological artifacts, formerly living organisms, or geological formations. This process is <strong>radiometric dating<\/strong> and has been responsible for many breakthrough scientific discoveries about the geological history of the earth, the evolution of life, and the history of human civilization. We will explore some of the most common types of radioactive dating and how the particular isotopes work for each type.<\/p>\r\n\r\n<section id=\"fs-idm78225712\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Radioactive Dating Using Carbon-14<\/h3>\r\n<p id=\"fs-idm42065648\">The radioactivity of carbon-14 provides a method for dating objects that were a part of a living organism. This method of radiometric dating, which is also called <strong>radiocarbon dating<\/strong> or carbon-14 dating, is accurate for dating carbon-containing substances that are up to about 30,000 years old, and can provide reasonably accurate dates up to a maximum of about 50,000 years old.<\/p>\r\n<p id=\"fs-idp34827216\">Naturally occurring carbon consists of three isotopes: [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex], which constitutes about 99% of the carbon on earth; [latex]{}_{\\phantom{1}6}{}^{13}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex], about 1% of the total; and trace amounts of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex]. Carbon-14 forms in the upper atmosphere by the reaction of nitrogen atoms with neutrons from cosmic rays in space:<\/p>\r\n\r\n<div id=\"fs-idp25204496\" data-type=\"equation\">[latex]{}_{\\phantom{1}7}{}^{14}\\text{N}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}+{}_{1}{}^{1}\\text{H}[\/latex]<\/div>\r\n<p id=\"fs-idp25818480\">All isotopes of carbon react with oxygen to produce CO<sub>2<\/sub> molecules. The ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] to [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] depends on the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] to [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] in the atmosphere. The natural abundance of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] in the atmosphere is approximately 1 part per trillion; until recently, this has generally been constant over time, as seen is gas samples found trapped in ice. The incorporation of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] and [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] into plants is a regular part of the photosynthesis process, which means that the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio found in a living plant is the same as the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in the atmosphere. But when the plant dies, it no longer traps carbon through photosynthesis. Because [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] is a stable isotope and does not undergo radioactive decay, its concentration in the plant does not change. However, carbon-14 decays by \u03b2 emission with a half-life of 5730 years:<\/p>\r\n\r\n<div id=\"fs-idp78792240\" data-type=\"equation\">[latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}7}{}^{14}\\text{N}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\r\n<p id=\"fs-idm11295264\">Thus, the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio gradually decreases after the plant dies. The decrease in the ratio with time provides a measure of the time that has elapsed since the death of the plant (or other organism that ate the plant). Figure 7\u00a0visually depicts this process.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214209\/CNX_Chem_21_03_CarbonDate.jpg\" alt=\"A diagram shows a cow standing on the ground next to a tree. In the upper left of the diagram, where the sky is represented, a single white sphere is shown and is connected by a downward-facing arrow to a larger sphere composed of green and white spheres that is labeled \u201csuperscript 14, subscript 7, N.\u201d This structure is connected to three other structures by a right-facing arrow. Each of the three it points to are composed of green and white spheres and all have arrows pointing from them to the ground. The first of these is labeled \u201cTrace, superscript 14, subscript 6, C,\u201d the second is labeled \u201c1 percent, superscript 13, subscript 6, C\u201d and the last is labeled \u201c99 percent, superscript 12, subscript 6, C.\u201d Two downward-facing arrows that merge into one arrow lead from the cow and tree to the ground and are labeled \u201corganism dies\u201d and \u201csuperscript 14, subscript 6, C, decay begins.\u201d A right-facing arrow labeled on top as \u201cDecay\u201d and on bottom as \u201cTime\u201d leads from this to a label of \u201csuperscript 14, subscript 6, C, backslash, superscript 12, subscript 6, C, ratio decreased.\u201d Near the top of the tree is a downward facing arrow with the label \u201csuperscript 14, subscript 6, C, backslash, superscript 12, subscript 6, C, ratio is constant in living organisms\u201d that leads to the last of the lower statements.\" width=\"975\" height=\"897\" data-media-type=\"image\/jpeg\" \/> Figure 7. Along with stable carbon-12, radioactive carbon-14 is taken in by plants and animals, and remains at a constant level within them while they are alive. After death, the C-14 decays and the C-14:C-12 ratio in the remains decreases. Comparing this ratio to the C-14:C-12 ratio in living organisms allows us to determine how long ago the organism lived (and died).[\/caption]\r\n<p id=\"fs-idm5880176\">For example, with the half-life of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] being 5730 years, if the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in a wooden object found in an archaeological dig is half what it is in a living tree, this indicates that the wooden object is 5730 years old. Highly accurate determinations of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratios can be obtained from very small samples (as little as a milligram) by the use of a mass spectrometer.<\/p>\r\n\r\n<div class=\"textbox\">Visit this <a href=\"http:\/\/phet.colorado.edu\/en\/simulation\/radioactive-dating-game\" target=\"_blank\" rel=\"nofollow\">website<\/a> to perform simulations of radiometric dating.<\/div>\r\n<div id=\"fs-idm12472048\" class=\"textbox shaded\" data-type=\"example\">\r\n<h3>Example 2<\/h3>\r\n<h4 id=\"fs-idm43395808\"><span data-type=\"title\">Radiocarbon Dating<\/span><\/h4>\r\nA tiny piece of paper (produced from formerly living plant matter) taken from the Dead Sea Scrolls has an activity of 10.8 disintegrations per minute per gram of carbon. If the initial C-14 activity was 13.6 disintegrations\/min\/g of C, estimate the age of the Dead Sea Scrolls.\r\n<h4 id=\"fs-idp4802464\"><span data-type=\"title\">Solution<\/span><\/h4>\r\nThe rate of decay (number of disintegrations\/minute\/gram of carbon) is proportional to the amount of radioactive C-14 left in the paper, so we can substitute the rates for the amounts, <em data-effect=\"italics\">N<\/em>, in the relationship:\r\n<div id=\"fs-idm27840624\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)\\longrightarrow t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{\\text{Rate}}_{t}}{{\\text{Rate}}_{0}}\\right)[\/latex]<\/div>\r\n<p id=\"fs-idp12237584\">where the subscript 0 represents the time when the plants were cut to make the paper, and the subscript <em data-effect=\"italics\">t<\/em> represents the current time.<\/p>\r\n<p id=\"fs-idm9913056\">The decay constant can be determined from the half-life of C-14, 5730 years:<\/p>\r\n\r\n<div id=\"fs-idm36556768\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{\\text{5730 y}}=1.21\\times {10}^{-4}{\\text{y}}^{-1}[\/latex].<\/div>\r\n<p id=\"fs-idp40647344\">Substituting and solving, we have:<\/p>\r\n\r\n<div id=\"fs-idm20370432\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{\\text{Rate}}_{t}}{{\\text{Rate}}_{0}}\\right)=-\\frac{1}{1.21\\times {10}^{-4}{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{10.8\\text{dis\/min\/g C}}{13.6\\text{dis\/min\/g C}}\\right)=\\text{1910 y}[\/latex].<\/div>\r\n<p id=\"fs-idp18759328\">Therefore, the Dead Sea Scrolls are approximately 1900 years old (Figure 8).<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1000\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214210\/CNX_Chem_21_03_DSScrolls.jpg\" alt=\"A photograph of six pages of ragged-edged paper covered in writing are shown.\" width=\"1000\" height=\"235\" data-media-type=\"image\/jpeg\" \/> Figure 8. Carbon-14 dating has shown that these pages from the Dead Sea Scrolls were written or copied on paper made from plants that died between 100 BC and AD 50.[\/caption]\r\n<h4 id=\"fs-idp79158784\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\r\nMore accurate dates of the reigns of ancient Egyptian pharaohs have been determined recently using plants that were preserved in their tombs. Samples of seeds and plant matter from King Tutankhamun\u2019s tomb have a C-14 decay rate of 9.07 disintegrations\/min\/g of C. How long ago did King Tut\u2019s reign come to an end?\r\n<div id=\"fs-idp861216\" data-type=\"note\">\r\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>about 3350 years ago, or approximately 1340 BC<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-idp143543968\">There have been some significant, well-documented changes to the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio. The accuracy of a straightforward application of this technique depends on the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in a living plant being the same now as it was in an earlier era, but this is not always valid. Due to the increasing accumulation of CO<sub>2<\/sub> molecules (largely [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2})[\/latex] in the atmosphere caused by combustion of fossil fuels (in which essentially all of the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] has decayed), the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the atmosphere may be changing. This manmade increase in [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] in the atmosphere causes the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio to decrease, and this in turn affects the ratio in currently living organisms on the earth. Fortunately, however, we can use other data, such as tree dating via examination of annual growth rings, to calculate correction factors. With these correction factors, accurate dates can be determined. In general, radioactive dating only works for about 10 half-lives; therefore, the limit for carbon-14 dating is about 57,000 years.<\/p>\r\n\r\n<\/section><section id=\"fs-idp103641152\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Radioactive Dating Using Nuclides Other than Carbon-14<\/h3>\r\n<p id=\"fs-idp55701552\">Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events. For example, uranium-238 (which decays in a series of steps into lead-206) can be used for establishing the age of rocks (and the approximate age of the oldest rocks on earth). Since U-238 has a half-life of 4.5 billion years, it takes that amount of time for half of the original U-238 to decay into Pb-206. In a sample of rock that does not contain appreciable amounts of Pb-208, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed. Therefore, by measuring and analyzing the ratio of U-238:Pb-206, we can determine the age of the rock. This assumes that all of the lead-206 present came from the decay of uranium-238. If there is additional lead-206 present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method. K-40 decays by positron emission and electron capture to form Ar-40 with a half-life of 1.25 billion years. If a rock sample is crushed and the amount of Ar-40 gas that escapes is measured, determination of the Ar-40:K-40 ratio yields the age of the rock. Other methods, such as rubidium-strontium dating (Rb-87 decays into Sr-87 with a half-life of 48.8 billion years), operate on the same principle. To estimate the lower limit for the earth\u2019s age, scientists determine the age of various rocks and minerals, making the assumption that the earth is older than the oldest rocks and minerals in its crust. As of 2014, the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4.4 billion years old.<\/p>\r\n\r\n<div id=\"fs-idp133528864\" class=\"textbox shaded\" data-type=\"example\">\r\n<h3>Example 3<\/h3>\r\n<h4 id=\"fs-idm84298880\"><span data-type=\"title\">Radioactive Dating of Rocks<\/span><\/h4>\r\nAn igneous rock contains 9.58 [latex]\\times [\/latex] 10<sup>\u20135<\/sup> g of U-238 and 2.51 [latex]\\times [\/latex] 10<sup>\u20135<\/sup> g of Pb-206, and much, much smaller amounts of Pb-208. Determine the approximate time at which the rock formed.\r\n<h4 id=\"fs-idp15526736\"><span data-type=\"title\">Solution<\/span><\/h4>\r\nThe sample of rock contains very little Pb-208, the most common isotope of lead, so we can safely assume that all the Pb-206 in the rock was produced by the radioactive decay of U-238. When the rock formed, it contained all of the U-238 currently in it, plus some U-238 that has since undergone radioactive decay.\r\n<p id=\"fs-idp16405632\">The amount of U-238 currently in the rock is:<\/p>\r\n\r\n<div id=\"fs-idp167610144\" data-type=\"equation\">[latex]9.58\\times {10}^{-5}\\cancel{\\text{g U}}\\times \\left(\\frac{\\text{1 mol U}}{238\\cancel{\\text{g U}}}\\right)=4.03\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\r\n<p id=\"fs-idp25854368\">Because when one mole of U-238 decays, it produces one mole of Pb-206, the amount of U-238 that has undergone radioactive decay since the rock was formed is:<\/p>\r\n\r\n<div id=\"fs-idp25663280\" data-type=\"equation\">[latex]2.51\\times {10}^{-5}\\cancel{\\text{g Pb}}\\times \\left(\\frac{1\\cancel{\\text{mol Pb}}}{206\\cancel{\\text{g Pb}}}\\right)\\times \\left(\\frac{\\text{1 mol U}}{1\\cancel{\\text{mol Pb}}}\\right)=1.22\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\r\n<p id=\"fs-idp142687024\">The total amount of U-238 originally present in the rock is therefore:<\/p>\r\n\r\n<div id=\"fs-idp51369808\" data-type=\"equation\">[latex]4.03\\times {10}^{-7}\\text{mol}+1.22\\times {10}^{-7}\\text{mol}=5.25\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\r\n<p id=\"fs-idm1352480\">The amount of time that has passed since the formation of the rock is given by:<\/p>\r\n\r\n<div id=\"fs-idm43256560\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)[\/latex]<\/div>\r\n<p id=\"fs-idp188052736\">with <em data-effect=\"italics\">N<\/em><sub>0<\/sub> representing the original amount of U-238 and <em data-effect=\"italics\">N<sub>t<\/sub><\/em> representing the present amount of U-238.<\/p>\r\n<p id=\"fs-idp178175456\">U-238 decays into Pb-206 with a half-life of 4.5 [latex]\\times [\/latex] 10<sup>9<\/sup> y, so the decay constant <em data-effect=\"italics\">\u03bb<\/em> is:<\/p>\r\n\r\n<div id=\"fs-idp138305008\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{4.5\\times {10}^{9}\\text{y}}=1.54\\times {10}^{-10}{\\text{y}}^{-1}[\/latex]<\/div>\r\n<p id=\"fs-idp46289376\">Substituting and solving, we have:<\/p>\r\n\r\n<div id=\"fs-idp171891856\" data-type=\"equation\">[latex]t=-\\frac{1}{1.54\\times {10}^{-10}{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{4.03\\times {10}^{-7}\\cancel{\\text{mol U}}}{5.25\\times {10}^{-7}\\cancel{\\text{mol U}}}\\right)=1.7\\times {10}^{9}\\text{y}[\/latex]<\/div>\r\n<p id=\"fs-idp8533088\">Therefore, the rock is approximately 1.7 billion years old.<\/p>\r\n\r\n<h4 id=\"fs-idp219511184\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\r\nA sample of rock contains 6.14 [latex]\\times [\/latex] 10<sup>\u20134<\/sup> g of Rb-87 and 3.51 [latex]\\times [\/latex] 10<sup>\u20135<\/sup> g of Sr-87. Calculate the age of the rock. (The half-life of the \u03b2 decay of Rb-87 is 4.7 [latex]\\times [\/latex] 10<sup>10<\/sup> y.)\r\n<div id=\"fs-idp66566112\" data-type=\"note\">\r\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>3.7 [latex]\\times [\/latex] 10<sup>9<\/sup> y<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"bcc-box bcc-success\">\r\n<h2>Key Concepts and Summary<\/h2>\r\nNuclei that have unstable n:p ratios undergo spontaneous radioactive decay. The most common types of radioactivity are \u03b1 decay, \u03b2 decay, \u03b3 emission, positron emission, and electron capture. Nuclear reactions also often involve \u03b3 rays, and some nuclei decay by electron capture. Each of these modes of decay leads to the formation of a new nucleus with a more stable n:p ratio. Some substances undergo radioactive decay series, proceeding through multiple decays before ending in a stable isotope. All nuclear decay processes follow first-order kinetics, and each radioisotope has its own characteristic half-life, the time that is required for half of its atoms to decay. Because of the large differences in stability among nuclides, there is a very wide range of half-lives of radioactive substances. Many of these substances have found useful applications in medical diagnosis and treatment, determining the age of archaeological and geological objects, and more.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Key Equations<\/h3>\r\n<ul>\r\n\t<li>decay rate = \u03bbN<\/li>\r\n\t<li>[latex]{t}_{1\\text{\/}2}=\\frac{\\text{ln 2}}{\\lambda }=\\frac{0.693}{\\lambda }[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Chemistry End of Chapter Exercises<\/h3>\r\n<ol>\r\n\t<li id=\"fs-idm10223552\">What are the types of radiation emitted by the nuclei of radioactive elements?<\/li>\r\n\t<li id=\"fs-idm33283872\">What changes occur to the atomic number and mass of a nucleus during each of the following decay scenarios?\r\n<ol>\r\n\t<li id=\"fs-idp137675104\">an \u03b1 particle is emitted<\/li>\r\n\t<li>a \u03b2 particle is emitted<\/li>\r\n\t<li id=\"fs-idp5160032\">\u03b3 radiation is emitted<\/li>\r\n\t<li id=\"fs-idp138318864\">a positron is emitted<\/li>\r\n\t<li id=\"fs-idp30872480\">an electron is captured<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp143657184\">What is the change in the nucleus that results from the following decay scenarios?\r\n<ol>\r\n\t<li id=\"fs-idm43346352\">emission of a \u03b2 particle<\/li>\r\n\t<li id=\"fs-idp177484800\">emission of a \u03b2<sup>+<\/sup> particle<\/li>\r\n\t<li id=\"fs-idp122950864\">capture of an electron<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp190856048\">Many nuclides with atomic numbers greater than 83 decay by processes such as electron emission. Explain the observation that the emissions from these unstable nuclides also normally include \u03b1 particles.<\/li>\r\n\t<li id=\"fs-idp26162912\">Why is electron capture accompanied by the emission of an X-ray?<\/li>\r\n\t<li id=\"fs-idp142495840\">Explain\u00a0how unstable heavy nuclides (atomic number &gt; 83) may decompose to form nuclides of greater stability (a) if they are below the band of stability and (b) if they are above the band of stability.<\/li>\r\n\t<li id=\"fs-idp201795280\">Which of the following nuclei is most likely to decay by positron emission? Explain your choice.\r\n<ol>\r\n\t<li id=\"fs-idp34995296\">chromium-53<\/li>\r\n\t<li id=\"fs-idp134866720\">manganese-51<\/li>\r\n\t<li id=\"fs-idp1557216\">iron-59<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp76332848\">The following nuclei do not lie in the band of stability. How would they be expected to decay? Explain your answer.\r\n<ol>\r\n\t<li id=\"fs-idp138950912\">[latex]{}_{15}{}^{34}\\text{P}[\/latex]<\/li>\r\n\t<li id=\"fs-idp143538880\">[latex]{}_{\\phantom{1}92}{}^{239}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n\t<li id=\"fs-idp134068000\">[latex]{}_{20}{}^{38}\\text{Ca}[\/latex]<\/li>\r\n\t<li id=\"fs-idp137322448\">[latex]{}_{1}{}^{3}\\text{H}[\/latex]<\/li>\r\n\t<li id=\"fs-idp179430752\">[latex]{}_{\\phantom{1}94}{}^{245}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idm5650336\">The following nuclei do not lie in the band of stability. How would they be expected to decay?\r\n<ol>\r\n\t<li id=\"fs-idp92587472\">[latex]{}_{15}{}^{28}\\text{P}[\/latex]<\/li>\r\n\t<li id=\"fs-idm20106176\">[latex]{}_{\\phantom{1}92}{}^{235}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n\t<li id=\"fs-idm22275216\">[latex]{}_{20}{}^{37}\\text{Ca}[\/latex]<\/li>\r\n\t<li id=\"fs-idp137311888\">[latex]{}_{3}{}^{9}\\text{L}\\text{i}[\/latex]<\/li>\r\n\t<li id=\"fs-idp137045664\">[latex]{}_{\\phantom{1}96}{}^{245}\\text{Cm}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp10180544\">Predict by what mode(s) of spontaneous radioactive decay each of the following unstable isotopes might proceed:\r\n<ol>\r\n\t<li id=\"fs-idm24027360\">[latex]{}_{2}{}^{6}\\text{H}\\text{e}[\/latex]<\/li>\r\n\t<li id=\"fs-idp7080592\">[latex]{}_{30}{}^{60}\\text{Zn}[\/latex]<\/li>\r\n\t<li id=\"fs-idm35115440\">[latex]{}_{\\phantom{1}91}{}^{235}\\text{Pa}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n\t<li id=\"fs-idm75261408\">[latex]{}_{\\phantom{1}94}{}^{241}\\text{Np}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\r\n\t<li id=\"fs-idp17960336\"><sup>18<\/sup>F<\/li>\r\n\t<li id=\"fs-idp7359744\"><sup>129<\/sup>Ba<\/li>\r\n\t<li id=\"fs-idm31917904\"><sup>237<\/sup>Pu<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp176277168\">Write a nuclear reaction for each step in the formation of [latex]{}_{\\phantom{1}84}{}^{218}\\text{Po}_{\\phantom{}}^{\\phantom{}}[\/latex] from [latex]{}_{\\phantom{1}98}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex], which proceeds by a series of decay reactions involving the step-wise emission of \u03b1, \u03b2, \u03b2, \u03b1, \u03b1, \u03b1 particles, in that order.<\/li>\r\n\t<li id=\"fs-idp25125904\">Write a nuclear reaction for each step in the formation of [latex]{}_{\\phantom{1}82}{}^{208}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex] from [latex]{}_{\\phantom{1}90}{}^{228}\\text{T}_{\\phantom{}}^{\\phantom{}}\\text{h,}[\/latex] which proceeds by a series of decay reactions involving the step-wise emission of \u03b1, \u03b1, \u03b1, \u03b1, \u03b2, \u03b2, \u03b1 particles, in that order.<\/li>\r\n\t<li id=\"fs-idm43525136\">Define the term half-life and illustrate it with an example.<\/li>\r\n\t<li id=\"fs-idm37225376\">A 1.00 [latex]\\times [\/latex] 10<sup>\u20136<\/sup>-g sample of nobelium, [latex]{}_{102}{}^{254}\\text{No}[\/latex], has a half-life of 55 seconds after it is formed. What is the percentage of [latex]{}_{102}{}^{254}\\text{No}[\/latex] remaining at the following times?\r\n<ol>\r\n\t<li id=\"fs-idp172348128\">\u00a05.0 min after it forms<\/li>\r\n\t<li id=\"fs-idm38933040\">1.0 h after it forms<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idm3633744\"><sup>239<\/sup>Pu is a nuclear waste byproduct with a half-life of 24,000 y. What fraction of the <sup>239<\/sup>Pu present today will be present in 1000 y?<\/li>\r\n\t<li id=\"fs-idm33356592\">The isotope <sup>208<\/sup>Tl undergoes \u03b2 decay with a half-life of 3.1 min.\r\n<ol>\r\n\t<li id=\"fs-idp143445632\">What isotope is produced by the decay?<\/li>\r\n\t<li id=\"fs-idp78376656\">How long will it take for 99.0% of a sample of pure <sup>208<\/sup>Tl to decay?<\/li>\r\n\t<li id=\"fs-idp211861968\">What percentage of a sample of pure <sup>208<\/sup>Tl remains un-decayed after 1.0 h?<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idm43874912\">If 1.000 g of [latex]{}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}[\/latex] produces 0.0001 mL of the gas [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex] at STP (standard temperature and pressure) in 24 h, what is the half-life of <sup>226<\/sup>Ra in years?<\/li>\r\n\t<li id=\"fs-idp126089680\">The isotope [latex]{}_{38}{}^{90}\\text{Sr}[\/latex] is one of the extremely hazardous species in the residues from nuclear power generation. The strontium in a 0.500-g sample diminishes to 0.393 g in 10.0 y. Calculate the half-life.<\/li>\r\n\t<li id=\"fs-idp219506992\">Technetium-99 is often used for assessing heart, liver, and lung damage because certain technetium compounds are absorbed by damaged tissues. It has a half-life of 6.0 h. Calculate the rate constant for the decay of [latex]{}_{43}{}^{99}\\text{Tc}[\/latex].<\/li>\r\n\t<li id=\"fs-idp53369552\">What is the age of mummified primate skin that contains 8.25% of the original quantity of <sup>14<\/sup>C?<\/li>\r\n\t<li>A sample of rock was found to contain 8.23 mg of rubidium-87 and 0.47 mg of strontium-87.\r\n<ol>\r\n\t<li id=\"fs-idm22718704\">Calculate the age of the rock if the half-life of the decay of rubidium by \u03b2 emission is 4.7 [latex]\\times [\/latex] 10<sup>10<\/sup> y.<\/li>\r\n\t<li id=\"fs-idp75993408\">If some [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] was initially present in the rock, would the rock be younger, older, or the same age as the age calculated in (a)? Explain your answer.<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp133778576\">A laboratory investigation shows that a sample of uranium ore contains 5.37 mg of [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex] and 2.52 mg of [latex]{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex]. Calculate the age of the ore. The half-life of [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex] is 4.5 [latex]\\times [\/latex] 10<sup>9<\/sup> yr.<\/li>\r\n\t<li id=\"fs-idp165416288\">Plutonium was detected in trace amounts in natural uranium deposits by Glenn Seaborg and his associates in 1941. They proposed that the source of this <sup>239<\/sup>Pu was the capture of neutrons by <sup>238<\/sup>U nuclei. Why is this plutonium not likely to have been trapped at the time the solar system formed 4.7 [latex]\\times [\/latex] 10<sup>9<\/sup> years ago?[latex]{}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex] has a half-life of 2.411 \u00d7 10<sup>4<\/sup> y. Calculate the value of <em data-effect=\"italics\">\u03bb<\/em> and then determine the amount of plutonium-239 remaining after 4.7 \u00d7 10<sup>9<\/sup> y:<em data-effect=\"italics\">\u03bbt<\/em> = <em data-effect=\"italics\">\u03bb<\/em>(2.411 \u00d7 10<sup>4<\/sup> y) = ln [latex]\\left(\\frac{1.0000}{0.5000}\\right)[\/latex] = 0.6931\r\n\r\n<em data-effect=\"italics\">\u03bb<\/em> = [latex]\\frac{0.6931}{2.411}[\/latex] \u00d7 10<sup>4<\/sup> y = 2.875 \u00d7 10<sup>\u20135<\/sup> y<sup>\u20131<\/sup>\r\n\r\nThen:\r\n\r\nln [latex]\\frac{{c}_{0}}{c}[\/latex] = <em data-effect=\"italics\">\u03bbt<\/em>\r\n\r\nln [latex]\\left(\\frac{1.000}{c}\\right)[\/latex] = 2.875 \u00d7 10<sup>\u20135<\/sup> y<sup>\u20131\u00a0<\/sup>\u00d7 4.7 \u00d7 10<sup>9<\/sup> y\r\n\r\nln <em data-effect=\"italics\">c<\/em> = \u20131.351 \u00d7 10<sup>5<\/sup>\r\n\r\n<em data-effect=\"italics\">c<\/em> = 0\r\n\r\nThis calculation shows that no Pu-239 could remain since the formation of the earth. Consequently, the plutonium now present could not have been formed with the uranium.<\/li>\r\n\t<li id=\"fs-idp217024288\">A [latex]{}_{4}{}^{7}\\text{Be}[\/latex] atom (mass = 7.0169 amu) decays into a [latex]{}_{3}{}^{7}\\text{L}\\text{i}[\/latex] atom (mass = 7.0160 amu) by electron capture. How much energy (in millions of electron volts, MeV) is produced by this reaction?<\/li>\r\n\t<li id=\"fs-idp71110896\">A [latex]{}_{5}{}^{8}\\text{B}[\/latex] atom (mass = 8.0246 amu) decays into a [latex]{}_{4}{}^{8}\\text{B}[\/latex] atom (mass = 8.0053 amu) by loss of a \u03b2<sup>+<\/sup> particle (mass = 0.00055 amu) or by electron capture. How much energy (in millions of electron volts) is produced by this reaction?<\/li>\r\n\t<li id=\"fs-idp131958896\">Isotopes such as <sup>26<\/sup>Al (half-life: 7.2 [latex]\\times [\/latex] 10<sup>5<\/sup> years) are believed to have been present in our solar system as it formed, but have since decayed and are now called extinct nuclides.\r\n<ol>\r\n\t<li><sup>26<\/sup>Al decays by \u03b2<sup>+<\/sup> emission or electron capture. Write the equations for these two nuclear transformations.<\/li>\r\n\t<li id=\"fs-idp179184448\">The earth was formed about 4.7 [latex]\\times [\/latex] 10<sup>9<\/sup> (4.7 billion) years ago. How old was the earth when 99.999999% of the <sup>26<\/sup>Al originally present had decayed?<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idp157539312\">Write a balanced equation for each of the following nuclear reactions:\r\n<ol>\r\n\t<li id=\"fs-idm52915712\">bismuth-212 decays into polonium-212<\/li>\r\n\t<li id=\"fs-idp90956192\">beryllium-8 and a positron are produced by the decay of an unstable nucleus<\/li>\r\n\t<li id=\"fs-idp18279664\">neptunium-239 forms from the reaction of uranium-238 with a neutron and then spontaneously converts into plutonium-239<\/li>\r\n\t<li id=\"fs-idp138334608\">strontium-90 decays into yttrium-90<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li id=\"fs-idm29139968\">Write a balanced equation for each of the following nuclear reactions:\r\n<ol>\r\n\t<li id=\"fs-idp133484688\">mercury-180 decays into platinum-176<\/li>\r\n\t<li id=\"fs-idp136801104\">zirconium-90 and an electron are produced by the decay of an unstable nucleus<\/li>\r\n\t<li id=\"fs-idp77337360\">thorium-232 decays and produces an alpha particle and a radium-228 nucleus, which decays into actinium-228 by beta decay<\/li>\r\n\t<li id=\"fs-idm29267360\">neon-19 decays into fluorine-19<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h4>Selected Answers<\/h4>\r\n1. \u03b1 (helium nuclei), \u03b2 (electrons), \u03b2<sup>+<\/sup> (positrons), and \u03b7 (neutrons) may be emitted from a radioactive element, all of which are particles; \u03b3 rays also may be emitted.\r\n\r\n3. (a) conversion of a neutron to a proton: [latex]{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{1}{}^{1}\\text{p}+{}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];\r\n\r\n(b) conversion of a proton to a neutron; the positron has the same mass as an electron and the same magnitude of positive charge as the electron has negative charge; when the n:p ratio of a nucleus is too low, a proton is converted into a neutron with the emission of a positron: [latex]{}_{1}{}^{1}\\text{p}\\longrightarrow {}_{0}{}^{1}\\text{n}+{}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];\r\n\r\n(c) In a proton-rich nucleus, an inner atomic electron can be absorbed. In simplifies form, this changes a proton into a neutron: [latex]{}_{1}{}^{1}\\text{p}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{0}{}^{1}\\text{p}[\/latex]\r\n\r\n5. The electron pulled into the nucleus was most likely found in the 1<i>s<\/i> orbital. As an electron falls from a higher energy level to replace it, the difference in the energy of the replacement electron in its two energy levels is given off as an X-ray.\r\n\r\n7. Manganese-51 is most likely to decay by positron emission. The n:p ratio for Cr-53 is [latex]\\frac{29}{24}[\/latex] = 1.21; for Mn-51, it is [latex]\\frac{26}{25}[\/latex] = 1.04; for Fe-59, it is [latex]\\frac{33}{26}[\/latex] = 1.27. Positron decay occurs when the n:p ratio is low. Mn-51 has the lowest n:p ration and therefore is most likely to decay by positron emission. Besides, [latex]{}_{24}{}^{53}\\text{Cr}[\/latex] is a stable isotope, and [latex]{}_{26}{}^{59}\\text{Fe}[\/latex] decays by beta emission.\r\n\r\n<span style=\"line-height: 1.5;\">9. (a) too many neutrons, \u03b2 decay; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(b) atomic number greater than 82, \u03b1 decay; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(c) too few neutrons, positron emission; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(d) too many neutrons, \u03b2 decay; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(e) atomic number greater than 83, \u03b1 decay<\/span>\r\n\r\n11.\u00a0[latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}90}{}^{234}\\text{Th}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex];\r\n\r\n[latex]{}_{\\phantom{1}90}{}^{234}\\text{Th}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}91}{}^{234}\\text{Pa}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];\r\n\r\n[latex]{}_{\\phantom{1}91}{}^{234}\\text{Pa}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}92}{}^{234}\\text{U}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];\r\n\r\n[latex]{}_{\\phantom{1}92}{}^{234}\\text{U}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}90}{}^{230}\\text{Th}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]\r\n\r\n[latex]{}_{\\phantom{1}90}{}^{230}\\text{Th}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]\r\n\r\n[latex]{}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex];\r\n\r\n[latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}84}{}^{218}\\text{Po}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]\r\n\r\n13. Half-life is the time required for half the atoms in a sample to decay. Example (answers may vary): For C-14, the half-life is 5770 years. A 10-g sample of C-14 would contain 5 g of C-14 after 5770 years; a 0.20-g sample of C-14 would contain 0.10 g after 5770 years.\r\n\r\n15. 1000 years is 0.04 half-lives. The fraction that remains after 0.04 half-lives is [latex]{\\left(\\frac{1}{2}\\right)}^{0.04}=0.973[\/latex] or 97.3%\r\n\r\n<span style=\"line-height: 1.5;\">17.\u00a0<\/span><em data-effect=\"italics\">PV<\/em> = <em data-effect=\"italics\">nRT<\/em>\r\n\r\n<em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> = [latex]\\frac{PV}{RT}=\\frac{\\left(\\text{1 atm}\\right)\\left(0.0001\\text{mL}\\times \\text{1 L\/}{10}^{3}\\text{mL}\\right)}{\\left(0.08206\\text{L atm}{\\text{mol}}^{-1}{\\text{K}}^{-1}\\right)\\left(273.15\\text{K}\\right)}[\/latex] = 4.4614 [latex]\\times [\/latex] 10<sup>\u20139<\/sup> mol\r\n\r\n<em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> produced = <em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> decayed\r\n\r\nmass Ra lost = 4.4614 [latex]\\times [\/latex] 10<sup>\u20139<\/sup> mol [latex]\\times [\/latex] [latex]\\frac{\\text{226 g}}{\\text{mol}}[\/latex] = 1.00827 [latex]\\times [\/latex] 10<sup>\u20136<\/sup> g\r\n\r\nmass Ra remaining after 24 h = 1 \u2013 (1.00827 [latex]\\times [\/latex] 10<sup>\u20136<\/sup> g) = 9.9999899 [latex]\\times [\/latex] 10<sup>\u20131<\/sup> g\r\n\r\nln [latex]\\frac{{c}_{0}}{c}=\\lambda t[\/latex] = ln [latex]\\frac{1.000}{9.9999899\\times {10}^{-1}}=\\lambda \\left(\\text{24 h}\\right)[\/latex] = 4.3785 [latex]\\times [\/latex] 10<sup>\u20137<\/sup>\r\n\r\n<em data-effect=\"italics\">\u03bb<\/em> = 4.2015 [latex]\\times [\/latex] 10<sup>\u20138<\/sup> h<sup>\u20131<\/sup>\r\n\r\n[latex]{t}_{1\\text{\/}2}=\\frac{0.693}{\\lambda }=\\frac{0.693}{4.2015\\times {10}^{-8}}[\/latex] = 1.6494 [latex]\\times [\/latex] 10<sup>7<\/sup> h\r\n\r\n= 1.6494 [latex]\\times [\/latex] 10<sup>7<\/sup> h [latex]\\times [\/latex] [latex]\\frac{\\text{1 d}}{\\text{24 h}}\\times \\frac{\\text{1 y}}{\\text{365 d}}[\/latex] = 1.883 [latex]\\times [\/latex] 10<sup>3<\/sup> y or 2 [latex]\\times [\/latex] 10<sup>3<\/sup> y\r\n\r\n<span style=\"line-height: 1.5;\">19.\u00a0<\/span>(Recall that radioactive decay is a first-order process.)\r\n\r\n\u03bb = [latex]\\frac{0.693}{{t}_{\\frac{1}{2}}}=\\frac{0.693}{6.0\\text{h}}[\/latex] = 0.12 h\u20131\r\n\r\n21.\u00a0(a) [latex]{}_{37}{}^{87}\\text{Rb}\\longrightarrow {}_{38}{}^{87}\\text{Sr}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]\r\n\r\n[latex]{}_{38}{}^{87}\\text{Sr}[\/latex] is a stable isotope and does not decay further. Calculate the value of the decay rate constant for [latex]{}_{37}{}^{87}\\text{Rb}[\/latex], remembering that all radioactive decay is first order:\r\n\r\n[latex]\\lambda =\\frac{0.693}{4.7\\times {10}^{10}\\text{y}}=1.47\\times {10}^{-11}{\\text{y}}^{-1}[\/latex]\r\n\r\nCalculate the number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] and [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] found in the sample at time <i>t<\/i>:\r\n\r\n[latex]\\begin{array}{l}\\\\ \\\\ 8.23\\text{mg}\\times \\frac{\\text{1 g}}{\\text{1000 mg}}\\times \\frac{\\text{1 mol}}{87.0\\text{g}}=9.46\\times {10}^{-5}\\text{mol of}{}_{37}{}^{87}\\text{Rb}\\\\ 0.47\\text{mg}\\times \\frac{\\text{1 g}}{\\text{1000 mg}}\\times \\frac{\\text{1 mol}}{87.0\\text{g}}=5.40\\times {10}^{-6}\\text{mol of}{}_{38}{}^{87}\\text{Sr}\\end{array}[\/latex]\r\n\r\nEach mol of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] that disappeared (by radioactive decay of the [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] initially present in the rock) produced 1 mol of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex]. Hence the number of moles of [latex]{}_{38}{}^{87}\\text{Rb}[\/latex] that disappeared by radioactive decay equals the number of moles of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] that were produced. This amount consists of the 5.40 [latex]\\times [\/latex] 10\u20136 mol of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] found in the rock at time <i>t<\/i> if all the [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] present at time <i>t<\/i> resulted from radioactive decay of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] and no strontium-87 was present initially in the rock. Using this assumption, we can calculate the total number of moles of rubidium-87 initially present in the rock:\r\n\r\nTotal number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] initially present in the rock at time <i>t<\/i> 0 = number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] at time <i>t<\/i> + number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] that decayed during the time interval t \u2013 t0 = number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] measured at time <i>t<\/i> + number of moles of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] measured at time <i>t<\/i> = 9.46 [latex]\\times [\/latex] 10\u20135 mol + 5.40 [latex]\\times [\/latex] 10\u20136 mol = 1.00 [latex]\\times [\/latex] 10\u20134 mol\r\n\r\nThe number of moles can be substituted for concentrations in the expression:\r\n\r\n[latex]\\text{ln}\\frac{{c}_{0}}{{c}_{t}}=\\lambda t[\/latex]\r\n\r\nThus:\r\n\r\n[latex]\\begin{array}{l}\\\\ \\\\ \\text{ln}\\frac{1.00\\times {10}^{-4}\\text{mol}}{9.46\\times {10}^{-5}\\text{mol}}=\\left(1.47\\times {10}^{-11}\\right)t\\\\ t=\\left(\\mathrm{ln}\\frac{1.00\\times {10}^{-4}}{9.46\\times {10}^{-5}}\\right)\\left(\\frac{1}{1.47\\times {10}^{-11}{\\text{y}}^{-1}}\\right)\\end{array}[\/latex]\r\n\r\n= 3.8 [latex]\\times [\/latex] 109 y = 3.8 billion years = age of the rock sample;\r\n\r\n(b) The rock would be younger than the age calculated in part (a). If Sr was originally in the rock, the amount produced by radioactive decay would equal the present amount minus the initial amount. As this amount would be smaller than the amount used to calculate the age of the rock and the age is proportional to the amount of Sr, the rock would be younger.\r\n\r\n<span style=\"line-height: 1.5;\">23.\u00a0<\/span>[latex]{}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex] has a half-life of 2.411 [latex]\\times [\/latex] 104 <em>y<\/em>. Calculate the value of \u03bb and then determine the amount of plutonium-239 remaining after 4.7 [latex]\\times [\/latex] 109 <em>y<\/em>:\r\n\r\n\u03bb<em>t<\/em> = \u03bb(2.411 [latex]\\times [\/latex] 104 y) = ln [latex]\\left(\\frac{1.0000}{0.5000}\\right)[\/latex] = 0.6931\r\n\r\n\u03bb = [latex]\\frac{0.6931}{2.411}[\/latex] [latex]\\times [\/latex] 104 <em>y<\/em> = 2.875 [latex]\\times [\/latex] 10\u20135 y\u20131\r\n\r\nThen:\r\n\r\nln [latex]\\frac{{c}_{0}}{c}[\/latex] = \u03bb<em>t<\/em>\r\n\r\nln [latex]\\left(\\frac{1.000}{c}\\right)[\/latex] = 2.875 [latex]\\times [\/latex] 10\u20135 y\u20131 [latex]\\times [\/latex] 4.7 [latex]\\times [\/latex] 109 <em>y<\/em>\r\n\r\nln c = \u20131.351 [latex]\\times [\/latex] 105\r\n\r\nc = 0\r\n\r\nThis calculation shows that no Pu-239 could remain since the formation of the earth. Consequently, the plutonium now present could not have been formed with the uranium.\r\n\r\n<span style=\"line-height: 1.5;\">25.\u00a0<\/span>Find the mass difference of the starting mass and the total masses of the final products. Then use the conversion for mass to energy to find the energy released:\r\n\r\n8.0246 \u2013 8.0053 \u2013 0.00055 = 0.01875 amu\r\n\r\n0.01875 amu [latex]\\times [\/latex] 1.6605 [latex]\\times [\/latex] 10\u201327 kg\/amu = 3.113 [latex]\\times [\/latex] 10\u201329 kg\r\n\r\n<em>E<\/em> = <em>mc<\/em><sup>2<\/sup> = (3.113 [latex]\\times [\/latex] 10\u201329 kg)(2.9979 [latex]\\times [\/latex] 108 m\/s)2\r\n\r\n= 2.798 [latex]\\times [\/latex] 10\u201312 kg m2\/s2 = 2.798 [latex]\\times [\/latex] 10\u201312 J\/nucleus\r\n\r\n2.798 [latex]\\times [\/latex] 10\u201312 J\/nucleus [latex]\\times [\/latex] [latex]\\frac{\\text{1 MeV}}{1.602177\\times {10}^{-13}\\text{J}}[\/latex] = 17.5 MeV\r\n\r\n<span style=\"line-height: 1.5;\">27. (a) [latex]{}_{\\phantom{1}83}{}^{212}\\text{Bi}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}84}{}^{212}\\text{Po}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(b) [latex]{}_{5}{}^{8}\\text{B}\\longrightarrow {}_{4}{}^{8}\\text{B}\\text{e}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(c) [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{\\phantom{1}93}{}^{239}\\text{Np}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{N}_{\\phantom{}}^{\\phantom{}}\\text{p}[\/latex], <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">[latex]{}_{\\phantom{1}93}{}^{239}\\text{Np}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span>\r\n\r\n<span style=\"line-height: 1.5;\">(d) [latex]{}_{38}{}^{90}\\text{Sr}\\longrightarrow {}_{39}{}^{90}\\text{Y}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/span>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\"><section id=\"glossary\">\r\n<h3>Glossary<\/h3>\r\n<strong>alpha (\u03b1) decay<\/strong>\r\nloss of an alpha particle during radioactive decay\r\n\r\n<strong>beta (\u03b2) decay<\/strong>\r\nbreakdown of a neutron into a proton, which remains in the nucleus, and an electron, which is emitted as a beta particle\r\n\r\n<strong>daughter nuclide<\/strong>\r\nnuclide produced by the radioactive decay of another nuclide; may be stable or may decay further\r\n\r\n<strong>electron capture<\/strong>\r\ncombination of a core electron with a proton to yield a neutron within the nucleus\r\n\r\n<strong>gamma (\u03b3) emission<\/strong>\r\ndecay of an excited-state nuclide accompanied by emission of a gamma ray\r\n\r\n<strong>half-life (t1\/2)<\/strong>\r\ntime required for half of the atoms in a radioactive sample to decay\r\n\r\n<strong>parent nuclide<\/strong>\r\nunstable nuclide that changes spontaneously into another (daughter) nuclide\r\n\r\n<strong>positron emission<\/strong>\r\n(also, \u03b2+ decay) conversion of a proton into a neutron, which remains in the nucleus, and a positron, which is emitted\r\n\r\n<strong>radioactive decay<\/strong>\r\nspontaneous decay of an unstable nuclide into another nuclide\r\n\r\n<strong>radioactive decay series<\/strong>\r\nchains of successive disintegrations (radioactive decays) that ultimately lead to a stable end-product\r\n\r\n<strong>radiocarbon dating<\/strong>\r\nhighly accurate means of dating objects 30,000\u201350,000 years old that were derived from once-living matter; achieved by calculating the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the object vs. the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the present-day atmosphere\r\n\r\n<strong>radiometric dating<\/strong>\r\nuse of radioisotopes and their properties to date the formation of objects such as archeological artifacts, formerly living organisms, or geological formations\r\n\r\n<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this module, you will be able to:<\/p>\n<ul>\n<li>Recognize common modes of radioactive decay<\/li>\n<li>Identify common particles and energies involved in nuclear decay reactions<\/li>\n<li>Write and balance nuclear decay equations<\/li>\n<li>Calculate kinetic parameters for decay processes, including half-life<\/li>\n<li>Describe common radiometric dating techniques<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-idp102942608\">Following the somewhat serendipitous discovery of radioactivity by Becquerel, many prominent scientists began to investigate this new, intriguing phenomenon. Among them were Marie Curie (the first woman to win a Nobel Prize, and the only person to win two Nobel Prizes in different sciences\u2014chemistry and physics), who was the first to coin the term \u201cradioactivity,\u201d and Ernest Rutherford (of gold foil experiment fame), who investigated and named three of the most common types of radiation. During the beginning of the twentieth century, many radioactive substances were discovered, the properties of radiation were investigated and quantified, and a solid understanding of radiation and nuclear decay was developed.<\/p>\n<p>The spontaneous change of an unstable nuclide into another is <strong>radioactive decay<\/strong>. The unstable nuclide is called the <strong>parent nuclide<\/strong>; the nuclide that results from the decay is known as the <strong>daughter nuclide<\/strong>. The daughter nuclide may be stable, or it may decay itself. The radiation produced during radioactive decay is such that the daughter nuclide lies closer to the band of stability than the parent nuclide, so the location of a nuclide relative to the band of stability can serve as a guide to the kind of decay it will undergo.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214159\/CNX_Chem_21_03_Reaction1.jpg\" alt=\"A diagram shows two spheres composed of many smaller white and green spheres connected by a right-facing arrow with another, down-facing arrow coming off of it. The left sphere, labeled \u201cParent nucleus uranium dash 238\u201d has two white and two green spheres that are near one another and are outlined in red. These two green and two white spheres are shown near the tip of the down-facing arrow and labeled \u201calpha particle.\u201d The right sphere, labeled \u201cDaughter nucleus radon dash 234,\u201d looks the same as the left, but has a space for four smaller spheres outlined with a red dotted line.\" width=\"975\" height=\"347\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. A nucleus of uranium-238 (the parent nuclide) undergoes \u03b1 decay to form thorium-234 (the daughter nuclide). The alpha particle removes two protons (green) and two neutrons (gray) from the uranium-238 nucleus.<\/p>\n<\/div>\n<section id=\"fs-idp242071680\" data-depth=\"1\">\n<div class=\"textbox\" data-type=\"title\">Although the radioactive decay of a nucleus is too small to see with the naked eye, we can indirectly view radioactive decay in an environment called a cloud chamber. Click <a href=\"https:\/\/www.youtube.com\/watch?v=pewTySxfTQk\" target=\"_blank\" rel=\"nofollow\">here<\/a> to learn about cloud chambers and to view an interesting Cloud Chamber Demonstration from the Jefferson Lab.<\/div>\n<h2 data-type=\"title\">Types of Radioactive Decay<\/h2>\n<p>Ernest Rutherford\u2019s experiments involving the interaction of radiation with a magnetic or electric field\u00a0helped him determine that one type of radiation consisted of positively charged and relatively massive \u03b1 particles; a second type was made up of negatively charged and much less massive \u03b2 particles; and a third was uncharged electromagnetic waves, \u03b3 rays. We now know that \u03b1 particles are high-energy helium nuclei, \u03b2 particles are high-energy electrons, and \u03b3 radiation compose high-energy electromagnetic radiation. We classify different types of radioactive decay by the radiation produced.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214200\/CNX_Chem_21_03_Radiation.jpg\" alt=\"A diagram is shown. A gray box on the left side of the diagram labeled \u201cLead block\u201d has a chamber hollowed out in the center in which a sample labeled \u201cRadioactive substance\u201d is placed. A blue beam is coming from the sample, out of the block, and passing through two horizontally placed plates that are labeled \u201cElectrically charged plates.\u201d The top plate is labeled with a positive sign while the bottom plate is labeled with a negative sign. The beam is shown to break into three beams as it passes in between the plates; in order from top to bottom, they are red, labeled \u201cbeta rays,\u201d purple labeled \u201cgamma rays\u201d and green labeled \u201calpha rays.\u201d The beams are shown to hit a vertical plate labeled \u201cPhotographic plate\u201d on the far right side of the diagram.\" width=\"975\" height=\"428\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Alpha particles, which are attracted to the negative plate and deflected by a relatively small amount, must be positively charged and relatively massive. Beta particles, which are attracted to the positive plate and deflected a relatively large amount, must be negatively charged and relatively light. Gamma rays, which are unaffected by the electric field, must be uncharged.<\/p>\n<\/div>\n<p id=\"fs-idm41800032\"><strong>Alpha (\u03b1) decay<\/strong> is the emission of an \u03b1 particle from the nucleus. For example, polonium-210 undergoes \u03b1 decay:<\/p>\n<div id=\"fs-idp134926448\" data-type=\"equation\">[latex]{}_{\\phantom{1}84}{}^{210}\\text{Po}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{2}{}^{4}\\text{He}+{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}84}{}^{210}\\text{Po}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{2}{}^{4}\\alpha+{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\n<p id=\"fs-idp202270256\">Alpha decay occurs primarily in heavy nuclei (A &gt; 200, Z &gt; 83). Because the loss of an \u03b1 particle gives a daughter nuclide with a mass number four units smaller and an atomic number two units smaller than those of the parent nuclide, the daughter nuclide has a larger n:p ratio than the parent nuclide. If the parent nuclide undergoing \u03b1 decay lies below the band of stability, the daughter nuclide will lie closer to the band.<\/p>\n<p id=\"fs-idm2875904\"><strong>Beta (\u03b2) decay<\/strong> is the emission of an electron from a nucleus. Iodine-131 is an example of a nuclide that undergoes \u03b2 decay:<\/p>\n<div id=\"fs-idm19827888\" data-type=\"equation\">[latex]{}_{\\phantom{1}53}{}^{131}\\text{I}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}54}{}^{131}\\text{X}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}53}{}^{131}\\text{I}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{-1}{}^{\\phantom{1}0}\\beta_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}54}{}^{131}\\text{Xe}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\n<p id=\"fs-idp100287120\">Beta decay, which can be thought of as the conversion of a neutron into a proton and a \u03b2 particle, is observed in nuclides with a large n:p ratio. The beta particle (electron) emitted is from the atomic nucleus and is not one of the electrons surrounding the nucleus. Such nuclei lie above the band of stability. Emission of an electron does not change the mass number of the nuclide but does increase the number of its protons and decrease the number of its neutrons. Consequently, the n:p ratio is decreased, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.<\/p>\n<p id=\"fs-idp8043424\"><strong>Gamma emission (\u03b3 emission)<\/strong> is observed when a nuclide is formed in an excited state and then decays to its ground state with the emission of a \u03b3 ray, a quantum of high-energy electromagnetic radiation. The presence of a nucleus in an excited state is often indicated by an asterisk (*). Cobalt-60 emits \u03b3 radiation and is used in many applications including cancer treatment:<\/p>\n<div id=\"fs-idp5264576\" data-type=\"equation\">[latex]{}_{27}{}^{60}\\text{Co*}\\longrightarrow {}_{0}{}^{0}\\gamma+{}_{27}{}^{60}\\text{Co}[\/latex]<\/div>\n<p id=\"fs-idp50313008\">There is no change in mass number or atomic number during the emission of a \u03b3 ray unless the \u03b3 emission accompanies one of the other modes of decay.<\/p>\n<p id=\"fs-idp18009856\"><strong>Positron emission (\u03b2<sup>+<\/sup> decay<\/strong>) is the emission of a positron from the nucleus. Oxygen-15 is an example of a nuclide that undergoes positron emission:<\/p>\n<div id=\"fs-idm9404976\" data-type=\"equation\">[latex]{}_{\\phantom{1}8}{}^{15}\\text{O}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}7}{}^{15}\\text{N}_{\\phantom{}}^{\\phantom{}}\\text{or}{}_{\\phantom{1}8}{}^{15}\\text{O}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{+1}{}^{\\phantom{1}0}\\beta_{\\phantom{}}^{\\phantom{}}+{}_{\\phantom{1}7}{}^{15}\\text{N}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\n<p id=\"fs-idp125268336\">Positron emission is observed for nuclides in which the n:p ratio is low. These nuclides lie below the band of stability. Positron decay is the conversion of a proton into a neutron with the emission of a positron. The n:p ratio increases, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.<\/p>\n<p id=\"fs-idp123320528\"><strong>Electron capture<\/strong> occurs when one of the inner electrons in an atom is captured by the atom\u2019s nucleus. For example, potassium-40 undergoes electron capture:<\/p>\n<div id=\"fs-idm22074816\" data-type=\"equation\">[latex]{}_{19}{}^{40}\\text{K}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{18}{}^{40}\\text{Ar}[\/latex]<\/div>\n<p id=\"fs-idm42894208\">Electron capture occurs when an inner shell electron combines with a proton and is converted into a neutron. The loss of an inner shell electron leaves a vacancy that will be filled by one of the outer electrons. As the outer electron drops into the vacancy, it will emit energy. In most cases, the energy emitted will be in the form of an X-ray. Like positron emission, electron capture occurs for \u201cproton-rich\u201d nuclei that lie below the band of stability. Electron capture has the same effect on the nucleus as does positron emission: The atomic number is decreased by one and the mass number does not change. This increases the n:p ratio, and the daughter nuclide lies closer to the band of stability than did the parent nuclide. Whether electron capture or positron emission occurs is difficult to predict. The choice is primarily due to kinetic factors, with the one requiring the smaller activation energy being the one more likely to occur.<\/p>\n<p id=\"fs-idp132245344\">Figure 3 summarizes these types of decay, along with their equations and changes in atomic and mass numbers.<\/p>\n<div style=\"width: 1310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214202\/CNX_Chem_21_03_RadioDecay.jpg\" alt=\"This table has four columns and six rows. The first row is a header row and it labels each column: \u201cType,\u201d \u201cNuclear equation,\u201d \u201cRepresentation,\u201d and \u201cChange in mass \/ atomic numbers.\u201d Under the \u201cType\u201d column are the following: \u201cAlpha decay,\u201d \u201cBeta decay,\u201d \u201cGamma decay,\u201d \u201cPositron emission,\u201d and \u201cElectron capture.\u201d Under the \u201cNuclear equation\u201d column are several equations. Each begins with superscript A stacked over subscript Z X. There is a large gap of space and then the following equations: \u201csuperscript 4 stacked over subscript 2 He plus superscript A minus 4 stacked over subscript Z minus 2 Y,\u201d \u201csuperscript 0 stacked over subscript negative 1 e plus superscript A stacked over subscript Z plus 1 Y,\u201d \u201csuperscript 0 stacked over subscript 0 lowercase gamma plus superscript A stacked over subscript Z Y,\u201d \u201csuperscript 0 stacked over subscript positive 1 e plus superscript A stacked over subscript Y minus 1 Y,\u201d and \u201csuperscript 0 stacked over subscript negative 1 e plus superscript A stacked over subscript Y minus 1 Y.\u201d Under the \u201cRepresentation\u201d column are the five diagrams. The first shows a cluster of green and white spheres. A section of the cluster containing two white and two green spheres is outlined. There is a right-facing arrow pointing to a similar cluster as previously described, but the outlined section is missing. From the arrow another arrow branches off and points downward. The small cluster to two white spheres and two green spheres appear at the end of the arrow. The next diagram shows the same cluster of white and green spheres. One white sphere is outlined. There is a right-facing arrow to a similar cluster, but the white sphere is missing. Another arrow branches off the main arrow and a red sphere with a negative sign appears at the end. The next diagram shows the same cluster of white and green spheres. The whole sphere is outlined and labeled, \u201cexcited nuclear state.\u201d There is a right-facing arrow that points to the same cluster. No spheres are missing. Off the main arrow is another arrow which points to a purple squiggle arrow which in turn points to a lowercase gamma. The next diagram shows the same cluster of white and green spheres. One green sphere is outlined. There is a right-facing arrow to a similar cluster, but the green sphere is missing. Another arrow branches off the main arrow and a red sphere with a positive sign appears at the end. The next diagram shows the same cluster of white and green spheres. One green sphere is outlined. There is a right-facing arrow to a similar cluster, but the green sphere is missing. Two other arrows branch off the main arrow. The first shows a gold sphere with a negative sign joining with the right-facing arrow. The secon points to a blue squiggle arrow labeled, \u201cX-ray.\u201d Under the \u201cChange in mass \/ atomic numbers\u201d column are the following: \u201cA: decrease by 4, Z: decrease by 2,\u201d \u201cA: unchanged, Z: increased by 1,\u201d \u201cA: unchanged, Z: unchanged,\u201d \u201cA: unchanged, Z: unchanged,\u201d \u201cA: unchanged, Z: decrease by 1,\u201d and \u201cA: unchanged, Z: decrease by 1.\u201d\" width=\"1300\" height=\"865\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. This table summarizes the type, nuclear equation, representation, and any changes in the mass or atomic numbers for various types of decay.<\/p>\n<\/div>\n<div id=\"fs-idp54313680\" class=\"chemistry everyday-life textbox shaded\" data-type=\"note\">\n<h3 data-type=\"title\">PET Scan<\/h3>\n<p id=\"fs-idp214075136\">Positron emission tomography (PET) scans use radiation to diagnose and track health conditions and monitor medical treatments by revealing how parts of a patient\u2019s body function (Figure 4). To perform a PET scan, a positron-emitting radioisotope is produced in a cyclotron and then attached to a substance that is used by the part of the body being investigated. This \u201ctagged\u201d compound, or radiotracer, is then put into the patient (injected via IV or breathed in as a gas), and how it is used by the tissue reveals how that organ or other area of the body functions.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214204\/CNX_Chem_21_03_PETScan.jpg\" alt=\"Three pictures are shown and labeled \u201ca,\u201d \u201cb\u201d and \u201cc.\u201d Picture a shows a machine with a round opening connected to an examination table. Picture b is a medical scan of the top of a person\u2019s head and shows large patches of yellow and red and smaller patches of blue, green and purple highlighting. Picture c also shows a medical scan of the top of a person\u2019s head, but this image is mostly colored in blue and purple with very small patches of red and yellow.\" width=\"975\" height=\"332\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. A PET scanner (a) uses radiation to provide an image of how part of a patient\u2019s body functions. The scans it produces can be used to image a healthy brain (b) or can be used for diagnosing medical conditions such as Alzheimer\u2019s disease (c). (credit a: modification of work by Jens Maus)<\/p>\n<\/div>\n<p id=\"fs-idp13254880\">For example, F-18 is produced by proton bombardment of <sup>18<\/sup>O [latex]\\left({}_{\\phantom{1}8}{}^{18}\\text{O}_{\\phantom{}}^{\\phantom{}}+{}_{1}{}^{1}\\text{p}\\longrightarrow {}_{\\phantom{1}9}{}^{18}\\text{F}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\right)[\/latex] and incorporated into a glucose analog called fludeoxyglucose (FDG). How FDG is used by the body provides critical diagnostic information; for example, since cancers use glucose differently than normal tissues, FDG can reveal cancers. The <sup>18<\/sup>F emits positrons that interact with nearby electrons, producing a burst of gamma radiation. This energy is detected by the scanner and converted into a detailed, three-dimensional, color image that shows how that part of the patient\u2019s body functions. Different levels of gamma radiation produce different amounts of brightness and colors in the image, which can then be interpreted by a radiologist to reveal what is going on. PET scans can detect heart damage and heart disease, help diagnose Alzheimer\u2019s disease, indicate the part of a brain that is affected by epilepsy, reveal cancer, show what stage it is, and how much it has spread, and whether treatments are effective. Unlike magnetic resonance imaging and X-rays, which only show how something looks, the big advantage of PET scans is that they show how something functions. PET scans are now usually performed in conjunction with a computed tomography scan.<\/p>\n<\/div>\n<\/section>\n<section id=\"fs-idp89067520\" data-depth=\"1\">\n<h2 data-type=\"title\">Radioactive Decay Series<\/h2>\n<p id=\"fs-idp54304864\">The naturally occurring radioactive isotopes of the heaviest elements fall into chains of successive disintegrations, or decays, and all the species in one chain constitute a radioactive family, or <strong>radioactive decay series<\/strong>. Three of these series include most of the naturally radioactive elements of the periodic table. They are the uranium series, the actinide series, and the thorium series. The neptunium series is a fourth series, which is no longer significant on the earth because of the short half-lives of the species involved. Each series is characterized by a parent (first member) that has a long half-life and a series of daughter nuclides that ultimately lead to a stable end-product\u2014that is, a nuclide on the band of stability (Figure 5). In all three series, the end-product is a stable isotope of lead. The neptunium series, previously thought to terminate with bismuth-209, terminates with thallium-205.<\/p>\n<div style=\"width: 1310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214206\/CNX_Chem_21_03_DecayS.jpg\" alt=\"A graph is shown where the x-axis is labeled \u201cNumber of neutrons, open parenthesis, n, close parenthesis\u201d and has values of 122 to 148 in increments of 2. The y-axis is labeled \u201cAtomic number\u201d and has values of 80 to 92 in increments of 1. Two types of arrows are used in this graph to connect the points. Green arrows are labeled as \u201calpha decay\u201d while red arrows are labeled \u201cbeta decay.\u201d Beginning at the point \u201c92, 146\u201d that is labeled \u201csuperscript 238, U,\u201d a green arrow connects this point to the second point \u201c90, 144\u201d which is labeled \u201csuperscript 234, T h.\u201d A red arrow connect this to the third point \u201c91, 143\u201d which is labeled \u201csuperscript 234, P a\u201d which is connected to the fourth point \u201c92, 142\u201d by a red arrow and which is labeled \u201csuperscript 234, U.\u201d A green arrow leads to the next point, \u201c90, 140\u201d which is labeled \u201csuperscript 230, T h\u201d and is connected by a green arrow to the sixth point, \u201c88, 138\u201d which is labeled \u201csuperscript 226, R a\u201d that is in turn connected by a green arrow to the seventh point \u201c86, 136\u201d which is labeled \u201csuperscript 222, Ra.\u201d The eighth point, at \u201c84, 134\u201d is labeled \u201csuperscript 218, P o\u201d and has green arrows leading to it and away from it to the ninth point \u201c82, 132\u201d which is labeled \u201csuperscript 214, Pb\u201d which is connected by a red arrow to the tenth point, \u201c83, 131\u201d which is labeled \u201csuperscript 214, B i.\u201d A red arrow leads to the eleventh point \u201c84, 130\u201d which is labeled \u201csuperscript 214, P o\u201d and a green arrow leads to the twelvth point \u201c82, 128\u201d which is labeled \u201csuperscript 210, P b.\u201d A red arrow leads to the thirteenth point \u201c83, 127\u201d which is labeled \u201csuperscript 210, B i\u201d and a red arrow leads to the fourteenth point \u201c84, 126\u201d which is labeled \u201csuperscript 210, P o.\u201d The final point is labeled \u201c82, 124\u201d and \u201csuperscript 206, P b.\u201d\" width=\"1300\" height=\"856\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Uranium-238 undergoes a radioactive decay series consisting of 14 separate steps before producing stable lead-206. This series consists of eight \u03b1 decays and six \u03b2 decays.<\/p>\n<\/div>\n<\/section>\n<section id=\"fs-idp38328768\" data-depth=\"1\">\n<h2 data-type=\"title\">Radioactive Half-Lives<\/h2>\n<p id=\"fs-idp135905344\">Radioactive decay follows first-order kinetics. Since first-order reactions have already been covered in detail in the kinetics chapter, we will now apply those concepts to nuclear decay reactions. Each radioactive nuclide has a characteristic, constant <strong>half-life<\/strong> (<em data-effect=\"italics\">t<\/em><sub>1\/2<\/sub>), the time required for half of the atoms in a sample to decay. An isotope\u2019s half-life allows us to determine how long a sample of a useful isotope will be available, and how long a sample of an undesirable or dangerous isotope must be stored before it decays to a low-enough radiation level that is no longer a problem.<\/p>\n<p id=\"fs-idm89252176\">For example, cobalt-60, an isotope that emits gamma rays used to treat cancer, has a half-life of 5.27 years (Figure 6). In a given cobalt-60 source, since half of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] nuclei decay every 5.27 years, both the amount of material and the intensity of the radiation emitted is cut in half every 5.27 years. (Note that for a given substance, the intensity of radiation that it produces is directly proportional to the rate of decay of the substance and the amount of the substance.) This is as expected for a process following first-order kinetics. Thus, a cobalt-60 source that is used for cancer treatment must be replaced regularly to continue to be effective.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214207\/CNX_Chem_21_03_HalfLife.jpg\" alt=\"A graph, titled \u201cC o dash 60 Decay,\u201d is shown where the x-axis is labeled \u201cC o dash 60 remaining, open parenthesis, percent sign, close parenthesis\u201d and has values of 0 to 100 in increments of 25. The y-axis is labeled \u201cNumber of half dash lives\u201d and has values of 0 to 5 in increments of 1. The first point, at \u201c0, 100\u201d has a circle filled with tiny dots drawn near it labeled \u201c10 g.\u201d The second point, at \u201c1, 50\u201d has a smaller circle filled with tiny dots drawn near it labeled \u201c5 g.\u201d The third point, at \u201c2, 25\u201d has a small circle filled with tiny dots drawn near it labeled \u201c2.5 g.\u201d The fourth point, at \u201c3, 12.5\u201d has a very small circle filled with tiny dots drawn near it labeled \u201c1.25 g.\u201d The last point, at \u201c4, 6.35\u201d has a tiny circle filled with tiny dots drawn near it labeled.\u201d625 g.\u201d\" width=\"975\" height=\"594\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. For cobalt-60, which has a half-life of 5.27 years, 50% remains after 5.27 years (one half-life), 25% remains after 10.54 years (two half-lives), 12.5% remains after 15.81 years (three half-lives), and so on.<\/p>\n<\/div>\n<p id=\"fs-idm84262544\">Since nuclear decay follows first-order kinetics, we can adapt the mathematical relationships used for first-order chemical reactions. We generally substitute the number of nuclei, <em data-effect=\"italics\">N<\/em>, for the concentration. If the rate is stated in nuclear decays per second, we refer to it as the activity of the radioactive sample. The rate for radioactive decay is:<\/p>\n<p id=\"fs-idp16246080\">decay rate = <em data-effect=\"italics\">\u03bbN<\/em> with <em data-effect=\"italics\">\u03bb<\/em> = the decay constant for the particular radioisotope<\/p>\n<p id=\"fs-idm22877072\">The decay constant, <em data-effect=\"italics\">\u03bb<\/em>, which is the same as a rate constant discussed in the kinetics chapter. It is possible to express the decay constant in terms of the half-life, <em data-effect=\"italics\">t<\/em><sub>1\/2<\/sub>:<\/p>\n<div id=\"fs-idp219598000\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{{t}_{1\\text{\/}2}}\\text{or}{t}_{1\\text{\/}2}=\\frac{\\text{ln 2}}{\\lambda }=\\frac{0.693}{\\lambda }[\/latex]<\/div>\n<p id=\"fs-idp20089360\">The first-order equations relating amount, <em data-effect=\"italics\">N<\/em>, and time are:<\/p>\n<div id=\"fs-idm4205984\" data-type=\"equation\">[latex]{N}_{t}={N}_{0}{e}^{-kt}\\text{or}t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)[\/latex]<\/div>\n<p id=\"fs-idp61393104\">where <em data-effect=\"italics\">N<\/em><sub>0<\/sub> is the initial number of nuclei or moles of the isotope, and <em data-effect=\"italics\">N<sub>t<\/sub><\/em> is the number of nuclei\/moles remaining at time <em data-effect=\"italics\">t<\/em>. Example 1 applies these calculations to find the rates of radioactive decay for specific nuclides.<\/p>\n<div id=\"fs-idp79391408\" class=\"textbox shaded\" data-type=\"example\">\n<h3>Example 1<\/h3>\n<h4 id=\"fs-idm1797296\"><span data-type=\"title\">Rates of Radioactive Decay<\/span><\/h4>\n<p>[latex]{}_{27}{}^{60}\\text{Co}[\/latex] decays with a half-life of 5.27 years to produce [latex]{}_{28}{}^{60}\\text{Ni}[\/latex].<\/p>\n<p id=\"fs-idp30533728\">(a) What is the decay constant for the radioactive disintegration of cobalt-60?<\/p>\n<p id=\"fs-idp132429920\">(b) Calculate the fraction of a sample of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] isotope that will remain after 15 years.<\/p>\n<p id=\"fs-idp176097216\">(c) How long does it take for a sample of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] to disintegrate to the extent that only 2.0% of the original amount remains?<\/p>\n<h4 id=\"fs-idm68606544\"><span data-type=\"title\">Solution<\/span><\/h4>\n<p>(a) The value of the rate constant is given by:<\/p>\n<div id=\"fs-idp17616736\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{5.27\\text{y}}=0.132{\\text{y}}^{-1}[\/latex]<\/div>\n<p id=\"fs-idp141655056\">(b) The fraction of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] that is left after time <em data-effect=\"italics\">t<\/em> is given by [latex]\\frac{{N}_{t}}{{N}_{0}}[\/latex]. Rearranging the first-order relationship <em data-effect=\"italics\">N<sub>t<\/sub><\/em> = <em data-effect=\"italics\">N<\/em><sub>0<\/sub><em data-effect=\"italics\">e<\/em><sup>\u2013<em data-effect=\"italics\">\u03bbt<\/em><\/sup> to solve for this ratio yields:<\/p>\n<div id=\"fs-idp16374688\" data-type=\"equation\">[latex]\\frac{{N}_{t}}{{N}_{0}}={e}^{-\\lambda t}={e}^{-\\left(0.132\\text{\/y}\\right)\\left(15.0\\text{\/y}\\right)}=0.138[\/latex]<\/div>\n<p id=\"fs-idm19598048\">The fraction of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] that will remain after 15.0 years is 0.138. Or put another way, 13.8% of the [latex]{}_{27}{}^{60}\\text{Co}[\/latex] originally present will remain after 15 years.<\/p>\n<p id=\"fs-idp82234704\">(c) 2.00% of the original amount of [latex]{}_{27}{}^{60}\\text{Co}[\/latex] is equal to 0.0200 [latex]\\times[\/latex] <em data-effect=\"italics\">N<\/em><sub>0<\/sub>. Substituting this into the equation for time for first-order kinetics, we have:<\/p>\n<div id=\"fs-idm21044912\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)=-\\frac{1}{0.132{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{0.0200\\times {N}_{0}}{{N}_{0}}\\right)=29.6\\text{y}[\/latex]<\/div>\n<h4 id=\"fs-idp7893440\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\n<p>Radon-222, [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex], has a half-life of 3.823 days. How long will it take a sample of radon-222 with a mass of 0.750 g to decay into other elements, leaving only 0.100 g of radon-222?<\/p>\n<div id=\"fs-idp123030208\" data-type=\"note\">\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>11.1 days<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-idp76857680\">Because each nuclide has a specific number of nucleons, a particular balance of repulsion and attraction, and its own degree of stability, the half-lives of radioactive nuclides vary widely. For example: the half-life of [latex]{}_{\\phantom{1}83}{}^{209}\\text{Bi}_{\\phantom{}}^{\\phantom{}}[\/latex] is 1.9 [latex]\\times[\/latex] 10<sup>19<\/sup> years; [latex]{}_{\\phantom{1}94}{}^{239}\\text{Ra}_{\\phantom{}}^{\\phantom{}}[\/latex] is 24,000 years; [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex] is 3.82 days; and element-111 (Rg for roentgenium) is 1.5 [latex]\\times[\/latex] 10<sup>\u20133<\/sup> seconds. The half-lives of a number of radioactive isotopes important to medicine are shown in the table below, and others are listed in <a href=\".\/chapter\/half-lives-for-several-radioactive-isotopes-missing-formulas\/\" target=\"_blank\">Half-Lives for Several Radioactive Isotopes<\/a>.<\/p>\n<table id=\"fs-idp14399952\" class=\"span-all\" summary=\"This table has four columns and six rows. The first row is a header row, and it labels each column: \u201cType,\u201d \u201cDecay Mode,\u201d \u201cHalf-life,\u201d and \u201cUses.\u201d Under the \u201cType\u201d column are the following: \u201cF - 18,\u201d \u201cC o - 60,\u201d \u201cT c - 99 m,\u201d \u201cI \u2013 131,\u201d and \u201cT l - 201.\u201d Under the \u201cDecay Mode\u201d column are the following: \u201clowercase beta superscript positive sign decay,\u201d \u201clowercase beta decay, lowercase gamma decay,\u201d \u201clowercase gamma decay,\u201d \u201clowercase beta decay,\u201d and \u201celectron capture.\u201d Under the \u201cHalf-life\u201d column are the following: 110. Minutes, 5.27 years, 8.01 hours, 8.02 days, and 73 hours. Under the \u201cUses\u201d column are the following: PET scans; concern treatment; scans of brain, lung heart bone, etc.; thyroid scans and treatment; heart and arteries scans and cardiac stress tests.\">\n<thead>\n<tr valign=\"middle\">\n<th style=\"text-align: center;\" colspan=\"4\">Half-lives of Radioactive Isotopes Important to Medicine<\/th>\n<\/tr>\n<tr valign=\"middle\">\n<th style=\"text-align: center;\">Type<a class=\"footnote\" title=\"The \u201cm\u201d in Tc-99m stands for \u201cmetastable,\u201d indicating that this is an unstable, high-energy state of Tc-99. Metastable isotopes emit \u03b3 radiation to rid themselves of excess energy and become (more) stable.\" id=\"return-footnote-3667-1\" href=\"#footnote-3667-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/th>\n<th style=\"text-align: center;\">Decay Mode<\/th>\n<th style=\"text-align: center;\">Half-Life<\/th>\n<th style=\"text-align: center;\">Uses<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"middle\">\n<td>F-18<\/td>\n<td>\u03b2<sup>+<\/sup> decay<\/td>\n<td>110. minutes<\/td>\n<td>PET scans<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>Co-60<\/td>\n<td>\u03b2 decay, \u03b3 decay<\/td>\n<td>5.27 years<\/td>\n<td>cancer treatment<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>Tc-99m<\/td>\n<td>\u03b3 decay<\/td>\n<td>8.01 hours<\/td>\n<td>scans of brain, lung, heart, bone<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>I-131<\/td>\n<td>\u03b2 decay<\/td>\n<td>8.02 days<\/td>\n<td>thyroid scans and treatment<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td>Tl-201<\/td>\n<td>electron capture<\/td>\n<td>73 hours<\/td>\n<td>heart and arteries scans; cardiac stress tests<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-idp25396560\" data-depth=\"1\">\n<h2 data-type=\"title\">Radiometric Dating<\/h2>\n<p id=\"fs-idm40273488\">Several radioisotopes have half-lives and other properties that make them useful for purposes of \u201cdating\u201d the origin of objects such as archaeological artifacts, formerly living organisms, or geological formations. This process is <strong>radiometric dating<\/strong> and has been responsible for many breakthrough scientific discoveries about the geological history of the earth, the evolution of life, and the history of human civilization. We will explore some of the most common types of radioactive dating and how the particular isotopes work for each type.<\/p>\n<section id=\"fs-idm78225712\" data-depth=\"2\">\n<h3 data-type=\"title\">Radioactive Dating Using Carbon-14<\/h3>\n<p id=\"fs-idm42065648\">The radioactivity of carbon-14 provides a method for dating objects that were a part of a living organism. This method of radiometric dating, which is also called <strong>radiocarbon dating<\/strong> or carbon-14 dating, is accurate for dating carbon-containing substances that are up to about 30,000 years old, and can provide reasonably accurate dates up to a maximum of about 50,000 years old.<\/p>\n<p id=\"fs-idp34827216\">Naturally occurring carbon consists of three isotopes: [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex], which constitutes about 99% of the carbon on earth; [latex]{}_{\\phantom{1}6}{}^{13}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex], about 1% of the total; and trace amounts of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex]. Carbon-14 forms in the upper atmosphere by the reaction of nitrogen atoms with neutrons from cosmic rays in space:<\/p>\n<div id=\"fs-idp25204496\" data-type=\"equation\">[latex]{}_{\\phantom{1}7}{}^{14}\\text{N}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}+{}_{1}{}^{1}\\text{H}[\/latex]<\/div>\n<p id=\"fs-idp25818480\">All isotopes of carbon react with oxygen to produce CO<sub>2<\/sub> molecules. The ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] to [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] depends on the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] to [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] in the atmosphere. The natural abundance of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\text{O}[\/latex] in the atmosphere is approximately 1 part per trillion; until recently, this has generally been constant over time, as seen is gas samples found trapped in ice. The incorporation of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] and [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] into plants is a regular part of the photosynthesis process, which means that the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio found in a living plant is the same as the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in the atmosphere. But when the plant dies, it no longer traps carbon through photosynthesis. Because [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] is a stable isotope and does not undergo radioactive decay, its concentration in the plant does not change. However, carbon-14 decays by \u03b2 emission with a half-life of 5730 years:<\/p>\n<div id=\"fs-idp78792240\" data-type=\"equation\">[latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}7}{}^{14}\\text{N}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/div>\n<p id=\"fs-idm11295264\">Thus, the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio gradually decreases after the plant dies. The decrease in the ratio with time provides a measure of the time that has elapsed since the death of the plant (or other organism that ate the plant). Figure 7\u00a0visually depicts this process.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214209\/CNX_Chem_21_03_CarbonDate.jpg\" alt=\"A diagram shows a cow standing on the ground next to a tree. In the upper left of the diagram, where the sky is represented, a single white sphere is shown and is connected by a downward-facing arrow to a larger sphere composed of green and white spheres that is labeled \u201csuperscript 14, subscript 7, N.\u201d This structure is connected to three other structures by a right-facing arrow. Each of the three it points to are composed of green and white spheres and all have arrows pointing from them to the ground. The first of these is labeled \u201cTrace, superscript 14, subscript 6, C,\u201d the second is labeled \u201c1 percent, superscript 13, subscript 6, C\u201d and the last is labeled \u201c99 percent, superscript 12, subscript 6, C.\u201d Two downward-facing arrows that merge into one arrow lead from the cow and tree to the ground and are labeled \u201corganism dies\u201d and \u201csuperscript 14, subscript 6, C, decay begins.\u201d A right-facing arrow labeled on top as \u201cDecay\u201d and on bottom as \u201cTime\u201d leads from this to a label of \u201csuperscript 14, subscript 6, C, backslash, superscript 12, subscript 6, C, ratio decreased.\u201d Near the top of the tree is a downward facing arrow with the label \u201csuperscript 14, subscript 6, C, backslash, superscript 12, subscript 6, C, ratio is constant in living organisms\u201d that leads to the last of the lower statements.\" width=\"975\" height=\"897\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. Along with stable carbon-12, radioactive carbon-14 is taken in by plants and animals, and remains at a constant level within them while they are alive. After death, the C-14 decays and the C-14:C-12 ratio in the remains decreases. Comparing this ratio to the C-14:C-12 ratio in living organisms allows us to determine how long ago the organism lived (and died).<\/p>\n<\/div>\n<p id=\"fs-idm5880176\">For example, with the half-life of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] being 5730 years, if the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in a wooden object found in an archaeological dig is half what it is in a living tree, this indicates that the wooden object is 5730 years old. Highly accurate determinations of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratios can be obtained from very small samples (as little as a milligram) by the use of a mass spectrometer.<\/p>\n<div class=\"textbox\">Visit this <a href=\"http:\/\/phet.colorado.edu\/en\/simulation\/radioactive-dating-game\" target=\"_blank\" rel=\"nofollow\">website<\/a> to perform simulations of radiometric dating.<\/div>\n<div id=\"fs-idm12472048\" class=\"textbox shaded\" data-type=\"example\">\n<h3>Example 2<\/h3>\n<h4 id=\"fs-idm43395808\"><span data-type=\"title\">Radiocarbon Dating<\/span><\/h4>\n<p>A tiny piece of paper (produced from formerly living plant matter) taken from the Dead Sea Scrolls has an activity of 10.8 disintegrations per minute per gram of carbon. If the initial C-14 activity was 13.6 disintegrations\/min\/g of C, estimate the age of the Dead Sea Scrolls.<\/p>\n<h4 id=\"fs-idp4802464\"><span data-type=\"title\">Solution<\/span><\/h4>\n<p>The rate of decay (number of disintegrations\/minute\/gram of carbon) is proportional to the amount of radioactive C-14 left in the paper, so we can substitute the rates for the amounts, <em data-effect=\"italics\">N<\/em>, in the relationship:<\/p>\n<div id=\"fs-idm27840624\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)\\longrightarrow t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{\\text{Rate}}_{t}}{{\\text{Rate}}_{0}}\\right)[\/latex]<\/div>\n<p id=\"fs-idp12237584\">where the subscript 0 represents the time when the plants were cut to make the paper, and the subscript <em data-effect=\"italics\">t<\/em> represents the current time.<\/p>\n<p id=\"fs-idm9913056\">The decay constant can be determined from the half-life of C-14, 5730 years:<\/p>\n<div id=\"fs-idm36556768\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{\\text{5730 y}}=1.21\\times {10}^{-4}{\\text{y}}^{-1}[\/latex].<\/div>\n<p id=\"fs-idp40647344\">Substituting and solving, we have:<\/p>\n<div id=\"fs-idm20370432\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{\\text{Rate}}_{t}}{{\\text{Rate}}_{0}}\\right)=-\\frac{1}{1.21\\times {10}^{-4}{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{10.8\\text{dis\/min\/g C}}{13.6\\text{dis\/min\/g C}}\\right)=\\text{1910 y}[\/latex].<\/div>\n<p id=\"fs-idp18759328\">Therefore, the Dead Sea Scrolls are approximately 1900 years old (Figure 8).<\/p>\n<div style=\"width: 1010px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/05\/23214210\/CNX_Chem_21_03_DSScrolls.jpg\" alt=\"A photograph of six pages of ragged-edged paper covered in writing are shown.\" width=\"1000\" height=\"235\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. Carbon-14 dating has shown that these pages from the Dead Sea Scrolls were written or copied on paper made from plants that died between 100 BC and AD 50.<\/p>\n<\/div>\n<h4 id=\"fs-idp79158784\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\n<p>More accurate dates of the reigns of ancient Egyptian pharaohs have been determined recently using plants that were preserved in their tombs. Samples of seeds and plant matter from King Tutankhamun\u2019s tomb have a C-14 decay rate of 9.07 disintegrations\/min\/g of C. How long ago did King Tut\u2019s reign come to an end?<\/p>\n<div id=\"fs-idp861216\" data-type=\"note\">\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>about 3350 years ago, or approximately 1340 BC<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-idp143543968\">There have been some significant, well-documented changes to the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio. The accuracy of a straightforward application of this technique depends on the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio in a living plant being the same now as it was in an earlier era, but this is not always valid. Due to the increasing accumulation of CO<sub>2<\/sub> molecules (largely [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2})[\/latex] in the atmosphere caused by combustion of fossil fuels (in which essentially all of the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] has decayed), the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the atmosphere may be changing. This manmade increase in [latex]{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}{\\text{O}}_{2}[\/latex] in the atmosphere causes the [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] ratio to decrease, and this in turn affects the ratio in currently living organisms on the earth. Fortunately, however, we can use other data, such as tree dating via examination of annual growth rings, to calculate correction factors. With these correction factors, accurate dates can be determined. In general, radioactive dating only works for about 10 half-lives; therefore, the limit for carbon-14 dating is about 57,000 years.<\/p>\n<\/section>\n<section id=\"fs-idp103641152\" data-depth=\"2\">\n<h3 data-type=\"title\">Radioactive Dating Using Nuclides Other than Carbon-14<\/h3>\n<p id=\"fs-idp55701552\">Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events. For example, uranium-238 (which decays in a series of steps into lead-206) can be used for establishing the age of rocks (and the approximate age of the oldest rocks on earth). Since U-238 has a half-life of 4.5 billion years, it takes that amount of time for half of the original U-238 to decay into Pb-206. In a sample of rock that does not contain appreciable amounts of Pb-208, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed. Therefore, by measuring and analyzing the ratio of U-238:Pb-206, we can determine the age of the rock. This assumes that all of the lead-206 present came from the decay of uranium-238. If there is additional lead-206 present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method. K-40 decays by positron emission and electron capture to form Ar-40 with a half-life of 1.25 billion years. If a rock sample is crushed and the amount of Ar-40 gas that escapes is measured, determination of the Ar-40:K-40 ratio yields the age of the rock. Other methods, such as rubidium-strontium dating (Rb-87 decays into Sr-87 with a half-life of 48.8 billion years), operate on the same principle. To estimate the lower limit for the earth\u2019s age, scientists determine the age of various rocks and minerals, making the assumption that the earth is older than the oldest rocks and minerals in its crust. As of 2014, the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4.4 billion years old.<\/p>\n<div id=\"fs-idp133528864\" class=\"textbox shaded\" data-type=\"example\">\n<h3>Example 3<\/h3>\n<h4 id=\"fs-idm84298880\"><span data-type=\"title\">Radioactive Dating of Rocks<\/span><\/h4>\n<p>An igneous rock contains 9.58 [latex]\\times[\/latex] 10<sup>\u20135<\/sup> g of U-238 and 2.51 [latex]\\times[\/latex] 10<sup>\u20135<\/sup> g of Pb-206, and much, much smaller amounts of Pb-208. Determine the approximate time at which the rock formed.<\/p>\n<h4 id=\"fs-idp15526736\"><span data-type=\"title\">Solution<\/span><\/h4>\n<p>The sample of rock contains very little Pb-208, the most common isotope of lead, so we can safely assume that all the Pb-206 in the rock was produced by the radioactive decay of U-238. When the rock formed, it contained all of the U-238 currently in it, plus some U-238 that has since undergone radioactive decay.<\/p>\n<p id=\"fs-idp16405632\">The amount of U-238 currently in the rock is:<\/p>\n<div id=\"fs-idp167610144\" data-type=\"equation\">[latex]9.58\\times {10}^{-5}\\cancel{\\text{g U}}\\times \\left(\\frac{\\text{1 mol U}}{238\\cancel{\\text{g U}}}\\right)=4.03\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\n<p id=\"fs-idp25854368\">Because when one mole of U-238 decays, it produces one mole of Pb-206, the amount of U-238 that has undergone radioactive decay since the rock was formed is:<\/p>\n<div id=\"fs-idp25663280\" data-type=\"equation\">[latex]2.51\\times {10}^{-5}\\cancel{\\text{g Pb}}\\times \\left(\\frac{1\\cancel{\\text{mol Pb}}}{206\\cancel{\\text{g Pb}}}\\right)\\times \\left(\\frac{\\text{1 mol U}}{1\\cancel{\\text{mol Pb}}}\\right)=1.22\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\n<p id=\"fs-idp142687024\">The total amount of U-238 originally present in the rock is therefore:<\/p>\n<div id=\"fs-idp51369808\" data-type=\"equation\">[latex]4.03\\times {10}^{-7}\\text{mol}+1.22\\times {10}^{-7}\\text{mol}=5.25\\times {10}^{-7}\\text{mol U}[\/latex]<\/div>\n<p id=\"fs-idm1352480\">The amount of time that has passed since the formation of the rock is given by:<\/p>\n<div id=\"fs-idm43256560\" data-type=\"equation\">[latex]t=-\\frac{1}{\\lambda }\\text{ln}\\left(\\frac{{N}_{t}}{{N}_{0}}\\right)[\/latex]<\/div>\n<p id=\"fs-idp188052736\">with <em data-effect=\"italics\">N<\/em><sub>0<\/sub> representing the original amount of U-238 and <em data-effect=\"italics\">N<sub>t<\/sub><\/em> representing the present amount of U-238.<\/p>\n<p id=\"fs-idp178175456\">U-238 decays into Pb-206 with a half-life of 4.5 [latex]\\times[\/latex] 10<sup>9<\/sup> y, so the decay constant <em data-effect=\"italics\">\u03bb<\/em> is:<\/p>\n<div id=\"fs-idp138305008\" data-type=\"equation\">[latex]\\lambda =\\frac{\\text{ln 2}}{{t}_{1\\text{\/}2}}=\\frac{0.693}{4.5\\times {10}^{9}\\text{y}}=1.54\\times {10}^{-10}{\\text{y}}^{-1}[\/latex]<\/div>\n<p id=\"fs-idp46289376\">Substituting and solving, we have:<\/p>\n<div id=\"fs-idp171891856\" data-type=\"equation\">[latex]t=-\\frac{1}{1.54\\times {10}^{-10}{\\text{y}}^{-1}}\\text{ln}\\left(\\frac{4.03\\times {10}^{-7}\\cancel{\\text{mol U}}}{5.25\\times {10}^{-7}\\cancel{\\text{mol U}}}\\right)=1.7\\times {10}^{9}\\text{y}[\/latex]<\/div>\n<p id=\"fs-idp8533088\">Therefore, the rock is approximately 1.7 billion years old.<\/p>\n<h4 id=\"fs-idp219511184\"><span data-type=\"title\">Check Your Learning<\/span><\/h4>\n<p>A sample of rock contains 6.14 [latex]\\times[\/latex] 10<sup>\u20134<\/sup> g of Rb-87 and 3.51 [latex]\\times[\/latex] 10<sup>\u20135<\/sup> g of Sr-87. Calculate the age of the rock. (The half-life of the \u03b2 decay of Rb-87 is 4.7 [latex]\\times[\/latex] 10<sup>10<\/sup> y.)<\/p>\n<div id=\"fs-idp66566112\" data-type=\"note\">\n<p style=\"text-align: right;\" data-type=\"title\"><strong>Answer:\u00a0<\/strong>3.7 [latex]\\times[\/latex] 10<sup>9<\/sup> y<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<h2>Key Concepts and Summary<\/h2>\n<p>Nuclei that have unstable n:p ratios undergo spontaneous radioactive decay. The most common types of radioactivity are \u03b1 decay, \u03b2 decay, \u03b3 emission, positron emission, and electron capture. Nuclear reactions also often involve \u03b3 rays, and some nuclei decay by electron capture. Each of these modes of decay leads to the formation of a new nucleus with a more stable n:p ratio. Some substances undergo radioactive decay series, proceeding through multiple decays before ending in a stable isotope. All nuclear decay processes follow first-order kinetics, and each radioisotope has its own characteristic half-life, the time that is required for half of its atoms to decay. Because of the large differences in stability among nuclides, there is a very wide range of half-lives of radioactive substances. Many of these substances have found useful applications in medical diagnosis and treatment, determining the age of archaeological and geological objects, and more.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Key Equations<\/h3>\n<ul>\n<li>decay rate = \u03bbN<\/li>\n<li>[latex]{t}_{1\\text{\/}2}=\\frac{\\text{ln 2}}{\\lambda }=\\frac{0.693}{\\lambda }[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Chemistry End of Chapter Exercises<\/h3>\n<ol>\n<li id=\"fs-idm10223552\">What are the types of radiation emitted by the nuclei of radioactive elements?<\/li>\n<li id=\"fs-idm33283872\">What changes occur to the atomic number and mass of a nucleus during each of the following decay scenarios?\n<ol>\n<li id=\"fs-idp137675104\">an \u03b1 particle is emitted<\/li>\n<li>a \u03b2 particle is emitted<\/li>\n<li id=\"fs-idp5160032\">\u03b3 radiation is emitted<\/li>\n<li id=\"fs-idp138318864\">a positron is emitted<\/li>\n<li id=\"fs-idp30872480\">an electron is captured<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp143657184\">What is the change in the nucleus that results from the following decay scenarios?\n<ol>\n<li id=\"fs-idm43346352\">emission of a \u03b2 particle<\/li>\n<li id=\"fs-idp177484800\">emission of a \u03b2<sup>+<\/sup> particle<\/li>\n<li id=\"fs-idp122950864\">capture of an electron<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp190856048\">Many nuclides with atomic numbers greater than 83 decay by processes such as electron emission. Explain the observation that the emissions from these unstable nuclides also normally include \u03b1 particles.<\/li>\n<li id=\"fs-idp26162912\">Why is electron capture accompanied by the emission of an X-ray?<\/li>\n<li id=\"fs-idp142495840\">Explain\u00a0how unstable heavy nuclides (atomic number &gt; 83) may decompose to form nuclides of greater stability (a) if they are below the band of stability and (b) if they are above the band of stability.<\/li>\n<li id=\"fs-idp201795280\">Which of the following nuclei is most likely to decay by positron emission? Explain your choice.\n<ol>\n<li id=\"fs-idp34995296\">chromium-53<\/li>\n<li id=\"fs-idp134866720\">manganese-51<\/li>\n<li id=\"fs-idp1557216\">iron-59<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp76332848\">The following nuclei do not lie in the band of stability. How would they be expected to decay? Explain your answer.\n<ol>\n<li id=\"fs-idp138950912\">[latex]{}_{15}{}^{34}\\text{P}[\/latex]<\/li>\n<li id=\"fs-idp143538880\">[latex]{}_{\\phantom{1}92}{}^{239}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<li id=\"fs-idp134068000\">[latex]{}_{20}{}^{38}\\text{Ca}[\/latex]<\/li>\n<li id=\"fs-idp137322448\">[latex]{}_{1}{}^{3}\\text{H}[\/latex]<\/li>\n<li id=\"fs-idp179430752\">[latex]{}_{\\phantom{1}94}{}^{245}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idm5650336\">The following nuclei do not lie in the band of stability. How would they be expected to decay?\n<ol>\n<li id=\"fs-idp92587472\">[latex]{}_{15}{}^{28}\\text{P}[\/latex]<\/li>\n<li id=\"fs-idm20106176\">[latex]{}_{\\phantom{1}92}{}^{235}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<li id=\"fs-idm22275216\">[latex]{}_{20}{}^{37}\\text{Ca}[\/latex]<\/li>\n<li id=\"fs-idp137311888\">[latex]{}_{3}{}^{9}\\text{L}\\text{i}[\/latex]<\/li>\n<li id=\"fs-idp137045664\">[latex]{}_{\\phantom{1}96}{}^{245}\\text{Cm}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp10180544\">Predict by what mode(s) of spontaneous radioactive decay each of the following unstable isotopes might proceed:\n<ol>\n<li id=\"fs-idm24027360\">[latex]{}_{2}{}^{6}\\text{H}\\text{e}[\/latex]<\/li>\n<li id=\"fs-idp7080592\">[latex]{}_{30}{}^{60}\\text{Zn}[\/latex]<\/li>\n<li id=\"fs-idm35115440\">[latex]{}_{\\phantom{1}91}{}^{235}\\text{Pa}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<li id=\"fs-idm75261408\">[latex]{}_{\\phantom{1}94}{}^{241}\\text{Np}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/li>\n<li id=\"fs-idp17960336\"><sup>18<\/sup>F<\/li>\n<li id=\"fs-idp7359744\"><sup>129<\/sup>Ba<\/li>\n<li id=\"fs-idm31917904\"><sup>237<\/sup>Pu<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp176277168\">Write a nuclear reaction for each step in the formation of [latex]{}_{\\phantom{1}84}{}^{218}\\text{Po}_{\\phantom{}}^{\\phantom{}}[\/latex] from [latex]{}_{\\phantom{1}98}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex], which proceeds by a series of decay reactions involving the step-wise emission of \u03b1, \u03b2, \u03b2, \u03b1, \u03b1, \u03b1 particles, in that order.<\/li>\n<li id=\"fs-idp25125904\">Write a nuclear reaction for each step in the formation of [latex]{}_{\\phantom{1}82}{}^{208}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex] from [latex]{}_{\\phantom{1}90}{}^{228}\\text{T}_{\\phantom{}}^{\\phantom{}}\\text{h,}[\/latex] which proceeds by a series of decay reactions involving the step-wise emission of \u03b1, \u03b1, \u03b1, \u03b1, \u03b2, \u03b2, \u03b1 particles, in that order.<\/li>\n<li id=\"fs-idm43525136\">Define the term half-life and illustrate it with an example.<\/li>\n<li id=\"fs-idm37225376\">A 1.00 [latex]\\times[\/latex] 10<sup>\u20136<\/sup>-g sample of nobelium, [latex]{}_{102}{}^{254}\\text{No}[\/latex], has a half-life of 55 seconds after it is formed. What is the percentage of [latex]{}_{102}{}^{254}\\text{No}[\/latex] remaining at the following times?\n<ol>\n<li id=\"fs-idp172348128\">\u00a05.0 min after it forms<\/li>\n<li id=\"fs-idm38933040\">1.0 h after it forms<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idm3633744\"><sup>239<\/sup>Pu is a nuclear waste byproduct with a half-life of 24,000 y. What fraction of the <sup>239<\/sup>Pu present today will be present in 1000 y?<\/li>\n<li id=\"fs-idm33356592\">The isotope <sup>208<\/sup>Tl undergoes \u03b2 decay with a half-life of 3.1 min.\n<ol>\n<li id=\"fs-idp143445632\">What isotope is produced by the decay?<\/li>\n<li id=\"fs-idp78376656\">How long will it take for 99.0% of a sample of pure <sup>208<\/sup>Tl to decay?<\/li>\n<li id=\"fs-idp211861968\">What percentage of a sample of pure <sup>208<\/sup>Tl remains un-decayed after 1.0 h?<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idm43874912\">If 1.000 g of [latex]{}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}[\/latex] produces 0.0001 mL of the gas [latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}[\/latex] at STP (standard temperature and pressure) in 24 h, what is the half-life of <sup>226<\/sup>Ra in years?<\/li>\n<li id=\"fs-idp126089680\">The isotope [latex]{}_{38}{}^{90}\\text{Sr}[\/latex] is one of the extremely hazardous species in the residues from nuclear power generation. The strontium in a 0.500-g sample diminishes to 0.393 g in 10.0 y. Calculate the half-life.<\/li>\n<li id=\"fs-idp219506992\">Technetium-99 is often used for assessing heart, liver, and lung damage because certain technetium compounds are absorbed by damaged tissues. It has a half-life of 6.0 h. Calculate the rate constant for the decay of [latex]{}_{43}{}^{99}\\text{Tc}[\/latex].<\/li>\n<li id=\"fs-idp53369552\">What is the age of mummified primate skin that contains 8.25% of the original quantity of <sup>14<\/sup>C?<\/li>\n<li>A sample of rock was found to contain 8.23 mg of rubidium-87 and 0.47 mg of strontium-87.\n<ol>\n<li id=\"fs-idm22718704\">Calculate the age of the rock if the half-life of the decay of rubidium by \u03b2 emission is 4.7 [latex]\\times[\/latex] 10<sup>10<\/sup> y.<\/li>\n<li id=\"fs-idp75993408\">If some [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] was initially present in the rock, would the rock be younger, older, or the same age as the age calculated in (a)? Explain your answer.<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp133778576\">A laboratory investigation shows that a sample of uranium ore contains 5.37 mg of [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex] and 2.52 mg of [latex]{}_{\\phantom{1}82}{}^{206}\\text{Pb}_{\\phantom{}}^{\\phantom{}}[\/latex]. Calculate the age of the ore. The half-life of [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}[\/latex] is 4.5 [latex]\\times[\/latex] 10<sup>9<\/sup> yr.<\/li>\n<li id=\"fs-idp165416288\">Plutonium was detected in trace amounts in natural uranium deposits by Glenn Seaborg and his associates in 1941. They proposed that the source of this <sup>239<\/sup>Pu was the capture of neutrons by <sup>238<\/sup>U nuclei. Why is this plutonium not likely to have been trapped at the time the solar system formed 4.7 [latex]\\times[\/latex] 10<sup>9<\/sup> years ago?[latex]{}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex] has a half-life of 2.411 \u00d7 10<sup>4<\/sup> y. Calculate the value of <em data-effect=\"italics\">\u03bb<\/em> and then determine the amount of plutonium-239 remaining after 4.7 \u00d7 10<sup>9<\/sup> y:<em data-effect=\"italics\">\u03bbt<\/em> = <em data-effect=\"italics\">\u03bb<\/em>(2.411 \u00d7 10<sup>4<\/sup> y) = ln [latex]\\left(\\frac{1.0000}{0.5000}\\right)[\/latex] = 0.6931\n<p><em data-effect=\"italics\">\u03bb<\/em> = [latex]\\frac{0.6931}{2.411}[\/latex] \u00d7 10<sup>4<\/sup> y = 2.875 \u00d7 10<sup>\u20135<\/sup> y<sup>\u20131<\/sup><\/p>\n<p>Then:<\/p>\n<p>ln [latex]\\frac{{c}_{0}}{c}[\/latex] = <em data-effect=\"italics\">\u03bbt<\/em><\/p>\n<p>ln [latex]\\left(\\frac{1.000}{c}\\right)[\/latex] = 2.875 \u00d7 10<sup>\u20135<\/sup> y<sup>\u20131\u00a0<\/sup>\u00d7 4.7 \u00d7 10<sup>9<\/sup> y<\/p>\n<p>ln <em data-effect=\"italics\">c<\/em> = \u20131.351 \u00d7 10<sup>5<\/sup><\/p>\n<p><em data-effect=\"italics\">c<\/em> = 0<\/p>\n<p>This calculation shows that no Pu-239 could remain since the formation of the earth. Consequently, the plutonium now present could not have been formed with the uranium.<\/li>\n<li id=\"fs-idp217024288\">A [latex]{}_{4}{}^{7}\\text{Be}[\/latex] atom (mass = 7.0169 amu) decays into a [latex]{}_{3}{}^{7}\\text{L}\\text{i}[\/latex] atom (mass = 7.0160 amu) by electron capture. How much energy (in millions of electron volts, MeV) is produced by this reaction?<\/li>\n<li id=\"fs-idp71110896\">A [latex]{}_{5}{}^{8}\\text{B}[\/latex] atom (mass = 8.0246 amu) decays into a [latex]{}_{4}{}^{8}\\text{B}[\/latex] atom (mass = 8.0053 amu) by loss of a \u03b2<sup>+<\/sup> particle (mass = 0.00055 amu) or by electron capture. How much energy (in millions of electron volts) is produced by this reaction?<\/li>\n<li id=\"fs-idp131958896\">Isotopes such as <sup>26<\/sup>Al (half-life: 7.2 [latex]\\times[\/latex] 10<sup>5<\/sup> years) are believed to have been present in our solar system as it formed, but have since decayed and are now called extinct nuclides.\n<ol>\n<li><sup>26<\/sup>Al decays by \u03b2<sup>+<\/sup> emission or electron capture. Write the equations for these two nuclear transformations.<\/li>\n<li id=\"fs-idp179184448\">The earth was formed about 4.7 [latex]\\times[\/latex] 10<sup>9<\/sup> (4.7 billion) years ago. How old was the earth when 99.999999% of the <sup>26<\/sup>Al originally present had decayed?<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idp157539312\">Write a balanced equation for each of the following nuclear reactions:\n<ol>\n<li id=\"fs-idm52915712\">bismuth-212 decays into polonium-212<\/li>\n<li id=\"fs-idp90956192\">beryllium-8 and a positron are produced by the decay of an unstable nucleus<\/li>\n<li id=\"fs-idp18279664\">neptunium-239 forms from the reaction of uranium-238 with a neutron and then spontaneously converts into plutonium-239<\/li>\n<li id=\"fs-idp138334608\">strontium-90 decays into yttrium-90<\/li>\n<\/ol>\n<\/li>\n<li id=\"fs-idm29139968\">Write a balanced equation for each of the following nuclear reactions:\n<ol>\n<li id=\"fs-idp133484688\">mercury-180 decays into platinum-176<\/li>\n<li id=\"fs-idp136801104\">zirconium-90 and an electron are produced by the decay of an unstable nucleus<\/li>\n<li id=\"fs-idp77337360\">thorium-232 decays and produces an alpha particle and a radium-228 nucleus, which decays into actinium-228 by beta decay<\/li>\n<li id=\"fs-idm29267360\">neon-19 decays into fluorine-19<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h4>Selected Answers<\/h4>\n<p>1. \u03b1 (helium nuclei), \u03b2 (electrons), \u03b2<sup>+<\/sup> (positrons), and \u03b7 (neutrons) may be emitted from a radioactive element, all of which are particles; \u03b3 rays also may be emitted.<\/p>\n<p>3. (a) conversion of a neutron to a proton: [latex]{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{1}{}^{1}\\text{p}+{}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];<\/p>\n<p>(b) conversion of a proton to a neutron; the positron has the same mass as an electron and the same magnitude of positive charge as the electron has negative charge; when the n:p ratio of a nucleus is too low, a proton is converted into a neutron with the emission of a positron: [latex]{}_{1}{}^{1}\\text{p}\\longrightarrow {}_{0}{}^{1}\\text{n}+{}_{+1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];<\/p>\n<p>(c) In a proton-rich nucleus, an inner atomic electron can be absorbed. In simplifies form, this changes a proton into a neutron: [latex]{}_{1}{}^{1}\\text{p}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{0}{}^{1}\\text{p}[\/latex]<\/p>\n<p>5. The electron pulled into the nucleus was most likely found in the 1<i>s<\/i> orbital. As an electron falls from a higher energy level to replace it, the difference in the energy of the replacement electron in its two energy levels is given off as an X-ray.<\/p>\n<p>7. Manganese-51 is most likely to decay by positron emission. The n:p ratio for Cr-53 is [latex]\\frac{29}{24}[\/latex] = 1.21; for Mn-51, it is [latex]\\frac{26}{25}[\/latex] = 1.04; for Fe-59, it is [latex]\\frac{33}{26}[\/latex] = 1.27. Positron decay occurs when the n:p ratio is low. Mn-51 has the lowest n:p ration and therefore is most likely to decay by positron emission. Besides, [latex]{}_{24}{}^{53}\\text{Cr}[\/latex] is a stable isotope, and [latex]{}_{26}{}^{59}\\text{Fe}[\/latex] decays by beta emission.<\/p>\n<p><span style=\"line-height: 1.5;\">9. (a) too many neutrons, \u03b2 decay; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(b) atomic number greater than 82, \u03b1 decay; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(c) too few neutrons, positron emission; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(d) too many neutrons, \u03b2 decay; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(e) atomic number greater than 83, \u03b1 decay<\/span><\/p>\n<p>11.\u00a0[latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}90}{}^{234}\\text{Th}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex];<\/p>\n<p>[latex]{}_{\\phantom{1}90}{}^{234}\\text{Th}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}91}{}^{234}\\text{Pa}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];<\/p>\n<p>[latex]{}_{\\phantom{1}91}{}^{234}\\text{Pa}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}92}{}^{234}\\text{U}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex];<\/p>\n<p>[latex]{}_{\\phantom{1}92}{}^{234}\\text{U}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}90}{}^{230}\\text{Th}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]<\/p>\n<p>[latex]{}_{\\phantom{1}90}{}^{230}\\text{Th}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]<\/p>\n<p>[latex]{}_{\\phantom{1}88}{}^{226}\\text{Ra}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex];<\/p>\n<p>[latex]{}_{\\phantom{1}86}{}^{222}\\text{Rn}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}84}{}^{218}\\text{Po}_{\\phantom{}}^{\\phantom{}}+{}_{2}{}^{4}\\text{He}[\/latex]<\/p>\n<p>13. Half-life is the time required for half the atoms in a sample to decay. Example (answers may vary): For C-14, the half-life is 5770 years. A 10-g sample of C-14 would contain 5 g of C-14 after 5770 years; a 0.20-g sample of C-14 would contain 0.10 g after 5770 years.<\/p>\n<p>15. 1000 years is 0.04 half-lives. The fraction that remains after 0.04 half-lives is [latex]{\\left(\\frac{1}{2}\\right)}^{0.04}=0.973[\/latex] or 97.3%<\/p>\n<p><span style=\"line-height: 1.5;\">17.\u00a0<\/span><em data-effect=\"italics\">PV<\/em> = <em data-effect=\"italics\">nRT<\/em><\/p>\n<p><em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> = [latex]\\frac{PV}{RT}=\\frac{\\left(\\text{1 atm}\\right)\\left(0.0001\\text{mL}\\times \\text{1 L\/}{10}^{3}\\text{mL}\\right)}{\\left(0.08206\\text{L atm}{\\text{mol}}^{-1}{\\text{K}}^{-1}\\right)\\left(273.15\\text{K}\\right)}[\/latex] = 4.4614 [latex]\\times[\/latex] 10<sup>\u20139<\/sup> mol<\/p>\n<p><em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> produced = <em data-effect=\"italics\">n<\/em><sub>Rn<\/sub> decayed<\/p>\n<p>mass Ra lost = 4.4614 [latex]\\times[\/latex] 10<sup>\u20139<\/sup> mol [latex]\\times[\/latex] [latex]\\frac{\\text{226 g}}{\\text{mol}}[\/latex] = 1.00827 [latex]\\times[\/latex] 10<sup>\u20136<\/sup> g<\/p>\n<p>mass Ra remaining after 24 h = 1 \u2013 (1.00827 [latex]\\times[\/latex] 10<sup>\u20136<\/sup> g) = 9.9999899 [latex]\\times[\/latex] 10<sup>\u20131<\/sup> g<\/p>\n<p>ln [latex]\\frac{{c}_{0}}{c}=\\lambda t[\/latex] = ln [latex]\\frac{1.000}{9.9999899\\times {10}^{-1}}=\\lambda \\left(\\text{24 h}\\right)[\/latex] = 4.3785 [latex]\\times[\/latex] 10<sup>\u20137<\/sup><\/p>\n<p><em data-effect=\"italics\">\u03bb<\/em> = 4.2015 [latex]\\times[\/latex] 10<sup>\u20138<\/sup> h<sup>\u20131<\/sup><\/p>\n<p>[latex]{t}_{1\\text{\/}2}=\\frac{0.693}{\\lambda }=\\frac{0.693}{4.2015\\times {10}^{-8}}[\/latex] = 1.6494 [latex]\\times[\/latex] 10<sup>7<\/sup> h<\/p>\n<p>= 1.6494 [latex]\\times[\/latex] 10<sup>7<\/sup> h [latex]\\times[\/latex] [latex]\\frac{\\text{1 d}}{\\text{24 h}}\\times \\frac{\\text{1 y}}{\\text{365 d}}[\/latex] = 1.883 [latex]\\times[\/latex] 10<sup>3<\/sup> y or 2 [latex]\\times[\/latex] 10<sup>3<\/sup> y<\/p>\n<p><span style=\"line-height: 1.5;\">19.\u00a0<\/span>(Recall that radioactive decay is a first-order process.)<\/p>\n<p>\u03bb = [latex]\\frac{0.693}{{t}_{\\frac{1}{2}}}=\\frac{0.693}{6.0\\text{h}}[\/latex] = 0.12 h\u20131<\/p>\n<p>21.\u00a0(a) [latex]{}_{37}{}^{87}\\text{Rb}\\longrightarrow {}_{38}{}^{87}\\text{Sr}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/p>\n<p>[latex]{}_{38}{}^{87}\\text{Sr}[\/latex] is a stable isotope and does not decay further. Calculate the value of the decay rate constant for [latex]{}_{37}{}^{87}\\text{Rb}[\/latex], remembering that all radioactive decay is first order:<\/p>\n<p>[latex]\\lambda =\\frac{0.693}{4.7\\times {10}^{10}\\text{y}}=1.47\\times {10}^{-11}{\\text{y}}^{-1}[\/latex]<\/p>\n<p>Calculate the number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] and [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] found in the sample at time <i>t<\/i>:<\/p>\n<p>[latex]\\begin{array}{l}\\\\ \\\\ 8.23\\text{mg}\\times \\frac{\\text{1 g}}{\\text{1000 mg}}\\times \\frac{\\text{1 mol}}{87.0\\text{g}}=9.46\\times {10}^{-5}\\text{mol of}{}_{37}{}^{87}\\text{Rb}\\\\ 0.47\\text{mg}\\times \\frac{\\text{1 g}}{\\text{1000 mg}}\\times \\frac{\\text{1 mol}}{87.0\\text{g}}=5.40\\times {10}^{-6}\\text{mol of}{}_{38}{}^{87}\\text{Sr}\\end{array}[\/latex]<\/p>\n<p>Each mol of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] that disappeared (by radioactive decay of the [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] initially present in the rock) produced 1 mol of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex]. Hence the number of moles of [latex]{}_{38}{}^{87}\\text{Rb}[\/latex] that disappeared by radioactive decay equals the number of moles of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] that were produced. This amount consists of the 5.40 [latex]\\times[\/latex] 10\u20136 mol of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] found in the rock at time <i>t<\/i> if all the [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] present at time <i>t<\/i> resulted from radioactive decay of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] and no strontium-87 was present initially in the rock. Using this assumption, we can calculate the total number of moles of rubidium-87 initially present in the rock:<\/p>\n<p>Total number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] initially present in the rock at time <i>t<\/i> 0 = number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] at time <i>t<\/i> + number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] that decayed during the time interval t \u2013 t0 = number of moles of [latex]{}_{37}{}^{87}\\text{Rb}[\/latex] measured at time <i>t<\/i> + number of moles of [latex]{}_{38}{}^{87}\\text{Sr}[\/latex] measured at time <i>t<\/i> = 9.46 [latex]\\times[\/latex] 10\u20135 mol + 5.40 [latex]\\times[\/latex] 10\u20136 mol = 1.00 [latex]\\times[\/latex] 10\u20134 mol<\/p>\n<p>The number of moles can be substituted for concentrations in the expression:<\/p>\n<p>[latex]\\text{ln}\\frac{{c}_{0}}{{c}_{t}}=\\lambda t[\/latex]<\/p>\n<p>Thus:<\/p>\n<p>[latex]\\begin{array}{l}\\\\ \\\\ \\text{ln}\\frac{1.00\\times {10}^{-4}\\text{mol}}{9.46\\times {10}^{-5}\\text{mol}}=\\left(1.47\\times {10}^{-11}\\right)t\\\\ t=\\left(\\mathrm{ln}\\frac{1.00\\times {10}^{-4}}{9.46\\times {10}^{-5}}\\right)\\left(\\frac{1}{1.47\\times {10}^{-11}{\\text{y}}^{-1}}\\right)\\end{array}[\/latex]<\/p>\n<p>= 3.8 [latex]\\times[\/latex] 109 y = 3.8 billion years = age of the rock sample;<\/p>\n<p>(b) The rock would be younger than the age calculated in part (a). If Sr was originally in the rock, the amount produced by radioactive decay would equal the present amount minus the initial amount. As this amount would be smaller than the amount used to calculate the age of the rock and the age is proportional to the amount of Sr, the rock would be younger.<\/p>\n<p><span style=\"line-height: 1.5;\">23.\u00a0<\/span>[latex]{}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}[\/latex] has a half-life of 2.411 [latex]\\times[\/latex] 104 <em>y<\/em>. Calculate the value of \u03bb and then determine the amount of plutonium-239 remaining after 4.7 [latex]\\times[\/latex] 109 <em>y<\/em>:<\/p>\n<p>\u03bb<em>t<\/em> = \u03bb(2.411 [latex]\\times[\/latex] 104 y) = ln [latex]\\left(\\frac{1.0000}{0.5000}\\right)[\/latex] = 0.6931<\/p>\n<p>\u03bb = [latex]\\frac{0.6931}{2.411}[\/latex] [latex]\\times[\/latex] 104 <em>y<\/em> = 2.875 [latex]\\times[\/latex] 10\u20135 y\u20131<\/p>\n<p>Then:<\/p>\n<p>ln [latex]\\frac{{c}_{0}}{c}[\/latex] = \u03bb<em>t<\/em><\/p>\n<p>ln [latex]\\left(\\frac{1.000}{c}\\right)[\/latex] = 2.875 [latex]\\times[\/latex] 10\u20135 y\u20131 [latex]\\times[\/latex] 4.7 [latex]\\times[\/latex] 109 <em>y<\/em><\/p>\n<p>ln c = \u20131.351 [latex]\\times[\/latex] 105<\/p>\n<p>c = 0<\/p>\n<p>This calculation shows that no Pu-239 could remain since the formation of the earth. Consequently, the plutonium now present could not have been formed with the uranium.<\/p>\n<p><span style=\"line-height: 1.5;\">25.\u00a0<\/span>Find the mass difference of the starting mass and the total masses of the final products. Then use the conversion for mass to energy to find the energy released:<\/p>\n<p>8.0246 \u2013 8.0053 \u2013 0.00055 = 0.01875 amu<\/p>\n<p>0.01875 amu [latex]\\times[\/latex] 1.6605 [latex]\\times[\/latex] 10\u201327 kg\/amu = 3.113 [latex]\\times[\/latex] 10\u201329 kg<\/p>\n<p><em>E<\/em> = <em>mc<\/em><sup>2<\/sup> = (3.113 [latex]\\times[\/latex] 10\u201329 kg)(2.9979 [latex]\\times[\/latex] 108 m\/s)2<\/p>\n<p>= 2.798 [latex]\\times[\/latex] 10\u201312 kg m2\/s2 = 2.798 [latex]\\times[\/latex] 10\u201312 J\/nucleus<\/p>\n<p>2.798 [latex]\\times[\/latex] 10\u201312 J\/nucleus [latex]\\times[\/latex] [latex]\\frac{\\text{1 MeV}}{1.602177\\times {10}^{-13}\\text{J}}[\/latex] = 17.5 MeV<\/p>\n<p><span style=\"line-height: 1.5;\">27. (a) [latex]{}_{\\phantom{1}83}{}^{212}\\text{Bi}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}84}{}^{212}\\text{Po}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(b) [latex]{}_{5}{}^{8}\\text{B}\\longrightarrow {}_{4}{}^{8}\\text{B}\\text{e}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(c) [latex]{}_{\\phantom{1}92}{}^{238}\\text{U}_{\\phantom{}}^{\\phantom{}}+{}_{0}{}^{1}\\text{n}\\longrightarrow {}_{\\phantom{1}93}{}^{239}\\text{Np}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{N}_{\\phantom{}}^{\\phantom{}}\\text{p}[\/latex], <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">[latex]{}_{\\phantom{1}93}{}^{239}\\text{Np}_{\\phantom{}}^{\\phantom{}}\\longrightarrow {}_{\\phantom{1}94}{}^{239}\\text{Pu}_{\\phantom{}}^{\\phantom{}}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]; <\/span><\/p>\n<p><span style=\"line-height: 1.5;\">(d) [latex]{}_{38}{}^{90}\\text{Sr}\\longrightarrow {}_{39}{}^{90}\\text{Y}+{}_{-1}{}^{\\phantom{1}0}\\text{e}_{\\phantom{}}^{\\phantom{}}[\/latex]<\/span><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<section id=\"glossary\">\n<h3>Glossary<\/h3>\n<p><strong>alpha (\u03b1) decay<\/strong><br \/>\nloss of an alpha particle during radioactive decay<\/p>\n<p><strong>beta (\u03b2) decay<\/strong><br \/>\nbreakdown of a neutron into a proton, which remains in the nucleus, and an electron, which is emitted as a beta particle<\/p>\n<p><strong>daughter nuclide<\/strong><br \/>\nnuclide produced by the radioactive decay of another nuclide; may be stable or may decay further<\/p>\n<p><strong>electron capture<\/strong><br \/>\ncombination of a core electron with a proton to yield a neutron within the nucleus<\/p>\n<p><strong>gamma (\u03b3) emission<\/strong><br \/>\ndecay of an excited-state nuclide accompanied by emission of a gamma ray<\/p>\n<p><strong>half-life (t1\/2)<\/strong><br \/>\ntime required for half of the atoms in a radioactive sample to decay<\/p>\n<p><strong>parent nuclide<\/strong><br \/>\nunstable nuclide that changes spontaneously into another (daughter) nuclide<\/p>\n<p><strong>positron emission<\/strong><br \/>\n(also, \u03b2+ decay) conversion of a proton into a neutron, which remains in the nucleus, and a positron, which is emitted<\/p>\n<p><strong>radioactive decay<\/strong><br \/>\nspontaneous decay of an unstable nuclide into another nuclide<\/p>\n<p><strong>radioactive decay series<\/strong><br \/>\nchains of successive disintegrations (radioactive decays) that ultimately lead to a stable end-product<\/p>\n<p><strong>radiocarbon dating<\/strong><br \/>\nhighly accurate means of dating objects 30,000\u201350,000 years old that were derived from once-living matter; achieved by calculating the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the object vs. the ratio of [latex]{}_{\\phantom{1}6}{}^{14}\\text{C}_{\\phantom{}}^{\\phantom{}}:{}_{\\phantom{1}6}{}^{12}\\text{C}_{\\phantom{}}^{\\phantom{}}[\/latex] in the present-day atmosphere<\/p>\n<p><strong>radiometric dating<\/strong><br \/>\nuse of radioisotopes and their properties to date the formation of objects such as archeological artifacts, formerly living organisms, or geological formations<\/p>\n<\/section>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3667\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Chemistry. <strong>Provided by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/openstaxcollege.org\">http:\/\/openstaxcollege.org<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3667-1\">The \u201cm\u201d in Tc-99m stands for \u201cmetastable,\u201d indicating that this is an unstable, high-energy state of Tc-99. Metastable isotopes emit \u03b3 radiation to rid themselves of excess energy and become (more) stable. <a href=\"#return-footnote-3667-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Chemistry\",\"author\":\"\",\"organization\":\"OpenStax College\",\"url\":\"http:\/\/openstaxcollege.org\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3667","chapter","type-chapter","status-publish","hentry"],"part":2950,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapters\/3667","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/wp\/v2\/users\/17"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapters\/3667\/revisions"}],"predecessor-version":[{"id":5527,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapters\/3667\/revisions\/5527"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/parts\/2950"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapters\/3667\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/wp\/v2\/media?parent=3667"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/pressbooks\/v2\/chapter-type?post=3667"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/wp\/v2\/contributor?post=3667"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-buffstate-chemistryformajorsxmaster\/wp-json\/wp\/v2\/license?post=3667"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}