How Does An Atom Change When It Undergoes Radioactive Decay?
The metastable sixtym Co nuclide has a half-life of 10.5 minutes. Since electromagnetic radiations carries neither accuse nor mass, the production of 
-ray emission by 60one thousand Co is 60Co.
Nuclides with atomic numbers of xc or more than undergo a course of radioactive decay known as spontaneous fission in which the parent nucleus splits into a pair of smaller nuclei. The reaction is usually accompanied by the ejection of i or more neutrons.
For all only the very heaviest isotopes, spontaneous fission is a very wearisome reaction. Spontaneous fission of 238U, for example, is almost two meg times slower than the rate at which this nuclide undergoes 
-decay.
Predict the products of the following nuclear reactions:
(a) electron emission by xivC (b) positron emission by eightB
(c) electron capture by 125I (d) blastoff emission by 210Rn
(e) gamma-ray emission by 56mNi
Click hither to cheque your respond to Practice Problem 3.
In 1934 Enrico Fermi proposed a theory that explained the three forms of beta decay. He argued that a neutron could decay to grade a proton by emitting an electron. A proton, on the other paw, could be transformed into a neutron past two pathways. Information technology can capture an electron or it can emit a positron. Electron emission therefore leads to an increase in the atomic number of the nucleus.
Both electron capture and positron emission, on the other hand, result in a decrease in the atomic number of the nucleus.
A plot of the number of neutrons versus the number of protons for all of the stable naturally occurring isotopes is shown in the figure beneath. Several conclusions can exist drawn from this plot.
 |
| A graph of the number of neutrons versus the number of protons for all stable naturally occurring nuclei. Nuclei that prevarication to the correct of this band of stability are neutron poor; nuclei to the left of the band are neutron-rich. The solid line represents a neutron to proton ratio of one:i. |
The near likely mode of disuse for a neutron-rich nucleus is one that converts a neutron into a proton. Every neutron-rich radioactive isotope with an atomic number smaller 83 decays by electron (�/i>-) emission. 14C, 32P, and 35Southward, for case, are all neutron-rich nuclei that decay by the emission of an electron.
Neutron-poor nuclides decay past modes that convert a proton into a neutron. Neutron-poor nuclides with atomic numbers less than 83 tend to disuse by either electron capture or positron emission. Many of these nuclides decay by both routes, simply positron emission is more oft observed in the lighter nuclides, such equally 22Na.
Electron capture is more common among heavier nuclides, such as 125I, because the anes electrons are held closer to the nucleus of an cantlet as the accuse on the nucleus increases.
A third mode of decay is observed in neutron-poor nuclides that take atomic numbers larger than 83. Although it is not obvious at offset, -decay increases the ratio of neutrons to protons. Consider what happens during the -decay of 238U, for case.
The parent nuclide (238U) in this reaction has 92 protons and 146 neutrons, which means that the neutron-to-proton ratio is 1.587. The daughter nuclide (234Thursday) has 90 protons and 144 neutrons, so its neutron-to-proton ratio is one.600. The girl nuclide is therefore slightly less likely to be neutron-poor, as shown in the effigy below.
Binding Energy Calculations
We should be able to predict the mass of an atom from the masses of the subatomic particles information technology contains. A helium atom, for case, contains two protons, two neutrons, and two electrons.
Example:
The mass of a helium atom should be 4.0329802 amu.
| two(1.0072765) amu | = | two.0145530 amu |
| 2(1.0086650) amu | = | ii.0173300 amu |
| 2(0.0005486) amu | = | 0.0010972 amu |
| Full mass | = | four.0329802 amu |
When the mass of a helium atom is measured, we discover that the experimental value is smaller than the predicted mass past 0.0303769 amu.
| Predicted mass | = | 4.0329802 amu |
| Observed mass | = | 4.0026033 amu |
| Mass defect | = | 0.0303769 amu |
The difference between the mass of an atom and the sum of the masses of its protons, neutrons, and electrons is called the mass defect. The mass defect of an atom reflects the stability of the nucleus. Information technology is equal to the free energy released when the nucleus is formed from its protons and neutrons. The mass defect is therefore besides known as the binding energy of the nucleus.
The binding energy serves the same function for nuclear reactions as 
H for a chemical reaction. It measures the deviation between the stability of the products of the reaction and the starting materials. The larger the binding energy, the more stable the nucleus. The bounden energy can besides be viewed as the amount of free energy information technology would accept to rip the nucleus autonomously to grade isolated neutrons and protons. It is therefore literally the free energy that binds together the neutrons and protons in the nucleus.
The binding energy of a nuclide can be calculated from its mass defect with Einstein's equation that relates mass and free energy.
E = mc two
Example:
Nosotros establish the mass defect of He to exist 0.0303769 amu. To obtain the binding free energy in units of joules, nosotros must convert the mass defect from diminutive mass units to kilograms.
Multiplying the mass defect in kilograms by the foursquare of the speed of calorie-free in units of meters per 2nd gives a binding energy for a single helium atom of four.53358 x ten-12 joules.
Multiplying the result of this calculation by the number of atoms in a mole gives a binding energy for helium of 2.730 x 1012 joules per mole, or 2.730 billion kilojoules per mole.
This calculation helps us sympathize the fascination of nuclear reactions. The energy released when natural gas is burned is about 800 kJ/mol. The synthesis of a mole of helium releases three.4 million times as much energy.
Since about nuclear reactions are carried out on very pocket-sized samples of fabric, the mole is not a reasonable basis of measurement. Binding energies are usually expressed in units of electron volts (eV) or 1000000 electron volts (MeV) per atom.
Instance:
The binding free energy of helium is 28.3 ten 106 eV/atom or 28.iii MeV/cantlet.
Calculations of the bounden energy tin be simplified past using the post-obit conversion cistron betwixt the mass defect in atomic mass units and the binding energy in million electron volts.
1 amu = 931.5016 MeV
Bounden energies gradually increment with atomic number, although they tend to level off near the stop of the periodic table. A more than useful quantity is obtained by dividing the binding free energy for a nuclide past the total number of protons and neutrons it contains. This quantity is known every bit the binding free energy per nucleon.
The binding energy per nucleon ranges from almost 7.5 to 8.8 MeV for most nuclei, as shown in the effigy below. Information technology reaches a maximum, nonetheless, at an atomic mass of almost sixty amu. The largest binding energy per nucleon is observed for 56Atomic number 26, which is the most stable nuclide in the periodic table.
The graph of binding energy per nucleon versus atomic mass explains why energy is released when relatively small nuclei combine to grade larger nuclei in fusion reactions.
Information technology likewise explains why free energy is released when relatively heavy nuclei separate autonomously in fission (literally, "to carve up or carve") reactions.
At that place are a number of small irregularities in the binding free energy bend at the low stop of the mass spectrum, as shown in the figure below. The fourHe nucleus, for example, is much more stable than its nearest neighbors. The unusual stability of the fourHe nucleus explains why -particle disuse is usually much faster than the spontaneous fission of a nuclide into two big fragments.
The Kinetics of Radioactive Decay
Radioactive nuclei disuse by first-order kinetics. The rate of radioactive decay is therefore the production of a rate constant (k) times the number of atoms of the isotope in the sample (N).
Charge per unit = kN
The rate of radioactive decay doesn't depend on the chemical land of the isotope. The rate of decay of 238U, for example, is exactly the same in uranium metal and uranium hexafluoride, or any other compound of this element.
The rate at which a radioactive isotope decays is called the activity of the isotope. The near common unit of measurement of activity is the curie (Ci), which was originally defined as the number of disintegrations per second in 1 gram of 226Ra. The curie is now divers as the amount of radioactive isotope necessary to attain an activity of 3.700 x 1010 disintegrations per second.
The relative rates at which radioactive nuclei decay tin be expressed in terms of either the charge per unit constants for the decay or the half-lives of the nuclei. Nosotros can conclude that 14C decays more rapidly than 238U, for example, by noting that the rate constant for the decay of xivC is much larger than that for 238U.
| 14C: | yard = 1.210 x 10-iv y-one | |
| 238U: | k = 1.54 10 10-x y-1 |
We can reach the same conclusion by noting that the one-half-life for the decay of 14C is much shorter than that for 235U.
| xivC: | t ane/2 = 5730 y | |
| 238U: | t 1/2 = 4.51 x 109 y |
The half-life for the disuse of a radioactive nuclide is the length of time it takes for exactly half of the nuclei in the sample to disuse. In our word of the kinetics of chemical reactions, we concluded that the one-half-life of a first-order process is inversely proportional to the rate constant for this process.
The half-life of a nuclide can be used to guess the corporeality of a radioactive isotope left after a given number of half-lives. For more complex calculations, information technology is easier to convert the half-life of the nuclide into a rate constant and and so apply the integrated course of the start-order rate police described in the kinetic section.
Dating By Radioactive Decay
The globe is constantly bombarded by cosmic rays emitted past the sun. The full free energy received in the form of cosmic rays is pocket-size 
no more than the energy received by the planet from starlight. But the energy of a single cosmic ray is very big, on the order of several billion electron volts (0 . 200 meg kJ/mol). These highly energetic rays react with atoms in the atmosphere to produce neutrons that then react with nitrogen atoms in the atmosphere to produce 14C.
The 14C formed in this reaction is a neutron-rich nuclide that decays by electron emission with a half-life of 5730 years.
Simply after World War II, Willard F. Libby proposed a way to employ these reactions to estimate the age of carbon-containing substances. The 14C dating technique for which Libby received the Nobel prize was based on the following assumptions.
- 14C is produced in the atmosphere at a more or less constant charge per unit.
- Carbon atoms circulate between the temper, the oceans, and living organisms at a rate very much faster than they decay. As a result, in that location is a constant concentration of 14C in all living things.
- Subsequently death, organisms no longer choice up 14C.
Thus, past comparison the activity of a sample with the action of living tissue we tin approximate how long it has been since the organism died.
The natural abundance of 14C is about ane part in 1012 and the average activeness of living tissue is 15.three disintegrations per infinitesimal per gram of carbon. Samples used for fourteenC dating tin can include charcoal, wood, material, paper, sea shells, limestone, flesh, hair, soil, peat, and bone. Since most iron samples too incorporate carbon, it is possible to gauge the time since iron was last fired by analyzing for xivC.
We at present know that i of Libby's assumptions is questionable: The amount of fourteenC in the atmosphere hasn't been constant with time. Because of changes in solar activity and the world'due south magnetic field, information technology has varied by as much as 5%. More than recently, contagion from the called-for of fossil fuels and the testing of nuclear weapons has caused significant changes in the amount of radioactive carbon in the temper. Radiocarbon dates are therefore reported in years earlier the nowadays era (B.P.). By convention, the present era is assumed to brainstorm in 1950, when xivC dating was introduced.
Studies of bristlecone pines allow u.s. to correct for changes in the abundance of fourteenC with time. These remarkable trees, which grow in the White Mountains of California, tin alive for up to five thousand years. By studying the xivC activity of samples taken from the almanac growth rings in these trees, researchers take developed a calibration curve for 14C dates from the present back to 5145 B.C.
After roughly 45,000 years (eight half-lives), a sample retains only 0.4% of the fourteenC activeness of living tissue. At that point it becomes too old to date by radiocarbon techniques. Other radioactive isotopes can be used to date rocks, soils, or archaeological objects that are much older. Potassium-argon dating, for case, has been used to date samples up to 4.3 billion years sometime. Naturally occurring potassium contains 0.0118% by weight of the radioactive 40K isotope. This isotope decays to fortyAr with a half-life of 1.3 billion years. The 40Ar produced afterward a stone crystallizes is trapped in the crystal lattice. It can be released, even so, when the stone is melted at temperatures up to 2000 C. By measuring the amount of 40Ar released when the stone is melted and comparing it with the amount of potassium in the sample, the time since the rock crystallized can exist determined.
Source: https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch23/modes.php
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