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The rate of radioactive decay is an intrinsic property of each radioactive isotope that is independent of the chemical and physical form of the radioactive isotope. In this section, we will describe radioactive decay rates and how half-lives can be used to monitor radioactive decay processes.In any sample of a given radioactive substance, the number of atoms of the radioactive isotope must decrease with time as their nuclei decay to nuclei of a more stable isotope.Using Activity is usually measured in disintegrations per second (dps) or disintegrations per minute (dpm).The activity of a sample is directly proportional to the number of atoms of the radioactive isotope in the sample: \[A = k N \label\] Here, the symbol is the same as the equation for the reaction rate of a first-order reaction, except that it uses numbers of atoms instead of concentrations.The most common method for measuring the age of ancient objects is carbon-14 dating.The carbon-14 isotope, created continuously in the upper regions of Earth’s atmosphere, reacts with atmospheric oxygen or ozone to form .
When the animal or plant dies, the carbon-14 nuclei in its tissues decay to nitrogen-14 nuclei by a radioactive process known as beta decay, which releases low-energy electrons (β particles) that can be detected and measured: \[ \ce \label\] The half-life for this reaction is 5700 ± 30 yr. Comparing the disintegrations per minute per gram of carbon from an archaeological sample with those from a recently living sample enables scientists to estimate the age of the artifact, as illustrated in Example 11.In a first-order reaction, every half-life is the same length of time. Calculate the half-life for the hydrolysis reaction under these conditions.If a freshly prepared solution of cis-platin has a concentration of 0.053 M, what will be the concentration of cis-platin after 5 half-lives? What is the percent completion of the reaction after 5 half-lives? Given: rate constant, initial concentration, and number of half-lives Asked for: half-life, final concentrations, and percent completion Strategy: Radioactivity, or radioactive decay, is the emission of a particle or a photon that results from the spontaneous decomposition of the unstable nucleus of an atom.In fact, radioactive decay is a first-order process and can be described in terms of either the differential rate law () or the integrated rate law: \[N = N_0e^ \] \[\ln \dfrac=-kt \label\] Because radioactive decay is a first-order process, the time required for half of the nuclei in any sample of a radioactive isotope to decay is a constant, called the half-life of the isotope.The half-life tells us how radioactive an isotope is (the number of decays per unit time); thus it is the most commonly cited property of any radioisotope.
SOLUTION We know the initial activity from the isotope’s identity (15 dpm/g), the final activity (8.0 dpm/g), and the half-life, so we can use the integrated rate law for a first-order nuclear reaction ( Exercise \(\Page Index\) It is believed that humans first arrived in the Western Hemisphere during the last Ice Age, presumably by traveling over an exposed land bridge between Siberia and Alaska.