{"id":905,"date":"2018-03-20T16:34:11","date_gmt":"2018-03-20T16:34:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-orgbiochemistry\/?post_type=chapter&p=905"},"modified":"2018-09-19T15:14:06","modified_gmt":"2018-09-19T15:14:06","slug":"11-2-half-life","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-monroecc-orgbiochemistry\/chapter\/11-2-half-life\/","title":{"raw":"11.2 Half-Life","rendered":"11.2 Half-Life"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
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  1. Define half-life<\/em>.<\/li>\r\n \t
  2. Determine the amount of radioactive substance remaining after a given number of half-lives.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    Whether or not a given isotope is radioactive is a characteristic of that particular isotope. Some isotopes are stable indefinitely, while others are radioactive and decay through a characteristic form of emission. As time passes, less and less of the radioactive isotope will be present, and the level of radioactivity decreases. An interesting and useful aspect of radioactive decay is half-life. The half-life<\/span><\/strong><\/span>\u00a0of a radioactive isotope is the amount of time it takes for one-half of the radioactive isotope to decay. The half-life of a specific radioactive isotope is constant; it is unaffected by conditions and is independent of the initial amount of that isotope.<\/p>\r\n

    Consider the following example. Suppose we have 100.0 g of 3<\/sup>H (tritium, a radioactive isotope of hydrogen). It has a half-life of 12.3 y. After 12.3 y, half of the sample will have decayed to 3<\/sup>He by emitting a beta particle, so that only 50.0 g of the original 3<\/sup>H remains. After another 12.3 y\u2014making a total of 24.6 y\u2014another half of the remaining 3<\/sup>H will have decayed, leaving 25.0 g of 3<\/sup>H. After another 12.3 y\u2014now a total of 36.9 y\u2014another half of the remaining 3<\/sup>H will have decayed, leaving 12.5 g of 3<\/sup>H. This sequence of events is illustrated in Figure 11.1 \"Radioactive Decay\"<\/a>.<\/p>\r\n\r\n

    \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1683\"]\"image\" Figure 11.1 Radioactive Decay.\u00a0<\/em>During each successive half-life, half of the initial amount will radioactively decay.[\/caption]\r\n

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    Determine the amount of a radioactive isotope remaining after a given number half-lives by using the following expression:<\/p>\r\n

    [latex]\\text{amount remaining}=\\text{initial amount}\\times{\\frac{1}{{2}^n}}[\/latex]<\/p>\r\n

    where n<\/em> is the number of half-lives, which is determined as follows:<\/p>\r\n

    [latex]n=\\frac{\\text{time elapsed}}{\\text{time for one half life}}[\/latex]<\/p>\r\n

    The expression for amount remaining works even if the number of half-lives is not a whole number, but it would be necessary to use a calculator to determine the value of [latex]\\frac{1}{{2}^n}[\/latex].<\/p>\r\n\r\n

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    Example 3<\/h3>\r\n

    The half-life of 20<\/sup>F is 11.0 s. If a sample initially contains 5.00 g of 20<\/sup>F, how much 20<\/sup>F remains after 44.0 s?<\/p>\r\n

    Solution<\/p>\r\n

    Comparing the time that has passed to the isotope\u2019s half-life, note that 44.0 s is exactly 4 half-lives, so using the previous equation, n<\/em> = 4. Substituting and solving results in the following:<\/p>\r\n

    [latex]\\text{amount remaining}=5.0\\text{ g}\\times{\\frac{1}{{2}^4}}=0.313\\text{ g}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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    Skill-Building Exercise<\/h3>\r\n
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      The half-life of 44<\/sup>Ti is 60.0 y. A sample initially contains 0.600 g of 44<\/sup>Ti. How much 44<\/sup>Ti remains after 180.0 y?<\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n

      Half-lives of isotopes range from fractions of a microsecond to billions of years. Table 11.2 \"Half-Lives of Various Isotopes\"<\/a> lists the half-lives of some isotopes.<\/p>\r\n\r\n

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      Table 11.2<\/span> Half-Lives of Various Isotopes<\/strong><\/h5>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Isotope<\/th>\r\nHalf-Life<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      3<\/sup>H<\/td>\r\n12.3 y<\/td>\r\n<\/tr>\r\n
      14<\/sup>C<\/td>\r\n5,730 y<\/td>\r\n<\/tr>\r\n
      40<\/sup>K<\/td>\r\n1.26 \u00d7 109<\/sup> y<\/td>\r\n<\/tr>\r\n
      51<\/sup>Cr<\/td>\r\n27.70 d<\/td>\r\n<\/tr>\r\n
      90<\/sup>Sr<\/td>\r\n29.1 y<\/td>\r\n<\/tr>\r\n
      131<\/sup>I<\/td>\r\n8.04 d<\/td>\r\n<\/tr>\r\n
      222<\/sup>Rn<\/td>\r\n3.823 d<\/td>\r\n<\/tr>\r\n
      235<\/sup>U<\/td>\r\n7.04 \u00d7 108<\/sup> y<\/td>\r\n<\/tr>\r\n
      238<\/sup>U<\/td>\r\n4.47 \u00d7 109<\/sup> y<\/td>\r\n<\/tr>\r\n
      241<\/sup>Am<\/td>\r\n432.7 y<\/td>\r\n<\/tr>\r\n
      248<\/sup>Bk<\/td>\r\n23.7 h<\/td>\r\n<\/tr>\r\n
      260<\/sup>Sg<\/td>\r\n4 ms<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
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      Looking Closer: Half-Lives of Radioactive Elements<\/h3>\r\n

      Many people think that the half-life of a radioactive element represents the amount of time an element is radioactive. In fact, it is the time required for half\u2014not all\u2014of the element to decay radioactively. The daughter isotope may also be radioactive, so its radioactivity must also be considered.<\/p>\r\n

      The expected working life of an ionization-type smoke detector (described in the opening essay) is about 10 years. In that time, americium-241, which has a half-life of about 432 y, loses less than 4% of its radioactivity. A half-life of 432 y may seem long, but it is not very long as half-lives go. Uranium-238, the most common isotope of uranium, has a half-life of about 4.5 \u00d7 109<\/sup> y, while thorium-232 has a half-life of 14 \u00d7 109<\/sup> y.<\/p>\r\n

      On the other hand, some nuclei have extremely short half-lives, presenting challenges to the scientists who study them. The longest-lived isotope of lawrencium, 262<\/sup>Lr, has a half-life of 3.6 h, while the shortest-lived isotope of lawrencium, 252<\/sup>Lr, has a half-life of 0.36 s. The largest atom ever produced by a nuclear reaction has atomic number 118, mass number 293, and a half-life of 120 ns. Can you imagine how quickly an experiment must be done to determine the properties of elements that exist for so short a time?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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      Concept Review Exercises<\/h3>\r\n<\/div>\r\n
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        Define half-life<\/em>.<\/p>\r\n\r\n<\/div><\/li>\r\n \t

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        Describe a way to determine the amount of radioactive isotope remaining after a given number of half-lives.<\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n

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        Answers<\/h3>\r\n

        [reveal-answer q=\"224282\"]Show Answer[\/reveal-answer]<\/p>\r\n\r\n

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        [hidden-answer a=\"224282\"]<\/p>\r\n

        1. Half-life is the amount of time needed for half of a radioactive material to decay.<\/p>\r\n

        2. take half of the initial amount for each half-life of time elapsed[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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        Key Takeaways<\/h3>\r\n