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fa-chevron-right divider"></i></li><li><a href="../354/atomic-nuclear-and-particles.html">Atomic, nuclear and particles</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../368/nuclear-physics.html">Nuclear physics</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">AHL Nuclear decay</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../quantum/popcorn.jpg" style="float: left; width: 250px; height: 166px;">Radioactive decay is a random process. This leads to an interesting link with mathematics.</p> <p>If the rate of change of a quantity is proportional to the quantity itself then the change is exponential.</p> <p> </p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Random processes</p> </div> </div> <div class="panel-body"> <div> <p>Nuclear decay is a random process. In a sample of unstable nuclei:</p> <ul> <li>We do not know <em>which</em> nucleus will be the next to decay</li> <li>We do not know <em>when</em> the next nucleus will decay</li> <li>There is a fixed probability that a given proportion of decays will occur in a given period of time. For a large sample of dice, one sixth would land on a six in a given roll.</li> </ul> <p>The probability of decay is related to the amount of energy that will be released when it happens.</p> <p>The consequence of the random nature of decay can be demonstrated with a more everyday substance: beer foam. Since the bubbles pop randomly, the amount of beer foam in a glass decays exponentially.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/185686478"></iframe></div> <p>Exponential decay graphs are an example of a <a href="../225/non-linear-graphs.html" title="Non-linear graphs">Non-linear graph</a>.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Activity</p> </div> </div> <div class="panel-body"> <div> <p>The number of decays per second in a sample of material is directly proportional to the number of nuclei. This is referred to as the activity of the sample.</p> <p style="text-align: center;"><span class="math-tex">\(A\propto N\)</span></p> <p style="text-align: center;"><span class="math-tex">\(A=\lambda N\)</span></p> <ul> <li><span class="math-tex">\(A\)</span> is the activity of the sample (Bq, equivalent to s<sup>-1</sup>)</li> <li><span class="math-tex">\(\lambda\)</span> is the decay constant (s<sup>-1</sup>)</li> <li><span class="math-tex">\(N\)</span> is the number of undecayed nuclei</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Half life</p> </div> </div> <div class="panel-body"> <div> <p>Half life gives a measure of the rate of decay of a sample. Protactinium is an isotope with a half life of about 1 minute. This makes it convenient for school experiments.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/185686488"></iframe></div> <p>There are methods for measuring short and long half lives:</p> <ul> <li>If the half life is short, a Geiger-Muller tube and counter can collect measurements for how count rate varies with time. The background count rate must be measured (without the presence of the sample) and deducted from each reading. A graph can be plotted of count rate against time, and the half life measured and confirmed using intervals on the count axis.</li> <li>The activity of isotopes with half lives of hundreds or thousands of years does not noticeably decrease in the time over which observations can be made. Instead, to find half life, a pure sample of the isotope can be isolated. The mass can be used to find the number of nuclei, which is combined with the activity to find decay constant.</li> </ul> <p style="text-align: center;"><img alt="" src="../../quantum/halflifegraph1.png" style="width: 400px; height: 255px;"></p> <p>Half life is often the quantity that is quoted for a given isotope in exam questions. A good tip for commencing further calculations is to use the half life to find the decay constant:</p> <p style="text-align: center;"><span class="math-tex">\(\lambda={\ln2\over t_{1\over 2}}\)</span></p> <ul> <li><span class="math-tex">\(\lambda\)</span> is the decay constant</li> <li><span class="math-tex">\(t_{1\over2}\)</span> is the half life</li> </ul> <p>The half life of a isotope can be used to determine whether the isotope may have uses for mankind:</p> <ul> <li>The long half life of carbon-14 (5730 years) means it can be used to date ancient objects</li> <li>The short half life of technetium-99m (6 hours) permits the medical use as a tracer in the human body </li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <h3>Exponential decay</h3> <p>If <span class="math-tex">\({\mathrm{d} N\over \mathrm{d}t} ∝ N\)</span> then the decay is exponential:</p> <p style="text-align: center;"><span class="math-tex">\(N =N_0\mathrm{e}^{-\lambda t}\)</span></p> <ul> <li><span class="math-tex">\(N\)</span> is the number of undecayed nuclei remaining</li> <li><span class="math-tex">\(N_0\)</span> is the number of undecayed nuclei when <span class="math-tex">\(t=0\)</span></li> <li><span class="math-tex">\(\lambda\)</span> is the decay constant, a fixed probability of a given proportion of nuclei decaying per unit time (units correspond to <span class="math-tex">\(t\)</span> e.g. s<sup>-1</sup>)</li> <li><span class="math-tex">\(t\)</span> is time (e.g. s)</li> </ul> <p>Since activity is proportional to the number of undecayed nuclei, it is also exponential, leading to equations that can often be more useful than the number of particles. After all, it's difficult to count individual nuclei!</p> <p style="text-align: center;"><span class="math-tex">\(A=\lambda N_0\mathrm{e}^{-\lambda t}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(A =A_0\mathrm{e}^{-\lambda t}\)</span></p> <ul> <li><span class="math-tex">\(A\)</span> is the activity of the source (s<sup>-1</sup>)</li> <li><span class="math-tex">\(A_0\)</span> is the activity when <span class="math-tex">\(t=0\)</span></li> </ul> <p>The exponential decay equation enables the calculation of the number of nuclei remaining at any time.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/185686477"></iframe></div> <p>A linear relationship can be produced by taking natural logarithms of each side of the equation:</p> <p style="text-align: center;"><span class="math-tex">\(\ln A=-\lambda t+\ln{A_0}\)</span></p> <p>This is the kind of mathematical manipulation that may required for your <a href="../854/investigation-survival-guide.html" title="Investigation survival guide">Investigation</a>.</p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use flashcards to practise your recall.</em></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1,3" data-mode="" data-topics="1022" data-subject-id="6" data-n-flashcards="9" style="text-align:center">Show flashcards</a></div><hr> <p><em>Use quizzes to practise application of theory.</em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#a4980ab4"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="a4980ab4"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Exponential decay <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-114-393" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The diagram shows 4 lines</p><p style="text-align: center;"><img alt="" height="258" src="../../mc-images/screen-shot-2017-01-04-at-15.40.58.png" width="265"></p><p>Which one is exponential?</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>The others are 1/x, 1/x<sup>2</sup> and a parabola</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The blue line below has the equation y = e<sup>-x</sup></p><p style="text-align: center;"><img alt="" height="284" src="../../mc-images/screen-shot-2017-01-04-at-16.14.33.png" width="269"></p><p>The equation of the red line is:</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> y = 3 e<sup>-2x</sup></label></p><p><label class="radio"><input class="c" type="radio"> y = 3 e<sup>-x</sup></label></p><p><label class="radio"><input type="radio"> y = 3 e<sup>-3x</sup></label></p><p><label class="radio"><input type="radio"> y = e<sup>-3x</sup></label></p></div><div class="q-explanation"><p>points on the blue line are simply multiplied by 3</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Radioactive decay is exponential because:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> The rate of decay is proportional of the number of nuclei</label></p><p><label class="radio"><input type="radio"> The rate of decay is inversely proportional to time</label></p><p><label class="radio"><input type="radio"> The rate of change of nuclei is proportional to time</label></p><p><label class="radio"><input type="radio"> The number of nuclei is inversely proportional to time</label></p></div><div class="q-explanation"><p>the gradient of an exponential curve is proportional to the x value</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph shows activity vs time for a given isotope</p><p style="text-align: center;"><img alt="" height="236" src="../../mc-images/screen-shot-2017-01-05-at-05.23.00.png" width="250"></p><p>The half life is</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 4.0 s</label></p><p><label class="radio"><input class="c" type="radio"> 3.5 s</label></p><p><label class="radio"><input type="radio"> 2.5 s</label></p><p><label class="radio"><input type="radio"> 3.0 s</label></p></div><div class="q-explanation"><p>time to decay to 250</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph shows the activity vs time for a given isotope</p><p style="text-align: center;"><img alt="" height="236" src="../../mc-images/screen-shot-2017-01-05-at-05.35.33.png" width="313"></p><p>If the initial activity (for the same isotope) had been 400 Bq how long would it take for the activity to reach 100 Bq</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2.7 s</label></p><p><label class="radio"><input class="c" type="radio"> 4 s</label></p><p><label class="radio"><input type="radio"> 2 s</label></p><p><label class="radio"><input type="radio"> 1.3 s</label></p></div><div class="q-explanation"><p>The half life is 2 s it doesn't matter that there is a lower starting point</p><p>400 to 100 is two half lives</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Some radioactive isotopes are used in medicine. They are injected into the blood and enable different organs to show up on scanners.</p><p>A suitable half life for such an application would be:</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 5 min</label></p><p><label class="radio"><input type="radio"> 5 years</label></p><p><label class="radio"><input type="radio"> 5 s</label></p><p><label class="radio"><input type="radio"> 5 days</label></p></div><div class="q-explanation"><p>Needs to be radioactive for long enough to use but not so that the body is radioactive for a long time afterwards</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Protactinium is an example of an isotope used to measure half life in school labs.</p><p>An isotope used for this experiment could have a half life of</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 50 years</label></p><p><label class="radio"><input type="radio"> 500 years</label></p><p><label class="radio"><input type="radio"> 5 μs</label></p><p><label class="radio"><input class="c" type="radio"> 5min</label></p></div><div class="q-explanation"><p>with a half life of 5 mins there will be a measurable decrease in activity during a lesson</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which combination of activity and half life would be best for an alpha emitter suitable for use in the school lab.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 100 MBq, 1 year</label></p><p><label class="radio"><input type="radio"> 0.01Bq, 1year</label></p><p><label class="radio"><input type="radio"> 500Bq, 1 day</label></p><p><label class="radio"><input class="c" type="radio"> 500 Bq, 1 year</label></p></div><div class="q-explanation"><p>Needs to have a measurable activity but not decay too quickly. 100 MBq would be too radioactive for school use.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation for exponential decay is</p><p style="text-align: center;">N = Noe<sup>-λt</sup></p><p>where λ is the decay constant.</p><p>The unit for λ is</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> s<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> Bq</label></p><p><label class="radio"><input type="radio"> no units</label></p><p><label class="radio"><input type="radio"> s</label></p></div><div class="q-explanation"><p>λt must be a number without units so if λ has units s<sup>-1</sup> then λt will have units s/s ⇨ no units</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>An isotope has a decay constant 0.1 s<sup>-1</sup>. If its original activity is 1 MBq what will its activity be after 10 s</p><p style="text-align: center;">A = Aoe<sup>-λt</sup></p><p style="text-align: center"><iframe height="420" scrolling="no" src="http://web2.0calc.com/widgets/minimal/?options=%7B%22angular%22%3A%22deg%22%2C%22options%22%3A%22hide%22%2C%22menu%22%3A%22show%22%7D" style="border: 1px solid silver; " width="320"></iframe><br><a href="http://web2.0calc.com/">Web 2.0 scientific calculator</a></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.04 MBq</label></p><p><label class="radio"><input type="radio"> 0.09 MBq</label></p><p><label class="radio"><input type="radio"> 5.0 Bq</label></p><p><label class="radio"><input type="radio"> 0.3 MBq</label></p></div><div class="q-explanation"><p>Substitute into the equation</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> 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