Carbon dating and half life
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Explaining the definition of an exponential curve and comparing it with one that looks similar but isn't exponential. Walking you through a realistic simulation of the decay of glad. Interesting insights are likely to come from such a group. The debate regarding this is outlined extensively in Renfrew 1972. However, radioisotope dating may not work so well in the future. In fact the radioactivity is directly proportional to the number of undecayed custodes. The level of proof demanded for such stories seems to be much less than for studies in the empirical sciences, such as physics, chemistry, molecular biology, physiology, etc. This means all living things have radioactive carbon-14 in them. Radiocarbon dating has been one of the most civil discoveries in 20th century science. Libby, Radiocarbon Dating, Univ. Comparing the chance of decay of different nuclei. Carbon has three main isotopes.
Radioactivity is measured in becquerels. The unit of radioactivity is named. Half-life: the time for the count rate to decrease by a half Radioactivity decreases with time. A given isotope always takes the same amount of time for the count rate to decrease by a half. Go forward in time and watch the radioactivity decrease by a half every half-life. Different samples of the same isotope all have the same half-life The half-life of a particular isotope is always the same. If we had a bigger sample of the same isotope then the count would be higher, say 200 becquerels. Using half-life in simple calculations What would its radioactivity be after 30 years? Each half-life is 10 years. So we imagine going in forward one half-life at a time from ZERO years: 10 years, 20 years, 30 years, etc. Then we halve the count for each half-life: 100 Bq after 10 years; 50 Bq after 20 years; 25 Bq after 30 years So we can see the radioactivity would be 25 becquerels afer 30 years. We can use the same idea to find out how long it would take for a sample with radioactivity 120 Bq to drop to 30 Bq. Carbon can be used to find out when living things died We can use radioactive decay to calculate the age of things. Carbon dating can only be used to find the age of things that were once alive, like wood, leather, paper and bones. If you have a wooden box, carbon dating can tell you when the tree to make it was cut down but not when the box was made. Carbon dating can be used to date things up to about 60 000 years old. So how does it work? Trees are made from reorganised air Many people think that plants grow by taking food from the soil through their roots but this is not true. All green plants make their own food in their leaves. They make it from carbon dioxide in the air. This process is called photosynthesis. You need energy from the Sun and lots of water. Carbon dioxide is made into simple sugars and it is these that are the building blocks that make up wood, bark and leaves. As it's produced carbon-14 reacts with oxygen to form carbon dioxide and some of it is taken in by green plants and made into sugars along with the 'normal' carbon. Carbon-14 is constantly created and constantly decays Carbon-14 is produced all the time but it also decays all the time back into nitrogen-14. An equilibrium is reached whereby about one in a trillion carbon atoms in the atmosphere is carbon-14. So all living things contain exactly the same proportion of carbon-14 compared to carbon-12: the proportion in the atmosphere. A living adult human body contains about a billionth of a gram of carbon-14. This means a human adult has a radioactivity of around 3000-4000 becquerels due to carbon-14. This is actually very small. When a living thing dies they stop eating so no new carbon-14 When a living thing dies the cells are no longer replaced so no new carbon enters it. The radioactivity of the carbon-14 begins to decrease. It halves about every 5600 years. By measuring the radioactivity we can tell how long ago the living thing died. Remember that the carbon-14 decays all the time whether the thing's alive or not. It's just that when it's living the carbon-14 is constantly replaced so the overall radioactivity stays constant. Preparing a sample for carbon dating Say we want to find the age of an old dead tree. Animals and plants have similar amounts of radioactive isotopes, particularly potassium-40, another beta emitter. The oldest samples have the lowest radioactivity. The radioactivity halves with each half-life. This means we can calculate the age of a sample. Counting carbon-14 atoms directly: another way of calculating age You can use a much smaller sample of the material you want to test if you rather than having to wait for them to decay. Even this kind of carbon dating can only be used to date things that were once alive and died less than about 60 000 years ago. Displaying radioactive decay on a graph We can plot a graph of radioactivity against time for our sample that had a half-life of 10 years. You can move forward in time and watch where on the graph you are. We can use our graph to show that it always takes 10 years for the radioactivity to drop by a half regardless of where you are on the graph. The decay of protactinium experiment A common school experiment is to find the half-life of an isotope called protactinium-234m. Walking you through a realistic simulation of the decay of protactinium. Take results, plot a graph and then calculate the half-life. It's useful because it has a half-life of the order of a minute and a pure sample. There's nothing special about half-life It's not that radioactive isotopes 'have' a half-life. They get less radioactive in a way that's called an exponential. Exponential decay means that equal periods of time give equal proportional changes in radioactivity. So you can pick any period of time, say 1 minute, and measure how much the radioactivity drops to in that minute. Say it drops to 63% of its initial value. After another minute it will drop to 63% of this value. Another minute, another 63% drop. Now 63% isn't a very obvious number so we pick a time where the drop is something simple, like 50%, or a half. We could equally well choose the one-third life or the four-fifths life. Remember the nucleus is just part of an atom, except that, when we talk about radioactivity, we tend to ignore the electrons that take up most of the volume. The atoms can't just vanish into nothingness and neither can its nucleus. The nucleus simply changes. Stepping through radioactive decay one second at a time to explain why radioactivity decreases with time. Let's think about a sample of a beta emitter. The sample consists of billions of atoms. The nucleus of each atom is unstable. Each nucleus will emit a single beta particle and then become stable. So every time a nucleus changes it gives out a beta particle. One nucleus, one particle. But as time passes there are fewer undecayed nuclei left TO decay. So the fewer undecayed nuclei you have, the slower you lose them, and the lower the radioactivity. There are lots of curves that look like exponentials but they don't have constant half-lives. Explaining the definition of an exponential curve and comparing it with one that looks similar but isn't exponential. Half-life is constant because every nucleus has a constant chance of decay each second. But the decay of a given nucleus is completely random. Nuclei never grow old The next point is slightly more subtle. A ninety year-old person is more likely to die this year than a sixteen year-old. The random nature of nuclear decay. Step forward in time and see that you can't predict when the nucleus will decay. At the start of every second it has exactly the same chance of decay. But if we have no idea at all exactly when a particular nucleus will decay how can we know how the radioactivity of a sample of trillions of nuclei will change with time? You know how many nuclei will decay, you just don't know which ones Imagine a large number of nuclei. Each nucleus has a 25% chance of decay each second, say. At the end of 1 second about 25% will have decayed but we have no idea which ones. The bigger the number we started with the closer we'll get to 25%. At the end of each second 25% of the number remaining have decayed. The PROPORTION depends only on the chance of decay. And this chance never changes. High chance of decay means short half-life But different isotopes have different chances of decay. In other words different isotopes have different half-lives. If you measured half-life with enough precision you could say that every half-life is unique. If you have three nuclei, each from different isotopes, then one will have the highest chance of decay and one will have the lowest. But you have no idea which one will decay first. It only makes sense to talk about likelihood when you have lots of nuclei for each isotope. Comparing the chance of decay of different nuclei. You still can't predict which one will decay first, only the probability. Radioactivity is proportional to the number of undecayed nuclei Imagine you have a sample of millions of nuclei of a beta emitter. Over time the undecayed nuclei decay. So the number of undecayed nuclei decreases. The rate of beta emission i. In fact the radioactivity is directly proportional to the number of undecayed nuclei. If you halve the number of undecayed nuclei, you halve the radioactivity. High chance of decay gives steep curve We can plot a graph of number of undecayed nuclei against time. This has an identical shape to the graph of radioactivity against time. This makes perfect sense because the fewer undecayed nuclei there are left, the fewer there are left to decay and give out a beta particle. Linking decay of nuclei with reduction in radioactivity. Change the chance of decay and the number of nuclei. Fit an exponential curve to the random decay. If the chance of decay is high, the nuclei decay quickly, the graph is steep and the half-life is short. The graph drops steeply because at the end of each second there are far fewer undecayed nuclei than there were. If there are fat fewer undecayed nuclei then the radioactivity must also be much lower. If the chance of decay is low, the nuclei decay slowly, the graph is shallow and the half-life is long. Half-life: the time for the number of undecayed nuclei to decrease by a half Every half-life the number of undecayed nuclei decreases by a half. So if the half-life is 10 seconds and there are 64 million undecayed nuclei, at the end of ten seconds there would 32 million, another ten seconds 16 million and so on. Comparing the graph of undecayed nuclei and radioactivity. Change the chance of decay and the number of undecayed nuclei and see what happens to each graph. The two graphs are deeply linked. The more undecayed nuclei there are, the more will decay, giving off a beta say each second. If you change the chance of decay by choosing different isotopes the number-of-nuclei curve always starts at the same place but the shape changes. This makes sense: you always start with 200 nuclei say no matter how fast they decay. The greater the chance of decay the greater the initial activity. This also makes sense. The curve after 60 is exactly the same as the curve of a different sample that STARTED with 60. The curves may not look as if they joined up if you plotted them because you tend to choose a scale that uses all your graph paper.
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