The math for calculating how old something is from a study of it's radioactive elements is often presented in a very obscure fashion when it is really quite simple.
A formula is a mathematical expression or 'sentence' that describes some phenomenon in nature (or not in nature). There is a straight forward expression of any formula - the one that can be understood intuitively. Then you can rearrange the formula (solve for) to make any of the other factors in the equation the subject. These 'derived' formula are often less easy to understand intuitively.
First, for any youngsters that haven't yet learned algebraic notation, let's divert for a moment. Skip this if you already know algebra.
Notation in Algebra
First why do we bother to make formulas with letters. Why not just put in the numbers. The reason is so that this gives us a general formula which can be used for the same phenomenon but for different numerical examples. Also with a different letter representing each different factor, it is much easier to make any one of the factors the subject of the formula.
ab
First what does it mean when two letters are presented beside and touching each other. For instance ab.
This means you are to multiply the value of 'a' by the value of 'b'. With letters, we don't separate them by a times sign such as axb because this can be confused for telling you to multiply a times x times b. On the other hand if you are using numbers you can include the times sign so 7 x 9. Alternatively, a dot is often used in algebra in the middle of the line like this 7.9. This notation says to multiply 7 by 9 while 7.9 with the dot at the bottom of the line means seven decimal (or point) 9 - nearly 8.
3a
How about 3a.
This instructs you to multiply 'a' by 3 so it is the same as a+a+a.
a3
And then we have a3.
This tells you to multiply 'a' by 'a' by 'a' which is also called 'a' to the third power
a3
We also have a3 also known as asub3. This isn't an operator. It doesn't tell you to do anything. Series are a powerful tool in math and this notation could mean the third 'a' in a series. You could also have a0. This could mean a at time 0. We will use this just now. Whatever the notation, generally we described what each of the factors in a formula means at the beginning of the calculation unless it is such a well known formula that an explanation is not needed
The Physical Situation
Math is used to describe something in nature. I fine it quite gob smacking that so many things in nature can be described by quite simple math. In our case we are looking at the breakdown of a radioactive isotope. Perhaps we should describe what an isotope is for the young ones. There are about 92 separate natural occurring elements on the periodic table and a whole bunch more that are artificially created. Their chemistry (how they react with other elements) and hence their identity is determined by the electrons whizzing about the nucleus and especially the outer shell of electrons. In the nucleus are positive particles called protons and the number of electrons in an uncharged atom* exactly match the number of protons. But there are also uncharged neutrons in the nucleus. They have almost the same mass as a proton. The sum of the protons and neutrons gives you the Atomic number of the element. The Electrons are very light compared to protons and neutrons. The number of neutrons approximately equal the number of protons but can vary quite a bit. The different forms of atoms of a given element, due to the different number of neutrons, are called isotopes of that element. All the atoms of a specific element contain the same number of protons but the neutrons can vary. Some of these isotopes are radioactive. All this means is that they break down spontaneously into simpler, lighter elements. They aren't stable. At break down, they give off alpha, beta or gamma radiation. I'll tell you about them in an appendix.
* Perhaps you have heard of ions and might confuse an ion with an isotope. An ion is an atom which is temporarily missing or has an excess of one or more electron. Ions have a very strong tendency to gain or loose electrons to bring them back into equilibrium. Actually ions have a part to play in the dating of radioactive isotopes as we shall see.
Here is where we get to the critical observation about these isotopes which allows us to date them. Early researchers noted that if you observed a given radioactive element for a while, at some time in the future, half of it will be gone - changed into something else. Nothing surprising here. Then if you continued to observe this isotope, for the that same time period, half of what was left will be gone. Starting with some initial amount, each time, that particular period of time (which is unique for each different isotope) elapses, you would have half, then a quarter, then an eighth, then a sixteenth and so forth until there is too little to observe. This period has, not surprisingly, being called its half life. Half lives vary from milliseconds to millions of years for different isotopes.
The Formula
So we can start to build a formula. We will put each formula into words as well.
Suppose you know how much of a radioactive isotope you have today and want to know how much you will have after one 'half life' has gone by.
A1/2 = 1/2A0 In words, the amount after one half life is equal to one half times the amount at time 0. This is what we observed in the physical world. So far pretty simple - no? Remember for this problem we define A1/2 to mean The Amount after one half life. We could put 't' instead of 1/2 meaning the Amount after time t has gone by after the initial condition.
Now suppose that two half lives have gone by. We have to multiply again by 1/2 so we have At = (1/2)(1/2)A0 Or we could write (1/2)2A0. ie. after two half lives one half squared times the Amount at time 0 which, of course, is one quarter as much as we started with.
We could go on like this or we could put n (number of half lives) in the formula so it becomes At = (1/2)nA0. In words, if you want to know how much of a radioactive element remains after a given time, raise 1/2 to the power of the number of half lives that have past and multiply this number by the original amount.
Now we will have a short break and I'll ask you a question. Suppose we have some radioactive isotope that has a half life of 5 (h = 5)years and 15(t = 15) years have gone by. How many half lives have gone by and what fraction of the original isotope is left.
If you have understood the concept, you calculated 3(n = 3) half lives and there is one eighth left of the original amount. Putting this into letters, n = t/h or the number of half lives that have passed equals the time elapsed divided by the half life. Since n= t/h we can put t/h where n appeared in our formula.
We now have At = (1/2)(t/h)A0 and that is the whole radioactive formula. One of the main uses of this formula is to find t, the time that has elapsed since, say, Carbon 14 was incorporated into a plant or animal or perhaps the time since a rock melted and solidified and reset the clock. Below a quick description of what is meant and then we will learn how to rearrange the formula to make t or h or A0 the subject of the formula instead of At
Note that your little hand held calculator isn't bothered at all if 't' is not an even number of half lives. Not an easy thing to do by hand without another branch of math called logarithms but the calculator takes it in its stride.
The physical situation
There are two main fields where radioactivity is used for dating. First Carbon fourteen.
Carbon 14
The usual, (common) non-radioactive form of Carbon is Carbon twelve. This means that the sum of the number of protons plus the number of neutrons in the nucleus is 12. In this case the number of neutrons and protons are equal. That is to say, 6 of each. This sort of Carbon is stable, it is not radioactive. However, when high energy particles know as cosmic rays hit the atmosphere from outer space and high energy particles from the sun do likewise, some of the Nitrogen 14 is changed into carbon 14. and this diffuses into the atmosphere. It becomes part of the biosphere and any plant or animal takes up some of this radioactive Carbon along with the non-radioactive type. The quantities are very small but the methods of detecting the relative proportions of radioactive and non radioactive carbon in a plant or animal are very very accurate. I will describe them later.
What this results in, is that any living animal or plant is more or less in equilibrium with its environment and to a close approximation all have the same proportion of radioactive carbon to non radioactive carbon in their bodies.
However, when an organism dies, it is no longer taking up either form of carbon and the proportion of Carbon 14 begins to decrease as it changes back into N14. The half life of C14 is 5730 years and at a pinch, the amount can be measured with modern techniques to about 10 half lives. Of course the accuracy decreases, the older the sample but this takes us back to about 50,000 years. To put this into perspective, we can date materials back to half way into the most recent glacial period but not to the most recent interglacial, the Eemian which occurred about 125,000 years ago.
Rocks
Rocks can also be dated using radioactive isotopes. Here you can do a wee experiment in your kitchen to illustrate the process. Get some Copper sulfate which is a blue crystal. Dissolve as much as you can in some water. Now add sugar and dissolve as much of this as you can in the same water. Pour off the clear liquid (colored blue) into a clean glass and suspend a piece of string that you have dipped in powdered crystals of both solutes. Watch what happens. The Copper Sulfate and the sugar will crystalize separately as the water evaporates so you will have sugar and copper sulfate crystals back again. The sugar crystalizes with the sugar and the Copper sulfate with the Copper sulfate. The same thing happens with rocks. If you melt them and let them cool and solidify, the separate mineral crystalize separately. Think of a granite rock. melting re-sets the clock. If there is a Uranium containing mineral in the melt, it will crystalize out separately. Then it will continue to break down and the final result of a series of radioactive decays is Pb (Lead). By measuring the relative proportions of U and Pb, you can date when the rock cooled from a molten state. However we need to re-arrange the formula to make 't' the subject of the formula. Let's do it.
Solving for 't'
We start with the basic formula At = (1/2)(t/h)A0
and we want to get 't' by itself and all the rest on the other side of the equation. This is called 'solving for t'. So let's divide both sides by A0. Remember, we can do anything we want to an equation as long as we do exactly the same to both sides. Of course, the trick is to choose the right thing to do.
At/A0 = (1/2)(t/h)
This cancels out Ao on the right side and leaves it in the denominator (bottom) of the left side.
Now we need a wee log identity. I will give you a hint at the bottom of this blog of how logs work but for the moment, take my word for it that:
logabc = clogab Or in words, Log to the base 'a' of 'b' raised to the 'c'th power equals c times log to the base a of b. ie You can move the power to the front. I'll explain more about this in an appendix. Just remember that we can do anything to one side of an equation if we do the same to the other side. So the formula becomes.
log At/A0 = log (1/2)(t/h) and this becomes
log At/A0 = (t/h)log(1/2)
Oh, I nearly forgot to mention something. When you use the log function without any explanation, 'log' it always means to the base 10. If you want it to a different base, you must note it and ln is the natural log to base 2.718. Both Log and ln are on your computer.
Now all we have to do is to divide both sides by log(1/2) to move log(1/2) to the other side and multiply both sides by h to move h to the left side. 't' remains in glorious isolation on the right side. It is conventional to put the subject of the formula on the left side so we can reverse them. After all if a = b then b = a
Our formula for t then becomes
t = (logAt/A0)
(log1/2)
The Analysis System
I suppose one could extract the lead from a rock and the Uranium and weigh how much of each there was. These figures could be inserted into the formula to age a rock. But that wouldn't work with Carbon. You are comparing the amounts of Carbon fourteen with the amount of Carbon twelve and both have the same chemistry. There is a better and very sensitive method that works much like the first television sets.
In the early TV sets, the screen was one side of a large tube which tapered to the back behind the screen. At the back of the tube was a filament which, when heated, gave off electrons. These were accelerated in an electrical field and then passed through two sets of magnets oriented at right angles to each other. A charged particle moving through a magnetic field is bent. One set of magnets bent the beam of electrons back and forth horizontally and the other set up and down. The magnets were varied in such a way that the beam of electrons sped back and forth over the inside of the screen, causing the layer of phosphorescent material on the inside of the screen to light up. The electrical field was varied to give a stronger field which would provide a brighter point or less strong field which would give a darker point. The 'refresh' rate was so fast that you saw moving picture.
A machine to measure, for instance, the relative amounts of Carbon 14 and Carbon 12 works pretty well the same way. The sample of carbon to be measured is heated so hot that charged atoms of carbon are produced. This can be done with a laser. The charged atoms are accelerated through an electrical field and then between electric magnets, one North and the other South. This bends the beam of Carbon atoms. The neat part is that the heavier atoms are bent less than the lighter atoms. Where the beams hit the side of the container are detectors and the electronics connected to them can measure individual atoms. The strength of the magnets can be varied to bend the beam more or less to ensure each beam hits a detector. Clearly, this is a very accurate method of measuring very tiny amounts of Carbon 14 and Carbon 12. The number of hits on each detector can be inserted into the formula.
Logarithms
I promised I would try to explain logarithms. First a bit of notation. If you see a notation log1000 = 3, it is understood that this is the log to the base 10. If you use another base, you have to state it. There is another notation ln. It means the log to the base 'e'. e is the base of the natural logarithms, whatever that means. I don't understand it. e is 2.71828. It is known as Euler's number. If anyone out there has a good explanation for the natural logarithms, please put it in the comments. Anyway the two following notations are equivalent.
105 = 100000
log 100000 = 5
You can see that the log of a number is the exponent that you have to raise 10 to in order to get the number. Note again that it is understood that if you use the term log, it is understood that it is to the base 10. You could also write it this way.
log10100000 = 5
No comments:
Post a Comment