# Half-Life and Mass Spectrometry

by Jared Rovny
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00:02 Lastly, there's one thing we should discuss about radioactivity which is something called the "Half-Life." First of all, and when we're talking about some material, may be we have a chunk of metal that is an unstable material.

00:14 And it is radioactively decaying.

00:16 What happens is as you can see in this graph, the number of radioactive particles decays as time goes on.

00:24 This is because of the number of particles is losing mass.

00:29 It's losing energy to this radiation as time goes on.

00:33 And so these radioactive particles are converting to more stable isotopes or more stable materials.

00:38 What we're doing in this graph is plotting the number of radioactive isotopes.

00:44 And so it isn't the total number of material that's there.

00:47 So it's not until our metal is disappearing.

00:49 We're just plotting the number of radioactive parts of whatever it is that we're referring to.

00:54 On the x-axis or the horizontal axis we have the time plotted.

00:57 So that as time goes on we see that the number of radioactive materials is decaying and is decaying is what we call an "Exponential Decay Law." If you look at that equation you'll see that we have the number with a parenthesis 't.' That's just means the number as a function of time.

01:13 So as time goes on the number will originally be equal to some original number of particles which we call "N 0 not and with a little zero.

01:21 This is the original number of radioactive particles that are present, times as exponential factor.

01:28 In this E to the minus lambda 't' is telling us how quickly the particles are decaying.

01:34 The lambda in this equation is the radioactive decay constant.

01:38 And is specifically telling us how quickly this curve is going down.

01:42 Or how quickly we are losing our radioactive material.

01:45 Looking at the image below the graph here, we've plotted a box.

01:49 This is representing our material.

01:51 Where the red amount of material is the amount that is radioactive.

01:55 As the material is radioactively decaying, we're losing that red material, that radioactive material.

02:01 And more and more of the boxes becoming blue.

02:05 The reason that this is an exponential decay, is that we start with a lot of radioactive material.

02:10 And so for that reason there's a lot of represented here by these red squiggly lines.

02:16 So we have three red squiggly lines meaning a lot of radioactive decay coming from this material.

02:22 Meaning that we're losing mass very quickly.

02:25 So looking at our graph this is why this is the steepest part of our graph.

02:29 Because originally when we still have all this radioactive material, we're losing that radioactive material very quickly as it converts to a more stable material.

02:38 Once we've lost more and more material as you can see as we go along in time, we have less and less radioactive material and less and less radioactive decay occurring.

02:47 And so the rate at which we lose material, as you can see from our graph dies away.

02:52 It becomes slower and slower that we're losing that radioactive decay.

02:56 We can find for ourselves what's called the "Half-Life." Or the time that it takes for us to lose exactly half of the original radioactive material.

03:07 It's not so difficult to solve for the half-life because what we would do is in the left hand side of this equation.

03:13 For the "N" of "t", the number of particles, the amount of materials that's left.

03:18 We simply say that the number is half of whatever the original number of the particles was.

03:24 So probably N and not and zero over 2 here for the number of particles we have left at the half life.

03:30 We can re-arrange this equation and solve for the half-life itself.

03:34 The t1/2.

03:35 What we would do to do this in fact, is to first cancel these N nots, these N zeroes.

03:41 And then we would have one half equals and then our exponential term.

03:44 E to the minus lambda, the half-life t1/2.

03:48 All we have to do to solve for the t1/2 in that kind of equation is to take the logarithm of both sides.

03:56 On the bottom here you can see that we've written logarithm as "ln." This is called the "Natural Logarithm." The natural logarithm is just like the regular logarithm that we discussed when we were talking about sound and decibels.

04:07 Except the base of the logarithm is base "e." Where "e" is this exponential number here which is about a little more than 2.

04:16 So this "e" number it's just a constant, it's just a number.

04:20 And the logarithm, the natural logarithm is log base e.

04:25 So by taking the natural logarithm of both sides we get the half-life is equal to the natural log of 2.

04:31 Which is just some number.

04:33 Divided by the lambda term which is again this term is telling us how fast our thing is decaying, whatever our object or whatever our radioactive material is.

04:43 There's one last thing about this half-life which is that if we're plotting this decay of radioactive material which is going to fall that exponential of all that we saw.

04:53 We can do something tricky with this equation to make it easier to plot.

04:58 What we do looking at this equation here, is on the left and the right, we will again take the natural logarithm.

05:05 Now there some mathematics about the natural logarithm that we are not going to go into depth here.

05:09 But I highly recommend that you review the basic properties of logarithms and how they work.

05:14 The two properties of logarithms that we're going to use here are first, that if you have the log of two numbers multiplied by each other, that can be split into the log of the sum of those two terms.

05:27 So for example, the log of A times B, would be the log of A plus the log of B.

05:33 And that's what we're doing with the right hand side of the equation.

05:36 Excuse me, right hand side of this equation we take the logarithm and then we have the log of N 0 times the exponential term.

05:44 Since they are multiplied as I've just said by this law.

05:47 We instead have the sum, the log of N 0 plus the log of the exponential term.

05:53 And then the exponential term simplifies.

05:55 The log of an exponential is simply equal to the argument of that exponential by definition.

06:00 And so with all of this simplification.

06:02 And then again don't worry about too much of mathematical rigor here.

06:05 What we have is a final equation.

06:07 The natural log of the number of particles that we have that are still radioactive, is equal to minus lambda times 't' the time.

06:15 Plus the natural log of the original number of particles that we have.

06:20 If you examine this equation you might recognize it as simply the equation for a line.

06:24 If you are familiar with the equation for the line as may be being y equals mx + b.

06:28 Or the argument is equal to the slope times the independent variable plus some offset for mere vertical axis.

06:38 This is exactly the equation that we see below here.

06:42 We have 'y' or the dependent variable in the vertical axis.

06:46 The log of the number of radioactive particles that has left, equals minus lambda.

06:52 This would be the slope of our line.

06:54 Since 't' is our horizontal axis.

06:56 Plus the log of the original number of particles.

06:59 And this will just be the vertical offset.

07:01 So this is again just the equation for a straight line.

07:05 So if we were able to plot using this natural logarithm method, the radioactive decay of our material, we can simply measure the slope of this straight line and find the decay rate of the material lambda.

07:17 So this is the way to really think about this equation is from an experimenter's point of view.

07:24 And he's watching the radioactive decay go on.

07:26 He has some data from that experiment.

07:29 And the best way to interpret that data might be to take the natural logarithm of the numbers that he was collecting as the experiment went on.

07:35 And then plotting those numbers in a graph like this.

07:39 In this case he would see a straight line.

07:41 And the slope of this line can then be measured and the slope would be equal to minus lambda, where lambda would be the decay rate of the material that was decaying.

07:51 The last thing we're going to discuss is Spectrometry.

07:54 With mass spectrometry what we do, is we're trying to measure how massive small particles are.

08:00 Or how many of a particular kind of particle or particular species is present in a particular sample.

08:07 What we do is actually quite simple.

08:08 We have here something you might recognize as our cyclotron setup with a magnetic field.

08:14 What we do with this magnetic field which again by our convention here is pointing out of the page, is first to ionize the particles in our sample.

08:22 Which simply again means to give them some sort of a charge, an electric charge.

08:26 We then from the source of ions shoot these particles into our magnetic field.

08:31 These particles were moved as we've already discussed in a circle in that magnetic field.

08:37 But the radius of that circle as we saw in our prior chapters is going to be different depending on the mass of those particles.

08:43 So for a more massive particle we would have a different radius of our curvature in the circle for a lighter particle.

08:51 So by doing this, by sending all these ions through this magnetic field, some of them will bend in one path and some of them will bend in a different path.

08:58 What we can do in our detector is following these steps using electric fields to shoot our ions through into the magnetic field following their paths, is in our detector is detecting exactly where the different particles arrive.

09:11 Since each one will follow it's own path.

09:14 May be a more massive particle will follow one path.

09:16 And the lighter particle will follow another path.

09:19 They will arrive at different locations in our detector.

09:22 Using this fact we can count the number of particles of a particular mass by how many arrived at a particular location.

09:29 So using are the known charge of these ions which is something we can put into this experiment.

09:35 We can find the mass of our material.

09:37 And that's mass spectrometry.

09:39 So this finishes our section on the Atomic Nucleus.

09:41 We are ready to go to the electronic structure of atoms.

09:44 And we'll do that before we cover and talk about the thermodynamics which will be very interesting for specially chemical and biological applications.

09:52 But until then thanks for listening.

The lecture Half-Life and Mass Spectrometry by Jared Rovny is from the course Atomic Nucleus.

### Included Quiz Questions

1. The initial amount of radioactive material
2. The final amount of radioactive material
3. The decay rate of the radioactive material
4. The difference between the initial and final amount of radioactive material
5. None of these
1. The time required to half the radioactive material
2. Half the time it takes to remove all the radioactive material
3. Half the time it takes to remove all the material, radioactive and not
4. The time required to half the entirety of the material, radioactive and otherwise
5. The time it takes to remove all the radioactive material
1. The cyclotron equation
2. Newton’s third law
3. The pendulum equation
4. The spectrometer equation
5. The ion equation
1. The location of particle arrival at a detector
2. The size of the ion
3. The shape of the ion
4. The number of molecules arriving at a detector
5. The number of photons
1. The slope of a semi-log plot of radioactive material versus time
2. The intercept of a semi-log plot of radioactive material versus time
3. The slope of a semi-log plot of half-life versus time
4. The intercept of a semi-log plot of half-life versus time
5. The logarithm of the intercept of a semi-log plot

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