Video: Beta Decay

In this lesson, we will learn how to solve nuclear equations involving beta decay.

14:58

Video Transcript

In this video, we’ll be talking about beta decay. Beta decay is one of the most common types of radioactive activity. All radioactivity involves taking a less stable atomic nucleus and making it more stable. And beta decay is no different. Since beta decay begins with atomic nuclei, we may as well start there. In this nucleus we have here, the blue dots we’ve drawn in our protons. And the green dots are neutrons. Let’s say that, as it is, this particular nucleus is fairly stable. It’s unlikely to decay into something else. But if we wanted to destabilize this nucleus, there’s a way we could do it. We could either add a whole lot more neutrons to the nucleus, like this. Or starting from that same nucleus, we could add many more protons to it, what we’ve represented as blue dots.

If we make these two types of modification to the original stable nucleus, then on the right-hand side, we’ll have a neutron rich nucleus. And on the left-hand side, we’ll have a neutron poor nucleus. In other words, on the right, the ratio of neutrons to protons is very high whereas on the left it’s very low. As we mentioned, making changes like this will destabilize the nucleus. These nuclei are set up then to undergo the radioactive event of beta decay. Beta decay can occur for both neutron poor and neutron rich nuclei. But it happens a little bit differently in each case.

In the case on the left, where we have lots of protons relative to neutrons, there’s a transformation that goes on where one of the protons, say this one here, turns into a neutron. The blue dot becomes a green dot. Not only that, but the nucleus emits two particles. One of these particles is called a positron. This is the antiparticle of the electron. That means it’s just like an electron. But it has the opposite charge to it, positive rather than negative.

The other particle emitted in this beta decay process is called a neutrino. As we might guess from the name, neutrinos have no electric charge. And also they have almost no mass. They’re very very light weight. So when we have a neutron poor nucleus, beta decay for that nucleus means a proton turns into a neutron. And the nucleus also emits what’s called a positron and a neutrino.

On the other hand, if we’re talking about a neutron-rich nucleus, like we have on the right, beta decay looks a bit different. In this case, rather than one of the protons turning into a neutron, the opposite happens. One of the neutrons, the green dots, turns into a proton, a blue dot. Just like before, two particles are emitted in this process. One is an electron. And the other particle, the counterpart of the neutrino, is called an antineutrino. Once again, this is a particle with no charge and very very little mass.

Both of these processes here are known as beta decay. But because they’re slightly different, they have different names. In the case on the left, where a proton turns into a neutron and the nucleus emits a positron and a neutrino, that’s called beta plus decay or beta plus radiation. On the right-hand side, where a neutron becomes a proton and the nucleus emits an electron and an antineutrino, that’s called beta minus decay.

One way to remember the difference is to remember that beta plus involves the emission of a positively charged object, a positron. Beta minus, on the other hand, involves the emission of a negatively charged object, an electron. Due to the conditions on Earth, matter on Earth is much more likely to have an overabundance of neutrons, compared to an overabundance of protons.

For that reason, beta minus decay in our environment is much more common than beta plus decay, so much so in fact that if someone talks about beta decay and doesn’t specify beta minus or beta plus, it’s reasonable to assume they’re talking about beta minus, the more common form of this decay. And indeed, that will be our assumption going forward, that when we talk about beta decay or specifically speaking of beta minus decay, that is, when a neutron converts into a proton and the nucleus emits an electron and an antineutrino.

While we have both these types of beta decay on screen though, it’s worth saying a bit about what these emission products look like in a nuclear equation. To see that, we can write out nuclear decay processes that involve both types. If we were to start out with a carbon ten isotope, that is, a nucleus that has six protons but only four neutrons, then because of the abundance of protons relative to neutrons in this isotope, it would be likely to decay. Now, if one of these six protons changed into a neutron, what element will we then have? We know it will be the element with the atomic number of five.

Looking this up on the periodic table, we see that this corresponds to boron, symbolized with a capital B. If we’ve lost one proton but gained one neutron, then that means the mass number of this product is the same as that of carbon, 10. Along with this change from the element of carbon to the element of boron, this process also involves the emission of a positron, which we said is the antiparticle of an electron as well as what’s called a neutrino.

Now, since neutrinos have no electric charge and negligibly small mass, they’re often not even written down as part of a nuclear equation. We might just see the equation written this way, without any mention of the neutrino since it has almost no effect. But this positron does have an impact on the overall equation. This decay product is known as a beta particle. And for that reason, we’ll typically see it written using the Greek letter 𝛽 to represent it. Along with this, the atomic number, as well as the mass number, of this beta particle are normally indicated.

Since the mass number, that’s the number that goes in the upper left, is equal to the sum of the number of protons plus the number of neutrons, we know that that will be zero since this beta particle has no neutrons or protons. It’s a positron. But then, on the bottom left, we’ll rewrite the atomic number. We might expect this to be zero as well. After all, a beta particle has no protons. But because our beta particle does have an electric charge and in fact that charge is equal to the charge of a proton, for this type of beta particle, we write that its atomic number is one. That doesn’t mean it has any protons. But it does mean that its charge is equal to the charge of one proton.

Notice, by the way, that having a one here, for the atomic number of our beta particle, makes the math of this equation work out. We started out with an atomic number of six. And then, our products have an atomic number of five plus one, six. So what we’re seeing here is a nuclear equation demonstrating beta plus decay. When it comes to beta minus decay, let’s say we start with a different isotope of carbon. We’ll start with carbon 14. This means we once again have six protons. But now, there are eight neutrons in the nucleus. It’s neutron heavy. This abundance of neutrons makes this isotope of carbon likely to decay. In beta minus decay, as we said, a neutron turns into a proton. That means that one of the reactions of this decay will be an element with an atomic number of seven.

If we look that up in the periodic table, we see that that corresponds to nitrogen, symbolized by a capital N. And then, since a neutron turned into a proton, we lost one neutron. But we gained one proton. And therefore, the mass number remains the same overall, still 14. And then, along with the change in nucleus, we have two emission products, an electron and an antineutrino. Just like for the neutrino, the antineutrino has a negligibly small impact on this process. For that reason, we often don’t write it down as part of a nuclear equation. This leaves us with carbon, nitrogen, and our electron.

We said that, in beta plus decay, the positron was called a beta particle. In the case of beta minus decay, the electron is known as the beta particle. For that reason, we symbolize it using that Greek letter, 𝛽. But because it’s not a positron, we write the atomic number a little bit differently. Instead of this beta particle having an atomic number of plus one, it has an atomic number of negative one. That’s because an electron, recall, has the opposite charge of a proton. In that sense, it’s like the reverse of a proton. So we write negative one for the atomic number of this beta particle. Then, for the mass number, we once again use zero because there are no protons or neutrons in this electron.

Looking at this reaction, we see that, once again, the math works out. Six, the atomic number on the left side, is equal to seven plus negative one, the sum of the atomic numbers on the right. And also mass number is conserved. We start out with 14. And we end up with the sum of 14. Now that we know a bit about beta decay, let’s get some practice with these ideas through an example.

Calcium-40 is created through the beta decay of potassium, as shown in the nuclear equation here. What are the values of 𝑝 and 𝑞 in the equation?

In this process, we see the element potassium, symbolized capital K, decaying into calcium and a beta particle. We see that, for both calcium as well as the beta particle, the atomic number and the mass number are given. For potassium though, the atomic number is given as 𝑞. And the mass number is given as 𝑝. It’s those values we want to solve for. To do that, we can recall that, in a nuclear equation, like the one we have here, both mass as well as electric charge is conserved. That means that the total mass on one side of the equation is equal to the total mass on the other side of the equation, and the same with charge.

The conservation of mass in this equation means that mass number is also conserved. In other words, we can say that the mass number on the left of the equation, 𝑝, is equal to the sum of the mass numbers on the right, 40 plus zero. This equation is saying that if we add up all the protons and neutrons on the left, that number is equal to the sum of all the protons and neutrons on the right. Since 𝑝 is equal to 40 plus zero, that means 𝑝 is equal to 40. That’s the mass number of this potassium isotope.

We can do something similar to solve for 𝑞. 𝑞 is the atomic number of potassium, which represents the number of protons in its nucleus. Since each proton has a relative electric charge of plus one, we could also say that 𝑞 represents the total electric charge in the nucleus. Since electric charge, like mass, is also conserved across this reaction, we can say that 𝑞 is equal to the sum of the relative charge of calcium, which since it has 20 protons as a relative charge of plus 20 plus the relative charge of the beta particle. That relative charge is indicated as negative one. That’s because this beta particle is an electron. It is a charge that has the same magnitude as a proton, but an opposite sign.

So 𝑞 is equal to 20 plus negative one. Adding negative one to 20, we find that 𝑞 is equal to positive 19. That’s the value of 𝑞, the atomic number of potassium in this equation.

Notice, by the way, that there’s a second way we could’ve solved for 𝑞 since 𝑞 is potassium’s atomic number. We could’ve looked up this symbol in the periodic table of elements and then seen what number element that is. That number would be the atomic number of this element, 19. So it would be possible to solve for 𝑞 that second way. 𝑝, on the other hand, couldn’t be solved for by looking at the table. We needed to use this equation to find that value.

Let’s now look at a second example involving beta decay.

When an atomic nucleus emits a beta particle, how much does the atomic number of the remaining nucleus change by?

Okay, in this scenario, we have an atomic nucleus. And this nucleus, we’re told, emits a beta particle. Now, in general, a beta particle can be either a beta plus or a beta minus particle. In the problem statement, we’re not told specifically which type it is. We just know it’s a beta particle. We do know however that beta minus radiation is much more common on Earth than beta plus radiation. So if we had to guess, we would say that this is a beta minus particle. In other words, what’s being emitted from this nucleus is an electron. Knowing that, we want to figure out how much the atomic number of the nucleus that remains after the emission of the electron changes.

One way to answer this question is to imagine an atomic nucleus of an element, we’ll just call element A. It could be any element. And like any element, it has an atomic number, we’ll call it 𝑁. And it also has a mass number, which we’ll call 𝑀. This is the symbol representing the nucleus that emits a beta particle. In other words, this element goes through a radioactive transformation. It emits a beta particle, an electron, which has a mass number of zero because there are no protons or neutrons in it. And it has an atomic number of negative one because the charge of this beta particle is equal and opposite the charge of a proton. And then, of course, along with the emitted electron, there’s the nucleus that’s left over. We’ll say it’s a nucleus of an element B.

In an equation like this, and in general in nuclear equations, both mass number as well as charge is conserved. So that means the mass number on the left-hand side of the equation, in this case 𝑀, is equal to the sum of the mass numbers on the right-hand side. That tells us that the mass number for element B must be 𝑀 in order for that equality to hold true. But then, what about atomic number, the number written to the lower left of these symbols.

Knowing that atomic number is also conserved, we can say that 𝑁 is equal to negative one, the atomic number of the beta particle, plus the atomic number of our element B. In order for this equality to be true, we can say what goes in the parentheses. It has to be 𝑁 plus one. That way, the plus one cancels with the minus one. And we just have 𝑁 is equal to 𝑁. So the atomic number of our product nucleus is 𝑁 plus one.

That’s not our final answer though because the question asks how much does the atomic number of the remaining nucleus change. In other words, when we go from element A to element B, how much of a change in atomic number is there. Since the atomic number of A is 𝑁 and the atomic number of B is 𝑁 plus one, we can see that the overall change is plus one. That’s the change in the atomic number of the remaining nucleus.

Let’s summarize now what we’ve learned in this lesson about beta decay. First and foremost, we saw that beta decay occurs in atomic nuclei that have an excess of protons or an excess of neutrons. And we saw that this second case, the excess of neutrons instance, is more common on Earth. Further, we saw that there are two types of beta decay. In beta plus decay, a proton decays into a neutron in the nucleus. And also the nucleus emits a positron, essentially a positively charged electron as well as a neutrino. While the neutrino has no electric charge and effectively no mass, the positron has a significant amount of electric charge.

In this decay reaction, the positron is referred to as a beta particle, symbolized using the Greek letter 𝛽. Because it has an overall charge of plus one, its atomic number is written as plus one whereas its mass number is written as zero. In beta minus decay, on the other hand, a neutron decays into a proton in the nucleus. And the nucleus emits an electron and an antineutrino. In this case, it’s the electron that’s referred to as the beta particle. It has a mass number of zero and an atomic number of negative one.

We learn that if no type is specified when we talk about beta decay, whether beta plus or beta minus, then it’s reasonable to assume beta minus decay is going on. That’s because this type of decay is more common in elements that have an abundance of neutrons, such as we find on Earth. Lastly, we saw that when we write these products out in nuclear equations, the neutrinos or the antineutrinos are often neglected. And that’s because of their negligible impact on beta decay.

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