Video: Solving Nuclear Equations Involving Beta Decay

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

06:22

Video Transcript

Calcium 40 is created through the beta decay of potassium, as shown in the nuclear equation 𝐾 𝑝 𝑞 goes to 𝐶𝑎 40 20 plus 𝛽 zero minus one. What are the values of 𝑝 and 𝑞 in the equation?

Let’s begin by underlining the important information given to us in the question. So we know that calcium 40 is being created by beta decay. And it starts out as potassium. We’ve got a large chunk of the nuclear equation given to us. And what we need to find out is what the values of 𝑝 and 𝑞 are in that equation. So we know that the atom we’re looking at begins its life as a potassium atom. And then it decays to a calcium atom plus a beta particle. We know that the calcium atom has an atomic number of 20. The atomic number of an atom simply measures the number of protons found in its nucleus. We also know that the calcium atom has an atomic mass of 40. The atomic mass measures the total number of protons and neutrons found in the nucleus of the atom. By the way, protons and neutrons together are known as nucleons because they can be found in the nucleus of an atom.

We also have the beta particle on the right-hand side of the equation. But we’ll come back to that in a second. Let’s look at the left-hand side of the equation where we’ve got the potassium atom. We can see that the atomic number of potassium is 𝑞. And its atomic mass is 𝑝. These are the two quantities we’re trying to work out. However, we can clearly see that in a nuclear equation, we have a certain convention that we use. And that is as follows.

Let’s say we have an element 𝑋, whatever it may be. It has an atomic number 𝑛 and an atomic mass 𝑚. Then this is how we write it. We write the symbol of the element in large, hence the large 𝑋. We write the atomic number on the bottom left. And we write the atomic mass on the top left.

So now let’s go back to our beta particle. We can clearly see that it has an atomic number of minus one and an atomic mass of zero. Does that make sense? Let’s think about it. Well, what does it mean, first of all, to have an atomic number of minus one? As we’ve said earlier, the atomic number is the number of protons in the nucleus of an atom. But, what even is a beta particle? Well, we know that a beta particle is an electron. And an electron is not really an atom. So it can’t have a nucleus. So what’s going on here?

Well, there is another convenient way to think about the atomic number, rather than just the number of protons in the nucleus of an atom. And that is that the atomic number measures the number of positive charges in whatever it is that we’re looking at. So in this case, we’re looking at a beta particle. Or, in other words, an electron. How many positive charges does an electron have? Well, zero, because it’s an electron. It’s negatively charged. But if we’re thinking about the atomic number as measuring the number of positive charges, then thinking about it logically, the electron or the beta particle being a negative charge has minus one positive charges because it’s a negative charge. And that’s why we can conveniently label the beta particle as having an atomic number of negative one.

As well as this, why does the beta particle have an atomic mass of zero? Well, if the atomic mass measures a number of protons and neutrons in the object that we’re looking at, then a beta particle, an electron, has no protons and neutrons in it. Hence, its atomic mass is zero. So now that we’ve figured that out, let’s get on and try and answer this question.

In order to find the values of 𝑝 and 𝑞, we need to remember two simple rules. The first rule goes as follows: the sum of atomic numbers on the left-hand side of the equation — 𝐿𝐻𝑆, left-hand side — is equal to the sum of the atomic numbers on the right-hand side, 𝑅𝐻𝑆. And the second rule is nearly identical, except this time it applies to the atomic masses rather than the atomic numbers. So to clarify: the sum of atomic masses on the left-hand side of the equation have to equal the sum of atomic masses on the right-hand side. So what does it mean to find the sum of atomic numbers or masses? Well, it’s actually quite simple.

Let’s start first with the atomic numbers, so rule number one. This is the one we’re looking at. In this case, the sum of atomic numbers on the left-hand side is just 𝑞 because there’s only one atomic number on the left-hand side, which is 𝑞. Let’s write that down here. And the sum of atomic numbers on the right-hand side is equal to 20, which is the atomic number of calcium, plus negative one, which is the atomic number of the beta particle. That’s what it means to find the sum of the atomic numbers. You add up all the atomic numbers on the left-hand side of the equation. And you add up all the atomic numbers on the right-hand side of the equation. And rule number one tells us that these two values should be equal to each other. The sum of atomic numbers on the left-hand side, that’s 𝑞, is equal to the sum of atomic numbers on the right-hand side. Now this is quite easy to solve. Quite simply, we get 𝑞 is equal to 20 plus negative one which happens to be 19. Let’s write that down on the side. And let’s crack on with rule number two.

Once again, we need to find the sum of atomic masses this time on the left-hand side. So the atomic mass on the left-hand side is 𝑝. There isn’t any other number. So we write 𝑝. And then we write our equal sign. And then we look at the right-hand side. The atomic masses that we have there, of course the atomic mass of calcium, which is 40, and the atomic mass of the beta particle, which is zero. And we can rejoice because this equation is also easily solved. It gives us 𝑝 is equal to 40.

And so the final answer to our problem is that 𝑝 is equal to 40 and 𝑞 is equal to 19.

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